Abstract
A fundamental assumption of linear acoustics is that the presence of a wave does not have an influence on the properties of the medium through which it propagates. By extension, the assumption of linearity also means that a waveform is stable since any individual wave does not interact with itself. Small modifications in the sound speed due to waveinduced fluid convection (particle velocity) and to the wave’s effect on sound speed through the equation of state can lead to effects that could not be predicted within the limitations imposed by the assumption of linearity. Although a wave’s influence on the propagation speed may be small, those effects are cumulative and create distortion that can produce shocks. These are nonlinear effects because the magnitude of the nonlinearity’s influence is related to the square of an individual wave’s amplitude (selfinteraction) or the product of the amplitudes of two interacting waves (intermodulation distortion). In addition, the timeaverage of an acoustically induced disturbance may not be zero. Sound waves can exert forces that are sufficient to levitate solid objects against gravity. The stability of such levitation forces will also be examined along with their relation to resonance frequency shifts created by the position of the levitated object.
Keywords
 Harmonic distortion
 Intermodulation distortion
 Shock waves
 Acoustic levitation
 Bernoulli pressure
 Rayleigh disk
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The goal of this chapter is to raise awareness of the limitation of linear analysis, not to create professional expertise in nonlinear acoustics. A fundamental assumption of linear acoustics is that the presence of a wave does not have an effect on the properties of the medium through which it propagates. Under that assumption, two sound waves can be superimposed when they occupy the same space at the same time, but one wave will have no effect on the other wave and once they part company there will be no evidence of their previous interaction. This is illustrated in Fig. 15.1. By extension, the assumption of linearity also means that a waveform is stable since any individual wave does not interact with itself.^{Footnote 1}
We already know that this assumption of the wave having no influence on the properties of the propagation medium cannot be strictly correct. The wave imparts a small particle velocity, v_{1}, to the fluid that adds to the sound speed when that velocity is in the direction of propagation and subtracts from the sound speed when the particle velocity is opposite to the direction of propagation. The local value of the sound speed, c(x, t), will vary in time and space due to the wave’s convective contribution so that c_{o} + v_{1}(x, t) ≥ c(x, t) ≥ c_{o} − v_{1}(x, t), where c_{o} is the equilibrium (thermodynamic) sound speed: \( {c}_o=\sqrt{{\left(\partial p/\partial \rho \right)}_s} \).
The wave also modulates the medium’s thermodynamic sound speed. For the case of an ideal gas undergoing adiabatic compressions and expansions, there is an accompanying temperature change of amplitude, T_{1}, given by Eq. (7.25), that is related to the amplitude of the pressure change, p_{1}(x, t): (∂T/∂p)_{s} = [(γ − 1)/γ](T_{m}/p_{m}). Since the sound speed in an ideal gas is dependent upon the temperature of the gas through Eq. (10.23), this implies that the change in sound speed, δc, due to a temperature change is given by (δc/c_{o}) = ½(T_{1}/T_{m}). In an ideal gas, the local sound speed is slightly faster than c_{o} when the acoustic pressure is positive since the gas is warmer and slightly slower than c_{o} when the acoustic pressure is negative since the gas is cooler.
As will be demonstrated, these small modifications in the sound speed due to waveinduced fluid convection and to the wave’s effect on sound speed through the equation of state can lead to interesting effects that could not be predicted within the limitations imposed by the assumption of linearity. Although their influence on the sound speed may be small, those effects are cumulative. These are called nonlinear effects because the magnitude of the nonlinearity’s influence is related to the square of an individual wave’s amplitude (selfinteraction) or the product of the amplitudes of two interacting waves (intermodulation distortion).
An additional consequence of the inclusion of nonlinearity is that the timeaverage of an acoustically induced disturbance may not be zero. In the linear case, the measure of a wave’s amplitude will be equally positive and negative around its undisturbed equilibrium value, so that the timeaverage of the wave’s influence will be zero. When the hydrodynamic equations and the equation of state were linearized, the terms in those equations that were discarded could lead to nonzero timeaveraged effects. For the linearized continuity equation, the ρ_{1}v_{1} term was discarded since Eq. (8.19) demonstrated that it was smaller than the ρ_{m}v_{1} term for small values of the acoustic Mach number, M_{ac} ≪ 1. A similar choice was made for the linearization of the Euler equation. The convective portion of the total derivative, \( \left({\overrightarrow{v}}_1\cdotp \nabla \right){\overrightarrow{v}}_1 \), was discarded when compared to \( \partial {\overrightarrow{v}}_1/\partial t \)in Eq. (8.38) under the same assumption of small acoustic Mach number. To complete the overall linearization, the Taylor series expansion of the equation of state in Eq. (10.3) was truncated after the firstderivative term.
In this chapter we will recover some of the interesting acoustical phenomena that were lost to the linearization of the phenomenological equations that describe both the dynamics and the medium itself.
1 Surf’s Up
When most people hear the term “wave,” it is likely that word will conjure mental images of surf breaking along a beach. (It is a most pleasant image!) The breaking of waves in shallow water is a dramatic nonlinear effect that is due to both the convective nonlinearity and the fact that the height of the wave modulates the propagation speed of a shallowwater gravity wave. The speed of a shallowwater gravity wave represents the competition between the water’s inertia and the restoring force of gravity. Figure 15.2 is a schematic representation of one cycle of such a wave on a fluid of equilibrium depth, h_{o}, with a peak wave height of magnitude h_{1} ≪ h_{o}.
The assumption that the fluid is “shallow” implies that the mean depth of the fluid, h_{o}, is much smaller than the wavelength of the disturbance, λ.
Since there is a free surface, we will assume that the fluid is incompressible. It is much more favorable (energetically) for the free surface to move up than it is for a pressure increase to increase the fluid’s density. The continuity equation can be written by recognizing that the rateofchange of the fluid’s height,\( {\dot{h}}_1\left(x,t\right) \), is determined by the difference in the amount of fluid that enters and leaves a “slab” of infinitesimal thickness, dx, shown in Fig. 15.2.
For a shallowwater gravity wave, the fluid’s particle velocity in the direction of propagation, v_{x}, is greater than the rateofchange of height of the free surface if h_{o} ≪ λ. This is an effect most of us have experienced while frolicking in the surf near the ocean’s shore—it is usually the “surge” that knocks us over, not \( {\dot{h}}_1 \).
Since gravity (not compressibility) provides the restoring force, Euler’s Eq. (7.34) relates the fluid’s velocity in the direction of propagation, v_{x}, to the gravitational pressure gradient.
The combination of Eqs. (15.2) and (15.3), with the assumption of a rightward traveling wave in Eq. (15.1), leads to a dispersion relation that generates the equilibrium values for propagation speed, c_{grav}, of a shallowwater gravity wave.^{Footnote 2}
Logarithmic differentiation of Eq. (15.4) provides the relationship between the local wave speed and the instantaneous depth of the fluid.
We would like to combine the effects of changing depth on the sound speed with the convective contribution to the local sound speed produced by v_{x}. The continuity Eq. (15.2) provides that necessary conversion.
The convective contribution to the local wave speed, c(v_{x}), can be combined with the change in local wave speed due to the changing fluid depth.
Both convection and the speed’s change with depth conspire to increase the local wave speed when h_{1}(x, t) > 0 and reduce the local wave speed when h_{1}(x, t) < 0. The wave’s crests travel faster than the zerocrossings (i.e., h_{1}(x, t) = 0) and its troughs travel slower than the zerocrossings. Figure 15.3 shows the cumulative consequences of the wave’s influence on its own local propagation speed. As the wave progresses, the crests will start to overtake the troughs.
In Fig. 15.3, the coordinate system was chosen to move with the equilibrium wave speed, c_{grav}, so that the distortion becomes evident. At the instant captured in Fig. 15.3, the slope of the zerocrossing has become vertical. To reach that condition, the crest of a sinusoidal waveform must have advanced by one radian length toward the zerocrossing, k^{−1} = λ/2π (see Prob. 1), and the trough must have lagged behind by the same amount. The time, T_{S}, it takes for the crest to advance by k^{−1} is given by the speed excess, 3v_{x}/2, calculated in Eq. (15.7). The distance traveled by the wave once the slope first becomes infinite is known as the shock inception distance, D_{S}.
For surf, the wave can continue beyond D_{S}. Since surf has free surface, h_{1} can actually become multivalued and will eventually “break,” sometimes with a spectacularly powerful display of sound and foaminess. Stokes was the first to recognize in 1848 that viscosity is the physical mechanism that prevents a sound wave from becoming multivalued. Stokes was also the first to draw a distorted waveform, like the one in Fig. 15.3, which he did in that same paper where he talked about the essential role of viscosity^{Footnote 3} [1].
1.1 The Grüneisen Parameter
The principles introduced to describe waveform distortion and the creation of a shock front for shallowwater gravity waves are common to all sound waves in fluids. A sound wave will influence the propagation speed of the medium due to a combination of the convective contribution and the fact that the wave’s amplitude also influences the propagation speed. Of course, the nature of that contribution and the relative importance of the convective and equation of state contributions will be differ depending upon the medium. The convenience of representing both contributions in terms of the local fluid particle velocity was demonstrated in the analysis of surf that produced Eq. (15.7). The strength of nonlinear distortion in any medium that supports a plane progressive wave will now be generalized by the introduction of the Grüneisen parameter, Γ.
The Grüneisen parameter is a designation taken from solidstate physics where it represents the nonlinearity of a solid’s elasticity that is responsible for the nonzero value of a solid’s thermal expansion coefficient.^{Footnote 4} The reader should be cautioned that calling this nonlinear distortion parameter the “Grüneisen parameter” and designating it as Γ is not a common choice in other treatments of nonlinear acoustics. For example, in a recent paper by Hamilton [2], Γ represents the Gol’dberg number that is abbreviated as G in this textbook (see Sect. 15.1.4). In Eq. (15.9), the general amplitude variable is simply written as “y,” and the equilibrium sound speed is designated c_{o} to distinguish it from the local amplitudedependent sound speed, c(v) = c_{o} +Γv.
If a medium’s sound speed depended upon the density of the medium, ρ, which obeyed the linear continuity equation, the Grüneisen parameter would be expressed in terms of the sound speed’s variation with density.
For an ideal gas, the sound speed depends upon the mean absolute temperature, T_{m}. As before, δc represents the change in the sound speed due to the change in local temperature.
The Grüneisen parameter for an ideal gas can be expressed in terms of the change in the speed of sound with temperature, the change in temperature with pressure, and the particle velocity amplitude, v_{1}, associated with the acoustic pressure amplitude, p_{1}, as related by the Euler equation for progressive plane wave propagation: p_{1} = (ρ_{m}c_{o})v_{1}.
Using the relationship between temperature and pressure for an adiabatic sound wave in Eq. (7.25), the Grüneisen parameter for an ideal gas can be calculated.
For noble gases, γ = 5/3 so Γ = 4/3. For diatomic gases and primarily diatomic gas mixtures like air, γ = 7/5, so Γ_{air} = 6/5. In both cases, it is the convective contribution that is most significant contributor for nonlinear distortion in a gas.
To start developing intuition regarding the formation of a shock wave, consider a sound wave in air that has an amplitude at the “threshold of feeling,” 120 dB_{SPL}, so p_{1} = 28 Pa. If the frequency of the sound wave is 1.0 kHz and the mean gas pressure is 100 kPa, then the acoustic Mach number for such a loud sound can be evaluated using the Euler equation.
When such a wave propagates down a duct of constant crosssection, the shock inception distance, D_{S}, can be expressed in terms of the wavelength of sound using Eq. (15.9).
At ten times that amplitude (140 dB_{SPL,} the “threshold of pain”) and for a frequency of 10 kHz, the shock inception distance would be 4.6 m. In the throat of the horn, for a hornloaded compression driver [3] or in a brass musical instrument (e.g., trumpet or trombone), the amplitude can be still larger by a factor of ten [4].
1.2 The Virial Expansion and B/2A
For the characterization of nonlinear behavior of sound waves in liquids, it is common to expand the equation of state in a Taylor series, known as a virial expansion, in powers of the relative deviation of the density from its equilibrium value, (δρ/ρ_{m}) = (ρ − ρ_{m})/ρ_{m}.
The coefficients in that expansion, A, B, C, etc. are called the virial coefficients and have the units of pressure. For an adiabatic process, they can be expressed in terms of progressively higherorder thermodynamic derivatives of pressure with respect to density.
It is useful to notice that B can be expressed in terms of the derivative of the sound speed with respect to density, which was related to the nonconvective contribution to the Grüneisen parameter in Eq. (15.10). The final form for B/A follows from the expansion of the sound speed derivative with respect to pressure, (∂c/∂p)_{s} = (∂c/∂p)_{T} + (∂T/∂p)_{s}(∂c/∂T)_{p}, and temperature, (∂p/∂T)_{s} = (∂ρ^{−1}/∂s)_{p} = (∂ρ^{−1}/∂T)_{p}/(∂s/∂T)_{p}, along with the introduction of the isobaric coefficient of thermal expansion, β_{p} = (1/V)(∂V/∂T)_{p} = ρ_{m}(∂ρ^{−1}/∂T)_{p}, and the introduction of the specific heat at constant pressure, c_{P} = (1/T_{m})(∂s/∂T)_{p} [5].
The sound speed can also be expressed in terms of these virial coefficients [6].
This result allows the Grüneisen parameter for liquids to be expressed in terms of B/A.
Some representative values of B/A for different substances is provided in Table 15.1. The values of B/A for liquids are generally greater than 2.0, which means that it is the equation of state’s nonlinearity that dominates the convective nonlinearity. This is reasonable since Euler’s equation implies that the particle velocity in a liquid is much less than that of a gas for equal pressure changes: (ρ_{m}c)_{liquid} ≫ (ρ_{m}c)_{gas}.
1.3 Anomalous Distortion*
Before moving on, it is interesting to consider the role that a nonzero value of C implies for the formation of shock waves. The behavior that is represented by the Grüneisen parameter causes the sound speed to be increased when the amplitude of the wave is positive and decrease when the amplitude is negative. The C coefficient makes a contribution that either always increases the sound speed, irrespective of the sign of the wave’s amplitude, or always decreases the sound speed, depending upon the sign of C.
Cormack and Hamilton have investigated shear waves with a cubic nonlinearity, C ≠ 0, using numerical simulations [8]. Figure 15.4 shows two plane waveforms that were initially sinusoidal (dotted lines) that have produced both leading and trailingedge shocks (solid lines); two shock fronts per wavelength, unlike Figs. 15.3 and 15.7, where only a quadratic nonlinearity was operative (e.g., 1 + B/2A ≠ 0 but C/A = 0).
A situation where both the quadratic and cubic nonlinearity play a role in superfluid helium sound propagation near absolute zero was identified for shockwave formation of compressional plane waves where the superfluid component velocity, v_{s}, is nonzero, but the (viscous) normal fluid is immobilized, v_{n} = 0. That sound wave mode in superfluids is known as 4^{th} sound (see Fig. 15.5). This creates a superfluid critical acoustic velocity amplitude, v_{d}, which can be defined in terms of the virial coefficients, to be the velocity amplitude where the contribution made by the wave distortion due to the (B/2A) term is equal to the influence of C/A [9].
For negative values of C, the wave is slowed whether the amplitude of the wave is positive or negative.
This doubleshock behavior, caused by C ≠ 0, is rather rare for compressional waves. Using values for (B/A) and (C/A) for water [10], v_{d} = 1.2c_{o}, corresponding to acoustic pressure swings of 26,000 atmospheres, well over 100 times greater than the highest cavitation threshold ever measured for pure water [11]. This doubleshock behavior has been observed for sound propagating through a liquid near its critical point [12].
In an ideal gas, the virial expansion can be expressed in terms of the ratio of specific heats, γ = c_{P} /c_{V}, also known as the polytropic coefficient.
For an ideal gas, (B/A) = (γ−1) and (C/A) = (γ−1) (γ −2) so the denominator of Eq. (15.22) vanishes making v_{d}/c_{o} = ∞; double shocks are an impossibility in gases.
Two other unusual results for the Grüneisen parameter arise from the propagation of sound in superfluid helium [13]. Superfluids are analogous to superconductors in that superfluids can flow without viscosity, just like electrical currents flowing without electrical resistance in superconductors. In addition, the superfluid component has both an elastic and a thermal “restoring force” [14]. In superfluid helium, there is a thermal sound mode, known as second sound, that is propagating, not diffusive, like the response governed by the Fourier heat diffusion Eq. (9.4) for classical liquids (see Sect. 9.3.1).^{Footnote 5} The temperature dependence of both second sound and the ordinary compressional wave speed (called first sound) are plotted in Fig. 15.5.
It is clear from the speed of second sound vs. temperature that there is a region where the second sound speed decreases with increasing temperature, behavior that is opposite to that of an ideal gas in Eq. (15.11). In that case, the convective contribution to the nonlinearity is opposite to the equation of state’s contribution. At T = 1.884 K, the two contributions cancel each other, and a large amplitude second sound wave can propagate without distortion [15].
A final anomalous example is provided by third sound in superfluid helium. Because the superfluid can flow without resistance, sound waves can propagate in adsorbed films as thin as two atomic layers of helium.^{Footnote 6} In very thin films, the dominant restoring force is the van der Waals attraction which varies inversely with the fourth power of the distance: f = α/h^{4}. Substituting the van der Waals force for the gravitational force in Eq. (15.4) and providing a correction for the thicknessaveraged mass density of the superfluid component, 〈ρ_{s}〉, unlike the surf, the speed of third sound, c_{3}, is inversely proportional to the film thickness, h_{o}.
For superfluid films that are less than 10 Å = 10^{−9} m thick or about three atomic layers of helium, the equation of state produces troughs that travel faster than the crests so the waves distort backward, as shown in Fig. 15.6, compared to ordinary distortion shown in Fig. 15.3.
1.4 The Gol’dberg Number
A wave of arbitrary amplitude will not necessarily form a shock. If the sound is attenuated, then the amplitude will decrease with distance, and the tendency to distort will be reduced, since the distortion is amplitude dependent. A dimensionless metric, known as the Gol’dberg number, G, compares the shock inception distance, D_{S}, to the exponential attenuation length, ℓ = α^{−1}, where α is the amplitude exponential attenuation constant that was examined in Chap. 14 [16].
As an example, the Gol’dberg number can be evaluated for a 2 kHz sound wave with pressure amplitude of \( \left\hat{\mathbf{p}}\right \) = 900 Pa (150 dB re: 20 μPa_{rms}) in dry air that propagates down a cylindrical waveguide with an inside diameter of 10.0 cm. For dry air at mean pressure, p_{m} = 100 kPa and T_{m} = 23 °C, c_{o} = 345 m/s with δ_{ν} = 50 μm and δ_{κ} = 59 μm. Using Eq. (15.15), with \( {M}_{ac}=\left\hat{\mathbf{p}}\right/\gamma {p}_m=0.64\% \) and λ = 17.3 cm, D_{S} = 3.6 m. Using Eq. (13.78), the attenuation length in that waveguide is ℓ = α_{tv}^{−1} = 20.6 m. The Gol’dberg number, given in Eq. (15.25), is G = 5.7 > 1. In this example, the wave will shock before the wave of that initial amplitude suffers sufficient attenuation.
For an initially sinusoidal plane wave in free space, far from any solid boundaries (i.e., not confined within a 10 cm diameter waveguide!), the attenuation length due to classical thermoviscous dissipation, including “bulk viscosity,” at 2 kHz in dry air at one atmosphere would be about 1.2 dB/km ≅ 1.4 × 10^{−4}m (see Fig. 14.5), resulting in an exponential attenuation distance of about 7 km making G ≅ 200. For a plane wave in free space with G = 5.7, there would be significant distortion, but a fully developed sawtooth shock would not be created. This is because the classical attenuation coefficient is proportional to frequency squared (see Sect. 14.3), so the attenuation of the second harmonic is four times that of the fundamental, rather than just \( \sqrt{2} \) larger for the waveguide, where the attenuation depends upon the square root of the frequency. Mark Hamilton has provided numerical simulations of the waveforms of such a plane progressive wave in free space for G = 5.7 that are shown in Fig. 15.7.
The Gol’dberg number is a dimensionless measure of the importance of nonlinearity relative to dissipation. In some circumstance, dissipation can be entirely ignored. For deepwater gravity waves, the primary source of dissipation is viscosity, and the Gol’dberg number is on the order of one million [17].
1.5 Stable Sawtooth Waveform Attenuation
For large values of the Gol’dberg number, an initially sinusoidal sound wave propagating in one dimension (i.e., ignoring spherical spreading) will steepen and ultimately become a repeated sawtooth waveform. At sufficiently high Gol’dberg numbers, even spherically spreading waveforms that are initially sinusoidal can form shocks [2]. In fact, any periodic waveform will steepen and ultimately form a repeated sawtooth shape, shown in Fig. 15.8, when the Gol’dberg number is sufficiently large and the wave has propagated well past the shock inception distance.
Once the sawtooth waveform has developed, the shock front produces a gradient in the temperature, particle velocity, and pressure that is very large. Such gradients produce large dissipation due to thermal conduction across the shock front and viscous shear. The amplitude of the sawtooth waveform must decrease due to the resulting energy dissipation. Calculation of the shock wave’s attenuation can be made by expressing the discontinuity of the entropy across the shock that is cubic in the pressure discontinuity [18]. For an ideal gas, the difference in entropy across the shock is expressed in terms of the universal gas constant, ℜ, and the mean molecular mass of the gas, M, by use of the RankineHugoniot shock relations [19].
Inspection of Fig. 15.8 suggests a simpler geometric approach [20]. If the particle velocity amplitude for the sawtooth waveform is u, then by Eq. (15.9), each portion of the waveform must advance, relative to the zerocrossing, by (Γu dt) during a time interval, dt. The coordinate system, as shown in Fig. 15.8, moves with c_{o}, by making the x axis be (x – c_{o}t). In that frame of reference, the fact that the back of the shock is a straight line, representing a linear increase in u, requires that the unshocked portion of the waveform undergo solid body rotation, as indicated by the curved arrow in Fig. 15.8.
Since the wave must remain singlevalued, the shock front must dissipate sufficient energy to keep the waveform from becoming multiplevalued. The two hashed triangles shown in Fig. 15.8 are similar triangles by Garrett’s First Law of Geometry, so the ratio of their heights to their bases must be equal.
Setting dt = dx/c_{o}, Eq. (15.27) can be integrated from a reference location, x_{o}, at which the acoustic Mach number is M_{o}, out to some arbitrary distance, x, from that reference location.
This result is both interesting and distinctly different from previous expressions for attenuation. First, the amplitude of the shock does not decay exponentially with distance. Second, although the dissipation is due to thermoviscous losses produced by the steep gradients across the shock front, the attenuation is independent of both the fluid’s shear viscosity, μ, and its thermal conductivity, κ, and depends instead upon the Grüneisen parameter.
This sawtooth waveform does not persist. Eventually, it “unshocks,” as shown in Fig. 15.30, as its amplitude decreases to the level where classical attenuation mechanisms are dominant [21].
2 Weak Shock Theory and Harmonic Distortion
In most fluids, the nonlinearity in the equation of state and the nonlinearity introduced by the acoustically induced convection conspire to cause waves to distort. That distortion increases with the propagation distance, if the amplitude of the wave is sufficient for such nonlinear effects to dominate thermoviscous attenuation (i.e., G ≫ 1). For waves of sufficiently large amplitude, this process will turn any periodic wave into a sawtooth wave. In this section, the focus will be on the initial stages of this distortion process.
If a wave is initially a sinusoidal “pure tone,” it will only contain a single Fourier component. That fundamental frequency can be designated f_{1}. The distortion will change the wave shape, but the wave will still be periodic with a period, T = (f_{1})^{−1}. The description of the distorted waveform will necessarily require additional Fourier components at harmonic multiples of the fundamental frequency, f_{n} = nf_{1}, with n = 2, 3, 4, etc. This section will focus on the growth of those harmonic components with distance and their dependence on the initial amplitude of the wave.
2.1 The Order Expansion
When linear acoustics was first developed in Chap. 8, the parameters that described the acoustic fields were expressed as the sum of an equilibrium value plus a firstorder deviation from equilibrium. Equation (8.1) expressed the pressure as \( p\left(\overrightarrow{x},t\right)={p}_m\left(\overrightarrow{x}\right)+{p}_1\left(\overrightarrow{x},t\right) \). Similar expansions were made for the mass density, \( \rho \left(\overrightarrow{x},t\right) \), in Eq. (8.2), temperature, \( T\left(\overrightarrow{x},t\right) \), in Eq. (8.3), and (specific) entropy per unit mass, \( s\left(\overrightarrow{x},t\right) \), in Eq. (8.4). In all cases, the firstorder deviations from equilibrium were assumed to be much smaller than the equilibrium values (e.g., p_{1} ≪ p_{m}).
This order expansion will now be extended to keep track of the effects of nonlinearity on propagation. For example, the particle velocity will be represented as the sum of the fluid’s mean equilibrium velocity, v_{m}, and the deviations from equilibrium that are proportional to successively higher powers of such deviations. These deviations will be subscripted to indicate their dependence on the amplitude of the disturbance. A subscript of “1” will indicate a firstorder contribution that is linear in the amplitude of the disturbance. A subscript of “2” will indicate a secondorder contribution that is quadratic in the amplitude of the disturbance or is the product of two firstorder contributions, possibly produced by the interaction of two different waves.
Since our attention will be focused on onedimensional propagation, x does not need to be a vector and because the fluids will not be subjected to any externally imposed mean flow, v_{m}(x) = 0. As was the case for linear acoustics, the firstorder contribution to the acoustical deviation from equilibrium, v_{1}(x, t), will be proportional to the amplitude of the disturbance from equilibrium. The secondorder contribution, v_{2}(x, t), will be proportional to the square of the amplitude of the disturbance from equilibrium or to the product of two firstorder disturbances, etc.
It will also be assumed that these individual contributions are “well ordered,” in that each successive higherorder contribution will be smaller than its lowerordered neighbor. In the case of particle velocity, all contributions will also be significantly smaller than the thermodynamic sound speed, c_{o}, in the weak shock limit.
2.2 Trigonometric Expansion of the Earnshaw Solution
The analysis of the distortion of an initially sinusoidal sound wave can generate a secondorder correction by allowing the speed of sound to be dependent upon the amplitude of the disturbance. This result was first exploited by Earnshaw (1805–1888) and was published in 1860 [22].
Here, Earnshaw solved the for a wave launched by a piston located at x = 0 that has a displacement, X(t), and velocity u(t) = dX/dt. The parameter, ϕ, represents the time a given point on a waveform left the piston’s face. Earnshaw was also the first to show that Γ_{gas} = (γ + 1)/2, for a sound wave in an gas obeying the Adiabatic Gas Law, as we did in Eq. (15.13).
We can exploit Earnshaw’s insight to calculate the growth of the second harmonic by successive approximation [23] if the initial disturbance is assumed to be a singlefrequency, rightward traveling wave with an initial particle velocity amplitude, v′.
A secondorder contribution will be generated by substitution of the “local” sound speed, as expressed in Eq. (15.9), for the thermodynamic sound speed that appears in Eq. (15.32), as was expressed by Earnshaw in Eq. (15.31).
In the weak shock limit, M_{ac} = v_{1}/c_{o} ≪ 1, so the denominator of the argument of the cosine function can be approximated by its binominal expansion.
The trigonometric identity for the cosine of the sum of two angles, a and b, is cosω(a + b) = cos (ωa) cos (ωb) − sin (ωa) sin (ωb). That identity can be used to separate Eq. (15.34) into two terms.
Since v_{1} (x, t) was defined in Eq. (15.32), the firstorder terms on both sides of Eq. (15.35) can be eliminated so that only the secondorder contribution remains. The firstorder contribution can also be substituted into the secondorder expression.
Using the doubleangle sine identity, sin(2a) = 2 sin (a) cos (a), it becomes clear that the trigonometric product introduces a second harmonic component that grows linearly with distance, x, scaled by the wavelength, λ, and is proportional to the square of the initial amplitude, (v’)^{2}.
The assumption regarding the relative amplitude of the terms in the order expansion, as asserted in Eq. (15.30), will be violated before v_{2} = v_{1}. To determine the limit of this solution’s applicability, those amplitudes can be equated to determine a distance, x_{1=2}, before which this assumption would be violated.
It is not surprising that this approximation would fail at a distance that is less than twice the shock inception distance, D_{S}. It is also true that this solution assumes that energy is transferred to the second harmonic with no reduction in the amplitude of the fundamental. That is clearly not possible, since the energy that appears as the second harmonic contribution was provided by the energy in the fundamental. The subsequent analysis will correct that difficulty.
2.3 Higher Harmonic Generation
It would be possible to continue the successive approximation procedure to calculate successively higher harmonics, but that procedure would quickly become algebraically messy. A simple and more intuitive approach is to use Eq. (15.9) to incorporate the local sound speed to deform the wave, as was done initially for shallowwater gravity waves in Fig. 15.3, and then simply use Fourier analysis to extract the amplitudes of the harmonics [24].
An undistorted wave can be parameterized by making its amplitude, y, be a function of a parameter, θ: y = sin (θ). To distort the wave, the plotted position can be advanced by an amount related to the propagation distance, d, scaled by the shock inception distance, D_{S}.
In Fig. 15.9, onehalf of a sine function has been plotted on the x axis at two different advanced locations in Eq. (15.40).
There is no additional information provided by the negative halfcycle, so the harmonic content of the distorted waveform can be Fourier analyzed between 0 ≤ θ < π.
The Fourier coefficients can be projected to obtain the amplitudes of the harmonics using the same procedure as applied to vibrating strings in Sect. 3.5.
Using the integral definition of Bessel functions of the 1^{st} kind in Eq. (C.26), Eq. (15.41) can be expressed as the sum of four Bessel functions.
Two successive applications of the recurrence relations in Eqs. (C.27) and (C.28) reduce the expression for the harmonic amplitude coefficients, C_{n}, to the compact form in Eq. (15.43), which is plotted in Fig. 15.10.
This result was originally obtained using algebraic methods by FubiniGhiron in 1935 [25].
The initial growth rate of the harmonics with propagation distance can be appreciated by expansion of the Bessel functions for small values of their arguments, nσ, as expressed in Eq. (C.12). As shown in Eq. (C.14), the J_{1}(x) Bessel function increases linearly with x = nσ. By Eq. (15.43), C_{1} ∝ J_{1}(σ)/(σ) so it is initially independent of distance. One nice feature of this solution is that as the higher harmonic amplitudes grow, the amplitude of the fundamental decreases. At d = D_{S}, the amplitude of the fundamental is only 88% of its original value.
The first terms in the expansion of the higherorder Bessel functions, J_{n} (x), all are proportional to x^{n}. As per Eq. (15.43), each Bessel function is divided by x = (nσ), so that each amplitude coefficient increases in proportion to the (n−1) power of the scaled distance, σ = x/D_{S}. This behavior is evident in Fig. 15.10. The second harmonic amplitude, C_{2}, initially grows linearly with distance, just as predicted by Airy [23] in the solution by successive approximation that led to Eq. (15.37). The third harmonic amplitude, C_{3}, has an initially quadratic dependence on distance, and the fourth harmonic amplitude, C_{4}, has an initially cubic dependence on the propagation distance.
A calculation by Fay [26] that included dissipation also produced an expression for the harmonic amplitudes, B_{n}, that describe a stabilized waveform where the Gol’dberg number includes thermoviscous attenuation, α_{TV}, in Eq. (14.29).
Note that the Fay solution produces the (stable) sawtooth waveform of Fig. 15.8 for distances that satisfy G ≫ n(1 + σ), where the hyperbolic sine function can be replaced by its argument to produce the Fourier amplitude coefficients of a sawtooth waveform (see Fig. 1.22 and Chap. 1, Prob. 12), \( {B}_n^{sawtooth}=2/n\left(1+\sigma \right) \). As shown by Blackstock [27], the Fay result for the harmonic amplitudes does not reduce to those of Fubini in Eq. (15.43), in the limit of vanishing viscosity since the Fubini coefficients are valid near the source, σ ≤ 1, and the Fay coefficients in Eq. (15.44) are valid in the sawtooth region, σ ≥ 3. Blackstock provides a solution that connects those two regimes in his paper that has become known as the “Blackstock bridging function.”
3 The Phenomenological Model
Hydrodynamics provides a complete description of the propagation of sound in fluids. All of the nonlinear behavior that has been introduced in this chapter thus far should be derivable from that hydrodynamic description. As will be demonstrated now, the hydrodynamic approach will also provide additional insights and motivate the description of additional nonlinear phenomena.
As discussed in Sect. 7.3, the dynamics of a singlecomponent homogeneous fluid can be described by two thermodynamic variables (e.g., ρ and s or p and T) and the three components of the velocity field.
As before, v_{x} is the x component of velocity, and \( {\hat{e}}_x \) is the unit vector in the x direction. The “system” is “closed” if there are five independent conservation equations that relate the variables to each other. Those equations should be familiar by now and are reproduced below:
The form of the entropy Eq. (15.47) is rather more general than will be required but includes the square of the viscous shear tensor, Φ, and the bulk viscosity, ζ, along with thermal conductivity, κ, all as potential sources of entropy production.
As before, those conservation laws contain both p and ρ as (mechanical) thermodynamic variables, so that an equation of state, p = p(ρ, s), describing each individual fluid’s properties, is required to “close” the set. In the absence of dissipation (i.e., κ = μ = ζ = 0), the equation of state can be combined with the continuity Eq. (15.46), and the entropy conservation Eq. (15.47) to demonstrate that the entropy will be conserved.
This simplifies the expansion of the equation of state in terms of the density deviation, ρ^{′} = ρ − ρ_{m}, since all of the derivatives can be evaluated at constant entropy.
3.1 The (Nondissipative) Nonlinear Wave Equation
As with Earnshaw’s solution and the calculation of the harmonic amplitude components in the weak shock limit, this analysis will be restricted to onedimensional propagation (i.e., v_{y} = v_{z} = 0), but at this point, there is no penalty for retaining the vector velocity for evaluation of the hydrodynamic equations and the equation of state up to terms of secondorder in the deviation from equilibrium.
The firstorder wave equation is homogeneous.
The firstorder terms from Sect. 7.2 that were combined to produce that linear wave equation can be subtracted from the combination of Eqs. (15.51), (15.52), and (15.53) to leave a wave equation for the spacetime evolution of the secondorder sound fields.^{Footnote 7}
This wave equation for the secondorder deviations of the density from equilibrium is not homogeneous; it has a source term that is driven by quadratic combinations of the firstorder sound fields. Using the Euler relation for the firstorder fields and Eq. (15.10), this secondorder wave equation can be rewritten in a more familiar form for plane progressive waves.
Not surprisingly, the strength of this nonlinear source term is proportional to the Grüneisen parameter, Γ.
3.2 Geometrical Resonance (Phase Matching)
The secondorder wave equation should reproduce the results obtained for second harmonic distortion in the weak shock limit that were generated by the trigonometric expansion of Earnshaw’s solution. That result can be recaptured by squaring the rightgoing sinusoidal traveling wave, ρ_{1} = ρ^{′} cos (ωt − kx), and then inserting it into the source term on the righthand side of Eq. (15.56).
The constant will disappear upon operation by the Laplacian, but the cos2(ω t − kx) term will drive the secondorder wave equation. What is crucially important is the recognition that the phase speed of the source term, c_{ph} = 2ω/2k, is identical to the phase speed of the secondorder density deviations, ρ_{2}, which propagates with speed, \( {c}_o=\sqrt{{\left(\partial p/\partial \rho \right)}_s} \).
This correspondence between the phase speed of the source and the phase speed of the disturbance it creates is called geometric resonance. In this case, the wavevectors representing the first and secondorder fields, \( {\overrightarrow{k}}_1\;\mathrm{and}\;{\overrightarrow{k}}_2 \), are colinear. Considering this process as the firstorder wave’s interaction with itself, the geometric resonance for these colinear propagation directions can be expressed as a wavevector sum.
Each infinitesimal fluid volume that is excited by quadratic combinations of the firstorder sound fields can be considered a source for the secondorder sound field. In Fig. 15.11, those fluid volumes are represented by individual loudspeakers with amplitudes that are proportional to ρ′^{2}. Because the phase velocity is also the thermodynamic sound speed, c_{o}, each of those “virtual loudspeakers” produces sound that sums in just the same way as the discrete endfire line array in Sect. 12.7.1. When the sound radiated by the first virtual loudspeaker propagates to the position of the second, the two will be inphase, and their amplitudes will add coherently. The sum then propagates to the third location and adds inphase and so on. This coherent addition along the direction of propagation produces the linear growth in the second harmonic’s amplitude that was described in Eqs. (15.37) and (15.43), as well as in Fig. 15.10. It also demonstrates the corresponding quadratic dependence on the amplitude of the firstorder field at any location.
3.3 Intermodulation Distortion and the Parametric EndFire Array
The distortion of a single, initially sinusoidal plane wave is due to the wave’s own influence on the medium through which it is propagating. The formalism of Eq. (15.56) makes it convenient to consider the nonlinear interaction of two plane waves propagating in the same direction but having different frequencies, ω_{↓} and ω_{↑}. For simplicity, let both sound waves have equal amplitudes, ρ_{1}.
At the linear level, they create a sound field that is simply their superposition.
The nonlinear source term in Eq. (15.56) is driven by the square of that linear superposition. Letting a = (ω_{↓}t − k_{↓}x) and b = (ω_{↑}t − k_{↑}x), the drive can be expressed as the sum of five contributions.
Again, the constant term in the square brackets will be eliminated from the driving term by the Laplacian in Eq. (15.56). The (2a) and (2b) terms represent the second harmonic distortion of the individual wave produced by their selfinteraction. The sum and difference terms, cos(a + b) and cos (a − b), are called intermodulation distortion products and represent the effect that one wave has on the medium that the other wave is passing through.
Having already analyzed the selfdistortion that creates the second harmonic distortion, our interest will now be focused on two interacting waves. Those interacting waves will be called the pump waves or primary waves. We will assume that their frequencies are closely spaced: ω_{↑} − ω_{↓} ≪ (ω_{↑} + ω_{↓})/2. These two colinear waves, as well as the products of their nonlinear interactions, are still all in geometric resonance.
In the absence of dispersion, if the two waves are not colinear, then the phase matching that is the consequence of geometrical resonance does not occur, and the interaction does not produce waves that propagate beyond the interaction volume [28].
Since the two “pump” or “primary” waves are assumed to be close in frequency, they have about the same thermoviscous spatial attenuation coefficient, α_{TV}, resulting in a characteristic exponential decay distance, ℓ = (α_{TV})^{−1}, as identified before in Sect. 15.1.3 for definition of the Gol’dberg number. Since the bulk attenuation coefficient is proportional to the square of the frequency, the two selfdistorted second harmonic components will suffer exponential decay over a distance that is only onefourth of ℓ, as will the wave that is produced by the nonlinear interaction that creates a wave at the sum of the two pump frequencies. The exponential decay of either of the pump waves is represented symbolically in Fig. 10.12 (Left). The growth and subsequent exponential decay of the second harmonics and sum waves are represented symbolically in Fig. 10.12 (right).
Although the waveform instability caused by nonlinear distortion had been understood since the time of the American Civil War, it was not until 1963 that Peter Westervelt recognized that highly directional receivers and transmitters of sound may be constructed by use of the nonlinearity in the equations of hydrodynamics^{Footnote 8} [29]. Although it had been known, both theoretically [30] and experimentally [31], that two plane waves of different frequencies propagating in the same direction generate two new waves at the sum and difference frequencies, it was not until Westervelt’s paper that the practical utility of that differencefrequency wave was recognized.
As we know from the analysis of the radiation from baffled circular pistons in Sect. 12.8, it is impossible to produce a narrow (i.e., directional) sound beam if the circumference of the radiating piston, 2πa, is on the order of the wavelength, λ, of the sound being radiated, or smaller. This makes it impossible to produce a directional sound beam at low frequencies from small vibrating surfaces. On the other hand, if 2πa ≫ λ, then the radiated sound will be confined to a narrow beam as quantified in Eq. (12.108). Westervelt recognized that it was possible to use nonlinear acoustics to create a narrow lowfrequency beam through the interaction of two narrow highfrequency, highamplitude sound beams of slightly different frequencies. If the highfrequency beams interacted over a distance that was much longer than the difference frequency wavelength, λ_{diff} = 2πc_{o}/(ω_{↑} − ω_{↓}), then the virtual array, like that depicted symbolically in Fig. 15.11, would produce a directional lowfrequency sound beam.
As long as the attenuation distance for the pump waves is longer than the wavelength of the difference frequency wave, the difference frequency will be produced by the endfire linear array from the nonlinear interaction of the two pump waves and will have the directionality characteristic of the pump wave’s directionality (see Fig. 15.15). The growth of the difference frequency wave will initially be linear with distance (as it was for the second harmonic distortion derived in Sect. 15.2.2), but due to the attenuation of the higherfrequency pump waves, depicted symbolically in Fig. 15.12 (left), the difference wave will reach some limiting amplitude as shown in Fig. 15.13.
An array consisting of 30 40 kHz piezoelectric transducers, shown in Fig. 15.14, was built to demonstrate the directionality of the differencefrequency beam. Fifteen of the transducers were wired electrically in parallel and driven at ω_{↓}/2π= 37.5 kHz, and the other 15 were wired in parallel and driven at ω_{↑}/2π=39.5 kHz to produce a parametric array that would create a difference wave at (ω_{↑} − ω_{↓})/2π= 2.0 kHz. These two subarrays were interlaced so that the nearest neighbors to any transducers driven at one of the frequencies would radiate at the other frequency.
That array is shown in Fig. 15.14. It has a height, h = 5 cm, and width, w = 21 cm. This produces a circularequivalent effective radius, a_{eff} = (h + w)/π ≅ 8 cm. At 40 kHz, the pump wavelength is λ_{pump} ≅ 0.9 cm, so ka_{eff} ≅ 60, making the pump waves very directional at that frequency. Using the directionality for a baffled piston in Eq. (12.108), the pump wave’s major lobe is confined within about ±3.6°. Since the array is rectangular rather than circular, the 40 kHz beam will be wider than this circular approximation in the vertical direction and narrower in the horizontal direction.
The attenuation of a 40 kHz sound wave in air is strongly dependent upon humidity (see Fig. 14.5). In dry air, the exponential absorption length, ℓ (0% RH) = 23 m, while for a relative humidity of 60%, ℓ (60% RH) = 2 m. The pump amplitude, p_{1} ≅ 20 Pa, so by Eq. (15.15), the shock inception distance is D_{S} = 8 m, assuming no spreading. A conservative estimate of the effective lowfrequency endfire array length, d_{eff}, might be 2 m, making the virtual line array’s value of k_{diff} d_{eff} ≅ 36 for the 2.0 kHz differencefrequency wave.
Although the directionality that can be achieved by the parametric array in this example is impressive, the electroacoustic energy conversion efficiency is very poor. The differencefrequency acoustic pressure amplitude, measured at 4 m from the source, was p_{2} = 0.14 Pa ≅ 74 dB re: 20 μPa_{rms}. At that distance, the beam’s crosssection was about 1 m^{2}. The intensity corresponding to p_{2} = 0.14 Pa is 22 μW/m^{2}. The electrical input power to the array was about 18 watts, so the net electroacoustic conversion efficiency is just about onepartpermillion or approximately 0.0001%.
This increase in differencefrequency directionality and the low conversion efficiency is illustrated in the directionality plots in Fig. 15.15 for a parametric endfire array operating at pump frequencies of 22 kHz and 27 kHz to produce a 5 kHz differencefrequency wave in water. The efficiency is better than in air due to the higher value of Γ in water and the higher acoustic pressures that could be produced, but the ratio of the amplitude of the differencefrequency to the pump is quite low, as demonstrated when both the pump and the difference frequency waves are plotted together in Fig. 15.15 (left). The smalleramplitude differencefrequency wave’s directionality is plotted by itself in Fig. 15.15 (right). Comparison of the two graphs shows that the differencefrequency beam is only slightly wider than the pump frequency beams.
The low conversion efficiencies of the parametric array are deemed acceptable for some niche applications. Parametric arrays for use in air are being produced commercially, but I have some trepidation about the possibility of detrimental physiological effects due to the very high pumpwave amplitudes at frequencies that are above the normal range of human hearing. I’m not of the opinion that “what you can’t hear, can’t hurt you.”
The ubiquity of such commercially available parametric arrays that are used to produce directional sound in air (i.e., “audio spotlights”) has renewed interest in the potentially detrimental health effects of highamplitude ultrasound exposure and led to the publication of a Special Issue of the Journal of the Acoustical Society of America that is focused on this subject [32].
We are currently in the undesirable situation where a member of the public can purchase a $20 device that can be used to expose another human to sound pressure levels that are > 50 dB in excess of the maximum permissible levels for public exposure.
Concern has been exacerbated by reports of the “weaponization” of highamplitude ultrasound that may have been used to injure diplomats at the US Embassy in Havana, Cuba [33], and elsewhere [34].
When I make measurements near such an ultrasound source (e.g., Fig. 15.14), I wear ear plugs and place soundattenuating earmuffs over my plugged ears. Other experimentalists who have not taken such precautions have exhibited symptoms like dizziness and nausea.
3.4 Resonant Mode Conversion
So far, the concept of geometrical resonance has restricted the evolution of harmonic distortion or the production of sum and difference waves to media that do not exhibit significant dispersion, as indicated by Eq. (15.61). If there is dispersion, so dc_{o}/df ≠ 0, then the some portions of the virtual array will start to become outofphase with other portions, and the uniform linear increase in amplitude with distance will become instead a “beating” where the amplitude would start growing and then start diminishing, possibly repeating that alternation if the interaction length were sufficiently long, as some portions of the virtual array subtract from the growth produced by other portions.
In this subsection, two beams that are not colinear are allowed to interact to produce another wave that travels at a different speed. That beating is illustrated by the measurements made in a waveguide of rectangular crosssection, made by Hamilton and TenCate [38], shown in Fig. 15.16.
If the propagation speed of the nonlinear product is greater than the propagation speed of the pumps and if the pump wavevectors are not colinear, there can be geometrical resonance (i.e., phase matching) at a unique interaction angle. I like to call this a “scissors effect.” If we assume that there are two waves of the same frequency, ω_{↑} = ω_{↓} = ω, but their wavevectors make a relative angle, θ, with each other, then the phase speed of the “sum” wave will be higher than the phase speed of either pump (primary) wave. This simple geometry is illustrated in Fig. 15.17.
This is similar to a scissors in that the speed of the intersection of the two blades moves faster than the speed at which the tips of the blades approach each other.
As introduced in Sect. 5.1.1, the speed of longitudinal waves in bulk solids is \( {c}_L=\sqrt{D/\rho } \), where D is the dilatational modulus, also known as the modulus of unilateral compression (see Sect. 4.2.2). Shear waves in bulk isotropic solids propagate at the shear wave speed, \( {c}_S=\sqrt{G/\rho } \), where G is the material’s shear modulus (see Sect. 4.2.3). The relationship between the moduli of any isotropic solid, summarized in Table 4.1, allows the relationship between those two sound speeds to be expressed in terms of the solid’s Poisson’s ratio, ν, and its Young’s modulus, E.
The stability criterion discussed in Sect. 4.2.3 restricts positive values of Poisson’s ratio to ν < ½, thus guaranteeing that c_{L} > c_{S}.
Based on the phase speed increase calculated for the interaction of two waves that are not colinear in Eq. (15.62) and the fact that c_{L} > c_{S}, it would be possible to have two shear waves interact though nonlinearity to produce the faster longitudinal wave where the modeconversion interaction angle, θ_{mc}, is determined by the Poisson’s ratio of the solid in which the two shear waves are interacting.
For polycrystalline aluminum, ν_{Al} = 0.345 [35], so cos (θ_{mc} /2) = 0.486. The required angle between the two shear wavevectors in aluminum must be θ_{mc} = 122° to make the interaction phase speed in Eq. (15.62) satisfies geometrical resonance for nonlinear mode conversion that couples two shear waves, each at frequency, ω, to a longitudinal wave with frequency 2ω.
Resonant mode conversion in solids was first described theoretically by Jones and Kobett [36] and observed experimentally shortly thereafter in aluminum, by Rollins, Taylor, and Todd, at the interaction angle predicted in Eq. (15.64) [37].
Another opportunity for resonant mode conversion is afforded by inspection of Fig. 15.5. From 1 K to 2 K, second (thermal) sound has a speed, c_{2}, of about 20 m/s, while the speed of first (compressional) sound, c_{1}, is around 230 m/s. Two second sound waves that are almost anticolinear could have an interaction phase speed equal to that of first sound. Using the geometry of Fig. 15.16, the mode conversion halfangle, at temperatures below T_{λ} = 2.172 K, depends upon the velocity ratio.
This suggests that θ_{mc} will be close to 180°.
A waveguide of rectangular crosssection affords an ideal geometry to provide a long interaction length while also affording precise control of the mode conversion angle for two plane waves of second sound. In a waveguide, the interaction angle of the two traveling plane waves (see Fig. 13.23) is controlled by the ratio of the drive frequency to the cutoff frequency. From Fig. 15.17 and Eq. (13.69), the modeconversion interaction halfangle, θ_{mc}/2, is related to the ratio of the second sound drive frequency, ω, to the cutoff frequency of the waveguide’s first nonplane wave mode, ω_{co}.
As shown in Fig. 15.18, if the height of the waveguide is ℓ_{z}, then the cutoff frequency would correspond to a single halfwavelength of second sound being equal to the waveguide’s height: ω_{co} = πc_{2}/ℓ_{z}. Substitution of Eq. (15.65) into Eq. (15.66) determines the ratio of the second sound frequency necessary for resonant mode conversion, ω_{mc}, to the waveguide’s cutoff frequency, ω_{co}.
Of course, it is necessary to do this experiment in superfluid helium at temperatures below T_{λ}, since second sound provides the pump (primary) waves, as well as to have an adequate nonlinear interaction length to observe this resonant mode conversion of second sound to first sound. Those two constraints led to the use of a spiral waveguide shown in Fig. 15.18 (left). Sum and difference waves generated by noncolinear waves in an airfilled waveguide of rectangular crosssection that were not geometrically resonant are shown in Fig. 15.16 that were measured by Hamilton and TenCate [38].
Landau’s twofluid description of superfluid helium requires eight variables [39]. In addition to the two thermodynamic variables, two separate velocity fields are necessary to describe the motion of the superfluid component and of the normal fluid component, \( {\overrightarrow{v}}_s\kern0.5em \mathrm{and}\kern0.5em {\overrightarrow{v}}_n \). This makes the secondorder wave equation for the nonlinear acoustic interactions more complicated than Eq. (15.56), but the inhomogeneous form, which provides a wave Eq. (15.68) to describe the spacetime evolution of the secondorder pressure, p_{2}, is still driven by quadratic combinations of the firstorder sound fields produced by first sound (\( {v}_1^2\kern0.5em \mathrm{and}\kern0.5em {p}_1^2 \)), second sound(\( {T}_1^2\kern0.5em \mathrm{and}\kern0.5em {w}_1^2 \)), or their interaction (p_{1}T_{1}) [15].
Since there are two velocity fields, Eq. (15.68) expresses the fluid’s motion in terms of the centerofmass velocity, \( \overrightarrow{v} \), which is nearly zero for second sound and \( \overrightarrow{w}={\overrightarrow{v}}_n{\overrightarrow{v}}_s \), which is nearly zero for first sound. Because w^{2} is a Galilean invariant (i.e., its value is not dependent on the motion of the coordinate system), it is also a thermodynamic variable, as evidenced by the partial derivative in the final term in Eq. (15.68).
Resonant mode conversion of second sound to first sound was observed experimentally from 1.15 K < T_{m} < 2.0 K using the spiral waveguide and heater shown in Fig. 15.19 (Right) [40].
4 Nonzero TimeAveraged Effects
Nonlinear acoustical effects are driven by quadratic combinations of firstorder sound fields. When the firstorder sound field was squared to produce Eq. (15.57), the constant term was ignored because it was operated upon by a Laplacian to produce the virtual sources that drove the inhomogeneous wave equation for the propagation of the secondorder sound field. In this subsection, the effects of that constant term will be explored, first with a focus on the square of the firstorder particle velocity, v_{1}, initially restricting our analysis to onedimensional propagating plane waves.
Since the firstorder acoustic fields have a sinusoidal time dependence, their timeaveraged values must vanish over times that are long compared to the periods of such disturbances, T ≫ 2π/ω.
The secondorder terms, like the squared velocity in Eq. (15.69), that contain a constant term, will produce timeaveraged secondorder pressures that will not vanish: 〈p_{2}〉_{t} ≠ 0. These secondorder nonzero timeaveraged pressures can produce substantial forces [41] and torques [42] on objects that are within the sound field. As early as the 1940s, Hillary St. Clair was able to levitate copper pennies (ρ_{Cu} = 8.9 gm/cm^{3}) [43]. Using an intense sound field produced by a siren and a reflector, Allen and Rudnick were able to repeat St. Clair’s demonstration:
“When a number of pennies are placed on a stretched silk screen, the parameters can be so adjusted that the pennies do somersaults with “Rockette“like precision; or so that a penny can be made to rise slowly to a vertical position, appearing all the while to be supporting, acrobatically, another which finally assumes a horizontal position above the first penny touching rim to rim. Also, coins resting on the silk screen can be flipped a distance of a few feet by varying the frequency of the siren rapidly.” [44]
4.1 The SecondOrder Pressure in an Adiabatic Compression
Nonlinear distortion, the generation of harmonics, and the “scattering of sound by sound” were attributed to the fact that a wave will modify the properties of the medium through which it is propagating. To start our investigation of nonzero timeaveraged effects, it will be instructive to consider the piston of area, A_{pist}, in a closefitted cylinder that is filled with an ideal gas at equilibrium pressure, p_{m}. With the piston in its equilibrium position, designated as x = 0, the equilibrium volume of the gas in the cylinder will be V_{o} = A_{pist} L, where L is the length of the cylinder from the rigid end located at x = L to the piston’s equilibrium position. This arrangement is identical to that depicted schematically in Fig. 8.5.
If the gas inside the cylinder obeys the Adiabatic Gas Law and if the motion of the piston is sinusoidal, with the piston’s position given by x(t) = x_{1} sin (ω t), then the pressure within the cylinder will be uniform throughout and given by the Adiabatic Gas Law as long as \( \sqrt{A_{pist}}\ll \lambda /2\pi \kern0.5em ={c}_o/\omega \kern0.5em \mathrm{and}\kern0.5em L\ll {c}_o/\omega \), so that the cylinder can be treated as a “lumped element,” where \( {c}_o=\sqrt{{\left(\partial p/\partial \rho \right)}_s} \)is the speed of sound under equilibrium conditions.
This is the familiar “linear” result; a sinusoidal variation in the piston’s position leads to a sinusoidal variation of the pressure within the cylinder. Such a result assumes that x_{1}/L ≪ 1, so the motion of the piston does not affect the volume, V_{o}, that appears in Eq. (15.71). Of course, that is not exactly true. As the ratio of x_{1} to L increases, the importance of the piston’s instantaneous position on the value of the volume of the gas becomes more influential. It is easy to take the change in the cylinder’s volume into account. When the piston moves inward, it sweeps out a volume, δV(t) = −A_{pist} x(t), which should be subtracted from the equilibrium volume, V_{o}.
Taking the timeaverage of the pressure over a period, T, the linear term vanishes, but the quadratic term produces a nonzero timeaveraged pressure, 〈p_{2}〉_{t}, since sin^{2}(ω t) = ½[1 − sin (2ω t)].
The integral over the component oscillating at 2ω will vanish but the constant component will not. That timeaveraged excess pressure will tend to push the piston away from the closed end of the cylinder. This effect produces “piston walk” in Stirling cycle machines.
This timeaveraged pressure can be expressed in terms of the firstorder pressure calculated in Eq. (15.71): \( {x}_1/L={p}_1/\gamma {p}_m={p}_1/{\rho}_m{c}_o^2 \), if the cylinder contains an ideal gas.
In this form, it is clear that the nonzero timeaveraged pressure is quadratic in the firstorder pressure. It is also useful to recognize that this result is equal to the potential energy density as derived from the energy conservation Eq. (10.35).
As with the results of weak shock theory in Sect. 15.2, it is the effects of the piston’s position on the volume that appears in the Adiabatic Gas Law of Eq. (15.72) that produces corrections to the linear result. The creation of a net secondorder pressure is due to the asymmetry produced by the fact that the average volume on compression of a piston is smaller than the average volume during expansion.
Application of this result to a onedimensional standing wave resonator is straightforward. Within the resonator, the firstorder pressure can be written as \( {p}_1\left(x,t\right)=\mathit{\Re e}\left[\hat{\mathbf{p}}\cos \left( n\pi x/L\right){e}^{j\omega\;t}\right] \). Close to the end at x = 0, the firstorder acoustic pressure is nearly independent of position, just as it is in the piston and cylinder example. By the Euler equation, the longitudinal particle velocity can be written as \( {v}_1\left(x,t\right)=\mathit{\Re e}\left[j\left(\hat{\mathbf{p}}/{\rho}_m{c}_o\right)\sin \left( n\pi x/L\right){e}^{j\omega\;t}\right]\equiv {v}_1\sin \left( n\pi x/L\right)\sin \left(\omega\;t+\varphi \right) \), so the particle velocity goes linearly to zero as x goes to L, just as it does in the piston and cylinder example. This situation near to the rigid end of the resonator (or close to any standing wave pressure antinode) can be represented by an imaginary line (i.e., a Lagrangian marker) that moves with the gas, acting as the piston while neglecting the remaining gas in the resonator.
In a sealed resonator, the total mass of the gas cannot change. If the static pressure at the rigid ends (as well as at any pressure antinode for higherorder longitudinal modes of the resonator, n > 1) increases by the amount specified in Eq. (15.74), then the density of the gas must also increase in those locations. For that to happen in a sealed system, the gas density (and pressure) must decrease elsewhere.
In a standing wave, the amplitude of the gas particle velocity at a pressure node (velocity antinode) is v_{1} = p_{1}/ρ_{m}c_{o}, so if the total mass of the gas cannot change, then the nonzero, secondorder, timeaveraged pressure at the firstorder pressure node must be equal and opposite to the value in Eq. (15.74) and can be rewritten in terms of v_{1}.
In this form, it is clear that the nonzero timeaveraged pressure is quadratic in the firstorder velocity amplitude. It is also useful to recognize that this result is equal to the kinetic energy density at the velocity antinode as derived from the energy conservation Eq. (10.35). It also has the functional form of the Bernoulli pressure.
The relationship between the secondorder timeaveraged pressure [45], also known as the radiation pressure, and the kinetic and potential energy densities will be derived from the hydrodynamic equations in Sect. 15.4.4 after examining a few examples of the acoustical consequences produced by the Bernoulli pressure in the following subsection and in Sect. 15.4.3.
4.2 The Bernoulli Pressure
The first introduction in this textbook to the Bernoulli pressure^{Footnote 9} was provided in the analysis of the Venturi tube (see Sect. 8.4.1) that was intended to aid in the understanding of the convective term in the total hydrodynamic derivative in Eq. (8.33). This resulted in the introduction of a pressure gradient produced in the tube that was driven by the square of the fluid’s velocity, v^{2}.
Since the Bernoulli pressure is proportional to the square of the fluid’s velocity, it is independent of the direction of flow. For the oscillatory velocities that are produced by sound waves, this means that the timeaveraged Bernoulli pressure will be nonzero.
The effects of the Bernoulli pressure for oscillatory flows produced by sound waves were recognized and understood by Lord Rayleigh. The Kundt’s tube was a popular piece of acoustic apparatus that produced highamplitude standing waves by stroking a rod that would excite longitudinal vibrations and couple those vibrations to the air contained in a transparent glass tube [46]. Cork dust or lycopodium seeds were commonly used to visualize the sound field by “decorating” velocity antinodes. Figure 8.14 shows cork dust striations in the neck of a resonator that is excited in its Helmholtz mode, f_{o} = 210 Hz (left), and at a frequency, f_{1} = 1240 Hz, that excited a halfwavelength standing wave in the neck (right) [47].
Rayleigh recognized that two small particles of sufficient mass to remain stationary within the oscillatory flow field, due to their inertia,^{Footnote 10} would be attracted to each other because the oscillatory air flow must accelerate as it passes between the constrictions produced by the adjacent particles. By Eq. (15.76), the increased fluid velocity between the particles results in a lower pressure so that the resultant pressure gradient would drive the particles together.
The figure taken from Rayleigh’s Theory of Sound that diagrams this attraction is shown in Fig. 15.20 (center). This effect, known as acoustic agglomeration, has been used in several applications where removal of larger clusters of smaller particles from a fluid is easier than the removal of smaller individual particles [48]. More recently, “acoustic agglomeration” has been used for separation of biological cells grown in bioreactors from their nutrient liquid [49].
Rayleigh makes a similar argument, as also illustrated in Fig. 15.20 (left), to explain the striations of the dust particles agglomerated in planes that are normal to the oscillatory flow. When two particles (or planes of particles) are separated along the direction of the oscillatory flow, the stagnation of the fluid between them produces an increase in the timeaveraged pressure that causes the particles (or planes of particles) to repel each other, as clearly visible in the striations seen in Fig. 8.14.
Another interesting manifestation of the Bernoulli pressure was mentioned by Rayleigh in regard to the forces on a Helmholtz resonator. The fluid’s velocity in the neck of a Helmholtz resonator is high. Based on Eq. (15.76), this suggests that the pressure in the neck must be reduced. Since the neck is in direct contact with an effectively infinite reservoir of atmospheric pressure, the only means by which the required pressure difference can be maintained is if the static timeaveraged pressure within the compliance (volume) of the Helmholtz resonator is greater than atmospheric pressure.
This secondorder, acoustically induced pressure difference, 〈p_{2}〉_{t}, will lead to a net force on the Helmholtz resonator since the pressure on the surfaces of the volume are unbalanced over the crosssectional area, \( \pi {a}_{neck}^2 \), of the neck:\( {F}_{net}=\pi {a}_{neck}^2{\left\langle {p}_2\left({v}_{neck}^2\right)\right\rangle}_t \).
“Among the phenomena of the second order which admit of a ready explanation, a prominent place must be assigned to the repulsion of resonators discovered independently by Dvořák [51] and Meyer [52]. These observers found that an air resonator of any kind when exposed to a powerful source experiences a force directed inwards from the mouth, somewhat after the manner of a rocket. A combination of four light resonators, mounted anemometer fashion upon a steel point, may be caused to revolve continuously.” [53]
Apparently, an acoustical demonstration of the nonlinear force on a resonator that resembles a lawn sprinkler, shown in Fig. 15.21, from [52], was well known to RaylTheir dimensionless eigh.^{Footnote 11} This effect can be observed in a quantitative way by placing a Helmholtz resonator on a sensitive balance and producing a large amplitude sound field in the vicinity using a loudspeaker driven at the Helmholtz resonance frequency and then observing the increase in the resonator’s apparent weight to do “the rocket.”
4.3 The Rayleigh Disk
The Bernoulli pressure of Eq. (15.76) can also exert torques, \( N\left({v}_1^2\right) \), on an extended object placed in an oscillatory flow field. Rayleigh’s diagram of such a disk that is aligned at about 45° with respect to the flow field is shown in Fig. 15.20 (right). That figure captures the flow at an instant when it is moving from right to left, as indicated by the arrows. The approaching flow stagnates between A and B where it diverges, and the receding flow stagnates on the other side of the disk between C and Q where it rejoins. On the inflow side of the desk, the flow must accelerate along AQC as the two flows converge on the outflow side at P. The stagnant flow between A and B on the inflow side and between C and Q on the outflow side has a higher pressure than the fastermoving flows at the same locations on the opposite sides of the disk. This produces a net torque that tends to orient the disk perpendicular to the flow, regardless of the flow direction.
An appreciation for the magnitude of this torque can be obtained by calculation of the moment of the Bernoulli pressure in Eq. (15.76) over both sides of a disk having radius, a, assuming the presence of the disk does not perturb the sound field.^{Footnote 12} The circle in Fig. 4.11 that was used to calculate the radius of gyration for beam flexure will provide the coordinate system for this integration.
If the disk is assumed to be suspended by a torsion fiber in the oscillatory flow, then the torque will be zero when the surface of the disk is perpendicular to the flow or when the surface of the disk is aligned with the flow. If the angle between the normal to the disk’s surface is designated θ, then the torque will be zero when θ = 0° (occluding the flow) or when θ = 90° (aligned with the flow), except that the θ = 90° orientation will be unstable. If the disk is aligned with the flow and its orientation deviates slightly from θ = 90°, then the torque will cause the disk to seek the θ = 0° orientation. If the disk is in the θ = 0°, any small deviation in θ will subject the disk to a torque that will tend to restore the θ = 0° orientation.
Based on the magnitude of the torque in Eq. (15.77) and the previous stability argument, the torque as a function of the square of the timeaveraged oscillatory velocity amplitude,\( {\left\langle {v}_1^2\right\rangle}_t \), and the orientation angle can be written in the form that appears in Theory of Sound, which Rayleigh attributes to König [54].
Rayleigh recognized that “Upon this principle an instrument may be constructed for measuring the intensities of aerial vibrations of selected pitch” and suggests that the disk be a mirror suspended by a silk thread so a light beam could be used as an optical lever (see Sect. 2.4.4) to determine the disk’s orientation [55].
Prior to the introduction of the reciprocity method for calibration of reversible transducers (see Sect. 10.7.2, 10.7.3 and 10.7.4), the Rayleigh disk was a primary technique for determination of acoustic sound field amplitudes [56]. Due to its importance, a detailed analysis of the torque was made by King to include corrections produced by the disk’s influence on the sound field [57]. The torque on a Rayleigh disk located at a velocity antinode in a standing wave field included wavelengthdependent corrections for the disk’s mass, m_{1}, as well as the disk’s hydrodynamic (inertial) entrained mass, m_{o} = (8/3)ρ_{m}a^{3}, as calculated in Eq. (12.126).
The indifference of the sign of the torque produced by flow in either direction was important in establishing the physical reality of Landau’s twofluid theory of superfluid hydrodynamics.^{5} As mentioned briefly in Sects. 15.1.2 and 15.3.4, there are two velocity fields necessary to characterize the dynamics of superfluid flow,\( {\overrightarrow{v}}_s\kern0.5em \mathrm{and}\kern0.5em {\overrightarrow{v}}_n \). In a thermally induced second sound wave, the superfluid’s centerofmass velocity is zero, but the counterflow of \( {\overrightarrow{v}}_s\kern0.5em \mathrm{and}\kern0.5em {\overrightarrow{v}}_n \)is nonzero.
Since the Rayleigh disk responds to the torque of both flow fields without respect to their direction, Pellam and Hanson were able to establish the physical existence of both velocity fields and make the first mechanical measurement of second sound in superfluid helium [58]. Later, Koehler and Pellam were also able to measure the superfluid fraction, ρ_{s}/ρ, as a function of temperature using their Rayleigh disk [59]. Both measurements employed a mirror as the disk to detect the disk’s deflection optically. Later measurements of torques in superfluids used a nonoptical method to determine the Rayleigh disk’s orientation [60].
4.4 Radiation Pressure
Restricting attention to one dimension, the Bernoulli pressure can be derived from the Euler Eq. (15.48).
The goal will be to express Eq. (15.80) entirely in terms of the gradient of a scalar, so it is useful to introduce the specific enthalpy (heat) function (see Sect. 14.2), h = ε + pV, where ε is the fluid’s internal energy per unit volume (see Sect. 7.1.2): dε = dU/V.
Using the definition of the internal energy from Eq. (7.10), dε = T ds – p dV, the pressure gradient can be expressed in terms of the specific enthalpy, dh = dp/ρ_{m}, and the product rule can be invoked to consolidate the convective contribution.
Having started with the Euler equation, the effects of viscosity have already been eliminated, so the Kelvin circulation theorem guarantees that the velocity field will be curl free; thus it can be expressed as the gradient of a scalar, ϕ, known as the velocity potential: \( \overrightarrow{v}=\overrightarrow{\nabla}\phi \) [61].
Since the argument within the gradient in Eq. (15.83) is equal to zero, the function within the gradient must be a constant everywhere throughout the fluid.
This is the “strong” form of Bernoulli’s equation, since it is not restricted only to streamlines, as it was for the version introduced in Eq. (15.76).
To retain accuracy to secondorder, the specific enthalpy must also be expanded to secondorder.
These thermodynamic derivatives can be evaluated for adiabatic processes, dS = 0, from the differential form of the specific enthalpy: dh = T dS − dp/ρ_{m}.
In this subsection, we are only interested in the parts of Eq. (15.86) which produce a nonzero timeaverage. As in Eq. (15.70), the timeaverage of firstorder variations will vanish: 〈∂ϕ/∂t〉_{t} = 〈p_{1}〉_{t} = 0.
The secondorder timeaveraged pressure is the difference between the potential and kinetic energy densities. In classical mechanics, that combination is known as the Lagrangian density [62].
For a collimated traveling wave of the usual form,^{Footnote 13} p_{1}(x, t) = p_{1} cos (ω t − kx), the linearized Euler’s equation provides the ubiquitous relationship between the firstorder acoustic field variables: p_{1} = ρ_{m}c_{o}v_{1}. That relationship can then be substituted into Eq. (15.87).
This result is oddly both philosophically significant and trivially obvious. If there were an object in the travelingwave field, it would scatter some portion of the sound (see Sects. 12.6.1 and 12.6.2), and the sum of the scattered and incident wave fields would produce a standing wave. If the field is entirely a traveling wave, then that wave field cannot include an “object” which would feel the force of a timeaveraged secondorder pressure based on the object’s density and/or compressibility contrast.
4.5 Acoustic Levitation in Standing Waves
The result for the timeaveraged secondorder pressure in Eq. (15.87) can also be evaluated for a standing wave.
Substitution of Eq. (15.89) into (15.87) produces the timeaveraged secondorder pressure distribution for a standing wave in an ideal gas where \( {\rho}_m{c}_o^2=\gamma {p}_m \).
A standing wave produces a timeaveraged (i.e., static) secondorder pressure distribution. Due to the spatial dependence on cos (2kx), there is a minimum in the secondorder pressure at the location of each pressure node, thus at each velocity antinode and a maximum onequarter wavelength from that minimum. This is consistent with the secondorder piston example used as an introduction to nonzero timeaveraged effects in Sect. 15.4.1.
This secondorder timeaveraged pressure distribution produces pressure gradients that are fixed in space and time and will exert forces on solid objects of nonzero thickness. The force on an object at either the maximum or the minimum in 〈p_{2}〉_{t} will be zero, but that equilibrium will be unstable at the maximum. If a levitated object is displaced slightly from the maximum, it will be forced toward the minimum in 〈p_{2}〉_{t} that occurs at a firstorder velocity antinode which has the lowest pressure, due to Bernoulli.
The integrated pressure over a small sphere of radius, a ≪ λ, will produce a force, F_{sphere}, that is, a function of the sphere’s location in the standing wave field.
Rudnick provided a clever confirmation of this result in a simple standing wave tube that measured the angle of displacement of small spheres suspended by “a hair” due to the standing wave [63]. The integrated pressure over a small disk of thickness, t, and radius, a, will produce a force, F_{disk}, that is, a function of the disk’s location
To levitate a small sphere made of a material with a mass density, ρ_{sphere}, against the force of gravity, the weight of the sphere must be cancelled by the levitation force. This requires that the square of the firstorder standing wave pressure field amplitude, p′^{2}, exceed a minimum value, \( {p^{\prime}}_{\mathrm{min}}^2 \).
When this criterion is satisfied, then the position of the sphere will adjust itself within the (vertical) standing wave field to make the net force on the sphere be zero at some location below a velocity antinode. The stability of that equilibrium will be the subject of Sect. 15.4.6.
Of course, the levitated object does not have to be either a sphere or a disk. As shown in Fig. 15.22, almost any small object can be suspended against the force of gravity if the amplitude of the standing wave sound field is sufficient.
4.6 Adiabatic Invariance and the Levitation Force
In the previous subsection, the influence of the object being levitated by the standing wave on the response of the resonator was ignored. As will now be demonstrated, the perturbation of the resonator’s normal mode frequency caused by an object will provide an alternative method to predict the levitation force by use of adiabatic invariance and without the necessity of integrating the pressure gradient around the object. In the subsequent subsection, the feedback between the radiation force and the object’s influence on the resonance frequency will also have significant impact on the stability of the levitated object in a resonator that is driven at a constant frequency.
Throughout this text, the concept of adiabatic invariance [64] has been utilized when it was convenient to relate changes in a system’s constraints (e.g., boundary conditions) to one or more of that system’s normal mode frequencies. Now adiabatic invariance will be applied in the same way (i.e., the “variable constraint” being the position of the object in the sound field) to a onedimensional standing wave tube’s resonance frequencies that are perturbed by an incompressible obstacle that can be placed anywhere within the resonator of length, L, and crosssectional area, A.
It is assumed that the obstacle of volume, V, shown as a small cube in Fig. 15.23 (left), has dimensions that are all much smaller than the wavelength, λ_{n}, of any normal mode of interest: \( \sqrt[3]{V}\ll {\lambda}_n \). Because the obstacle is located at a pressure antinode (velocity node) in Fig. 15.23 (left), the excluded (incompressible) volume “stiffens” the gas springiness at that rigid end and raises the unperturbed (empty resonator) frequency, f_{1}, of the fundamental (n = 1) mode:f_{1} = c_{o}/2L.
To estimate the increase in frequency caused by the obstacle when it is near a pressure antinode (velocity node), we can use the same trick that simplified the calculation of the frequency shift caused by the deposition of a thin layer of gold that lowered the fundamental frequency of a quartz microbalance in Sect. 5.1.2 due to its additional mass loading. Let the cube be made of wax. If the wax were melted with the tube in the vertical orientation, then the volume of wax would remain unchanged, but it would be spread uniformly over the resonator’s endcap as shown in Fig. 15.23 (right). Since the slope of the pressure at the endcap is zero, the cube and slab versions of the obstacle produce the same stiffening of the gas (i.e., exclude the same amount of resonator volume). The perturbed frequency, f_{1}′, is then that of the slightly shorter resonator shown in Fig. 15.23 (right): f_{1}^{′} = c_{o}/2L_{eff}.
If the same obstacle was moved to the center of the resonator, as shown in Fig. 15.24, then it would lower the resonance frequency below the unperturbed frequency, f_{1}. This is because the obstacle has created a constriction in the resonator’s crosssectional area, A, at a position within the fundamental mode that is located at a velocity antinode (pressure node). The highspeed gas near the resonator’s midplane must accelerate to go around the obstacle, thus increasing the kinetic energy without affecting the potential energy stored at the ends of the resonator (see Sect. 13.3.4).
By Rayleigh’s method (see Sect. 2.3.2), this means that the fundamental normal mode frequency must be reduced. The amount of that frequency reduction is dependent upon the shape of the obstacle, so it is not as easy to make a quantitative estimate of the frequency reduction as it was for the case where the incompressible obstacle was located at a pressure antinode (velocity node). Fortunately, the use of adiabatic invariance provides a method to measure the effect of an obstacle of any shape and any location within the standing wave then relate that frequency shift produced to the levitation force, as described in the next subsection.
For the resonator’s second mode (n = 2), the obstacle is located at a pressure antinode and thus raises the resonance frequency of that mode. Again, since the resonator’s midplane contributes gas stiffness in the second mode (along with the gas stiffnesses at both ends), the volume exclusion produced by the obstacle increases the gas stiffness. The “melted wax” trick would work again by symmetry, treating the resonator as two halfresonators, each shortened by the appropriate amount: L_{eff}/2 < L/2.
It is worthwhile to notice that when this obstacle is located at the center of the resonator, it has ruined the harmonicity of the modes of the closedclosed resonator of uniform crosssection: f_{n} ≠ nf_{1}. All of the n = odd modes will be “flattened” (^{b}) (i.e., their normal mode frequencies will be lowered), and all of the n = even modes will be sharpened (^{#}), as long as the \( \sqrt[3]{V}\ll {\lambda}_n \) constraint is satisfied so the obstacle can be consider to be “small.” This strategy is regularly employed to suppress the formation of shock waves in standing wave resonators that are used in highamplitude applications like thermoacoustic refrigerators [65] and sonic compressors [66].
In previous applications of adiabatic invariance, the work that was done against (or by) the radiation pressure was used to estimate normal mode frequencies of resonators with shapes that did not conform to the 11 separable geometries (see Sect. 13.1). It will now be easy, using Eq. (15.87), to demonstrate the connection between frequency changes and work done against the radiation force. If the resonator, shown schematically in Fig. 15.25, has an initial length, L, the standing wave pressure distribution is related to the velocity distribution that satisfied the rigid (impenetrable) boundary conditions as provided in Eq. (15.89). Here, we will focus on the fundamental mode, n = 1.
The timeaveraged energy in the first mode, E_{1}, can be expressed as the timeaverage of the sum of the kinetic and potential energies, or by the virial theorem (Sect. 2.3.1), as the maximum potential energy,(PE)_{max}, by integrating the potential energy density of Eq. (10.35) throughout the resonator’s volume.
The rightmost result again assumes an ideal gas. The radiation force on the piston at the left of Fig. 15.23 is given by Eq. (15.90). The work increment, dW, done by the piston against the radiation force is just the force, F_{rad} = 〈p_{2}〉_{t}A, times the displacement, dx.
Adiabatic invariance requires that the ratio of the energy in a mode to its frequency remains constant if the system’s constraints are changed slowly (i.e., we don’t “jerk” the piston).
This is exactly the frequency change that would be due to a decrease in the resonator’s length by an amount, dx, based on the simplest result: f_{1} = c_{o}/2 L. In fact, the triviality of this result can be interpreted as a check on the expression (or a derivation) of the radiation pressure, 〈p_{2}〉_{t}, in Eqs. (15.87) and (15.90).
It is now possible to combine adiabatic invariance and the normal mode frequency change, related to the change in an obstacle’s position in a standing wave resonator, to calculate the levitation force from an alternative perspective [67]. A DeltaEC model of a resonator is provided in Fig. 15.26. The DeltaEC model makes use of a “MasterSlave Link” between Segments #2c and #6c that keeps the total length of the resonator fixed as the constriction, produced by Segments #3, #4, and #5, is moved from one end of the resonator to the other by changing the length of the DUCT in Seg. #2c from 0.0 m to 0.97 m.
The resonance frequencies of the first and second standing waves that are plotted in Fig. 15.27 as a function of the constriction’s location were produced using DeltaEC’s incremental plotting function (see Sect. 8.6.12). Plots of the normal mode frequency shifts, similar to those in Fig. 15.27, appeared in the literature for the fundamental mode and for the n = 2 mode [68], although it was not recognized at that time that those shifts were related to the levitation forces by adiabatic invariance.
The mobile constriction in the DeltaEC model removes 40 cm^{3} of resonator’s unperturbed 10 liter volume (10,000 cm^{3}). This is approximately equivalent to a resonator of uniform crosssectional area, A = 1.0 × 10^{−2} m^{2}, that contains an incompressible sphere of radius, a_{sphere} = 2.12 cm, or to a disk of radius, a_{disk} = 2.52 cm and thickness, t = 2.0 cm. Again, the DeltaEC model will not be exact because the shift in the frequency due to a kinetic energy perturbation is shape dependent, even if the obstacle is small compared to the wavelength. Although the constriction in the DeltaEC model is trapezoidal and not a sphere or disk, it provides a plausible approximation of the change in the resonator’s crosssectional area that would be caused by the sphere or disk that provides the same volume exclusion.
Adiabatic invariance requires that the ratio of the modal energy to the modal frequency, E_{n}/f_{n}, be a constant as long as the motion of the obstacle is slow compared to the period of the standing wave, T_{n} = (f_{n})^{−1}. As shown in Fig. 15.27, the resonance frequency is a function of the constriction’s location within the resonator. The energy of the mode must also be a function of position so the radiation force on the sphere, F_{sphere}, or an obstacle of some other shape must be equal to the gradient in that energy (see Sect. 1.2.1).
Adiabatic invariance guarantees that E_{n}/f_{n} = constant, so by log differentiation (Sect. 1.1.3), dE_{n}/df_{n} = E_{n}/f_{n} = constant.
The value of df_{n}/dx will depend upon the obstacle’s position within the resonator. That slope will have its maximum value at locations equidistant between the nodes and the antinodes of the firstorder standing wave fields. Using the results for the second standing wave mode produced by the DeltaEC model and plotted in Fig. 15.25, the maximum slope is just π/2 times the difference between the maximum frequency (f_{2+} = 348.4 Hz) and the minimum frequency (f_{2−} = 345.09 Hz), divided by the separation between the location of those two extrema, Δx = 0.235 m: df_{2}/dx = 21.1 Hz/m.
For convenience, the constant, E_{2}/f_{2}, can be evaluated with the obstacle located at the driven end of resonator (i.e., Seg. #2c = 0.0 m), where E_{2} is given by Eq. (15.94), with p’ = 2.0 kPa (Seg. #0d), p_{m} = 100 kPa (Seg. #0a), γ = (7/5), and (AL) ≅ 0.01 m^{3}. At that location, f_{2} = f_{2+} = 348.4 Hz, and E_{2} = 7.14 x 10^{−2} J, so E_{2}/f_{2} = 2.05 x 10^{−4} J/Hz. Substitution of these two results into Eq. (15.97) provides the radiation force due to the constriction, F_{rad}, at a position equidistant between the nodes and the antinodes of the firstorder standing wave fields, which is a consequence of adiabatic invariance for a trapezoidalshaped obstacle.
This result can be compared to the radiation force at the same position for the n = 2 mode, under the same conditions, if the pressure at either antinode (i.e., rigid end) is set to p’ = 2.0 kPa, equivalent to 157 dB re: 20 μPa_{rms}, for a sphere, F_{sphere}, in Eq. (15.91), or a disk, F_{disk}, in Eq. (15.92). To make a reasonable comparison, the volume of the sphere is set equal to the volume excluded by the trapezoidal constriction: V = 40 cm^{3}.
From Eq. (15.92), the result would be the same for a disk of the same volume.
Our estimate of the levitation force based on the DeltaEC model and adiabatic invariance is reasonably close to that result, given that the frequency shift computation was based on a constriction rather than an obstruction.
At this point, the serious reader will pause to marvel at the elegance and beauty that adiabatic invariance has demonstrated by its ability to circumvent the difficulties of integrals of secondorder pressure fields over objects of arbitrary shape in favor of a simple measurement of the resonant frequency shift as a function of position of the object to be levitated within the resonator. Putterman claims that adiabatic invariance is “the cornerstone of modern physics” [69]. Similar results can be obtained for determination of the torque on a Rayleigh disk by measuring the shift in the resonance frequency as a function of the disk’s orientation [60].
4.7 Levitation Superstability (“Acoustic Molasses”)
Most acoustic levitation systems are driven at a fixed frequency [70]. Since the position of the levitated object can change, the ratio of the drive frequency to the resonator’s resonance frequency, ω/ω_{o}, will also change. That frequency shift at fixed drive frequency produces an effect referred to as “detuning” that is illustrated in Fig. 15.27. The frequency shift causes the amplitude of the standing wave to change resulting in a change in the radiation pressure acting on the levitated object. This modifies the Hooke’s law “stiffness” of the radiation force acting on the object. If the object did not influence the tuning, then the object would be levitated at the equilibrium position within the standing wave where the radiation force and the gravitational force would be equal and opposite. The fact that the object’s position also changes the tuning would change the trapping stiffness constant because its position influences the amplitude of the sound in the resonator when driven at fixed frequency.
This change in stiffness can be understood by examining the three shifted response curves illustrated in Fig. 15.28. All the resonance curves in Fig. 15.28 correspond to a quality factor of Q = 10. Assume that the resonator is driven at a fixed frequency that was 5% above the resonance frequency of the empty resonator so that ω/ω_{o} = 1.05. The value of v_{1}^{2} would be 51.2% of the maximum that occurs at ω/ω_{o} = 1.00, if the resonator was empty. If the object moved up from its equilibrium position (i.e., toward the closest velocity antinode), then the resonator’s resonance frequency would become lower, corresponding to the dotted resonance curve. The force on the object would decrease because the acoustic standing wave amplitude would decrease, since the value of v_{1}^{2} would be 32.3% of the maximum that occurs at ω/ω_{o} = 1.00.
If the object moved down from its equilibrium position (i.e., toward the closest velocity node), then the resonator’s resonance frequency would become higher, and the drive frequency would be closer to the resonance frequency. The value of v_{1}^{2} in Fig. 15.28 would be 80.4% of the maximum that occurs at ω/ω_{o} = 1.00, if the resonator was empty. This corresponds to the dashed resonance curve in Fig. 15.28, and the force on the object would increase because the acoustic standing wave amplitude would have increased. The combined effect would be an increase in the stiffness.
If the empty resonator was initially tuned ω/ω_{o} = 0.95, then the effective stiffness would be less by the same argument except that the object’s influence on the sound amplitude would be determined by its “motion” along the vertical line in Fig. 15.28 above ω/ω_{o} = 0.95, instead of the previous discussion that had the object’s motion causing changes to the acoustic amplitude represented by “motion” along the vertical line above ω/ω_{o} = 1.05 in Fig. 15.28.
If this influence of the object’s position on the effective stiffness of its capture around its equilibrium position in the standing wave occurred instantaneously in response to the object’s change in position, then any displacement of the object would simply oscillate at a slightly different frequency about the equilibrium position than it would if the detuning was neglected. Viscous effects (i.e., Stokes drag) would eventually damp those oscillations, corresponding to a mechanical resistance, R_{m}, in the simple harmonic oscillator equation.
Because we are considering the standing wave resonator as a driven resonant system with Q ≠ 0, the exponential relaxation time, τ, required for the resonator to achieve its steadystate response after its tuning is changed is nonzero (see Sect. 2.5.4): Q = (½)ω_{ο}τ. The resonator’s response time is much longer than the period, T = 2π/ω_{o}, of the standing wave: τ = (Q/π)T. That delay in the resonator’s response to the position of the levitated object means that there will be a component of the force modulated by the object’s position that will not be inphase with the object’s position but that will be inphase or outofphase with the object’s velocity. The influence of the detuning will be retarded by a time, τ, so the current radiation force acting on the object will depend upon the position of that object at an earlier time, t − τ.
If this retardation produces a component of the excess (i.e., detuning) force that is outofphase with the velocity of the object’s displacement from its equilibrium position, dx/dt, then this force will behave like mechanical damping in addition viscous “Stokes drag,” in Eq. (15.100). If this retardation produces a component of the excess (i.e., detuning) force that is inphase with the velocity of the object’s displacement from its equilibrium position, dx/dt, then this force will behave like a negative mechanical resistance.
When the magnitude of that negative resistance is less than the ordinary viscous resistance, R_{m}, in Eq. (15.100), then oscillations will take longer to damp out. If the magnitude of that negative resistance is greater than R_{m}, then the amplitude of the object’s oscillations will grow exponentially with time until some other effect limits the oscillation’s amplitude. In some important cases this detuning/dephasing instability will throw the levitated object out of the equilibrium position and possible propel the object against the resonator’s boundaries [71].
The two possible scenarios are illustrated symbolically in Fig. 15.29. If the natural frequency of the resonator is lower than the drive frequency, ω/ω_{o} > 1 (sharp tuning), then motion of the levitated object toward a pressure antinode (i.e., away from a velocity antinode) will raise ω_{o} and bring the drive frequency closer to the resonance frequency. This will produce an excess force, F_{excess}, that will be increased, thus inphase with the velocity of the object as it is moving up from its lowest position, since the force will depend upon the previous position of the object at a time, τ, earlier. When the object reaches its maximum vertical position, the natural frequency of the standing wave resonator will be farther outoftune, and the radiation force is reduced, so gravity will provide an excess force. Again, due to the delay, that excess force will be acting in the downward direction and is again inphase with the (now downward) velocity of the object. This scenario is depicted in Fig. 15.29 (left).
The net effect for the “sharp tuning” case has the excess force doing work on the object, thus increasing the amplitude of its oscillations during each cycle. If the effect is sufficiently large, it can overcome viscous damping making the amplitude of the object’s oscillations grow linearly with time until some other effect limits the amplitude of the oscillations or the object is flung too far from the equilibrium position that it is no longer trapped or bangs against the walls or ends of the resonator.
If the natural frequency of the resonator is higher than the drive frequency, ω/ω_{o} < 1 (flat tuning), then motion of the levitated object toward a pressure antinode (i.e., away from a velocity antinode) will raise ω_{o} and bring the drive frequency farther from the resonance frequency. This will reduce the excess radiation force, F_{excess}, making the influence of gravity more important. That will produce an additional force that is outofphase with the velocity of the object as it is moving up from its lowest position, since the force depends upon the previous position of the object at a time, τ, earlier.
When the object reaches its maximum vertical position, the natural frequency of the standing wave resonator will be closer to the drive frequency, and the radiation force will be increased. Again, due to the delay, that excess force will be acting in the upward direction and is again outofphase with the velocity of the object which will be moving downward. This scenario is depicted in Fig. 15.29 (right).
The net effect for the “flat tuning” case has the excess force adding to the viscous resistance and thus increases the damping. The amplitude of the object’s oscillations, if displaced from equilibrium, will decay more quickly than it would if the damping was due only to the Stokes drag. This additional damping causes superstability [72].
This same damping effect is observed in optics where it is known as “optical molasses” and was responsible for Stephen Chu sharing the Nobel Prize in Physics in 1997 with Claude CohenTannoudji and William Phillips “for development of methods to cool and trap atoms with laser light” [73].
5 Beyond the Linear Approximation
Most ordinary acoustical phenomena can be analyzed from the linear perspective that has been the focus of every other chapter of this textbook. Linear acoustics and vibrations provide many useful and convenient simplifications. As we have seen, such simplifications are applicable to a large range of interesting problems. That said, this chapter has introduced a few interesting and useful phenomena that are not contained within a linear analysis. Waveform distortion, harmonic generation, shock wave formation and dissipation, parametric endfire arrays, and mode conversion all rely upon incorporation of effects that a wave has on its propagation medium which are ignored in the linear limit. Inclusion of nonlinear effects leads to an interesting “life cycle” for a large amplitude acoustic disturbance: distortion → shocking → dissipation → classical attenuation [21]. That evolution in an ordinary fluid is depicted symbolically in Fig. 15.30.
By restricting the analysis to onedimensional propagation of plane wave, many of the nonlinear behaviors have been demonstrated while avoiding more complicated mathematics and still being able to appreciate the cumulative influence of convection and of the medium’s own nonlinearity.
The inclusion of nonlinear contributions also provided an introduction to the ability of a sound wave to exert nonzero timeaveraged forces and torques on objects that are exposed to highamplitude sound waves. Acoustic radiation forces are generally much larger than forces that can be exerted by electromagnetic radiation used for trapping atoms [73]. Much of our understanding of these effects can be attributed directly to the Bernoulli pressure that provides an intrinsically secondorder contribution to the linear (firstorder) pressure field. Once again, exploitation of adiabatic invariance provided a means of avoiding complicated mathematics while providing useful quantitative results.
Finally, it is important to recognize that this chapter was only the “tip of the iceberg.” Many important nonlinear acoustical phenomena have not even been mentioned. Among the most significant are thermoacoustic engines, refrigerators [74], pulsetube cryocoolers, and sonic mixture separators [75], as well as other important cases of acoustically driven mass streaming [76]. Another area that has been entirely ignored is nonlinear bubble oscillations that can be so violent that they convert sound into light by a process referred to as “sonoluminescence” [77]. The nonlinear distortion of pulses and the propagation of Nwaves [78], like those which produce a “sonic boom” [79], are other important phenomena also worthy of investigation.
Topics in the area of nonlinear vibrations also abound. As mentioned in the beginning of this textbook, the inclusion of nonHookean elasticity leads to the violation of Galilean isochronous independence of period and amplitude. Much like the harmonic distortion produced in highamplitude wave propagation, a driven nonlinear oscillator will respond at frequencies that are not just the driving frequency. In fact, the response of a nonlinear oscillator can be at subharmonic frequencies or can become entirely chaotic rather than deterministic [80].
The purpose of this chapter was to raise awareness of the limitation of linear analysis, not to create professional expertise in nonlinear acoustics. If the reader can recognize the “symptoms” of nonlinear behavior and understand how they arise, then the goals of this final chapter will have been realized.
Talk Like an Acoustician
Convective nonlinearity  Phase matching 
Selfinteraction  Resonant mode conversion 
Intermodulation distortion  Pump waves 
Shock inception distance  Primary waves 
Grüneisen parameter  Bernoulli pressure 
Virial expansion  Radiation pressure 
Second sound  Kundt’s tube 
Gol’dberg number  Internal energy 
Order expansion  Enthalpy 
Blackstock bridging function  Velocity potential 
Geometric resonance  Acoustic levitation 
Intermodulation distortion  Detuning/dephasing instability 
Exercises

1.
Shock inception distance. The derivation of Eq. (15.8) used the fact that the crest of a plane sinusoidal wave advances “by one radian length,” k^{−1}, toward the zerocrossing when the slope of the zerocrossing, dv/dx, becomes infinite. By using the excess velocity, Γv, defined in Eq. (15.9), show this is true in the case where dissipation can be neglected, with an initial waveform, \( v\left(x,t\right)=\left\hat{\mathbf{v}}\right\sin \left(\omega t kx\right) \). For the waveform to become the fully developed sawtooth shock, the crest of the initially sinusoidal wave must advance by λ/4, placing the crest directly over the zerocrossing (see Fig. 15.6). Express that distance, D_{saw}, in terms of D_{S}, again for the case where dissipation can be neglected.

2.
Waveform distortion. A 19.2 m long waveguide of circular crosssection with inside diameter, D = 5.21 cm, is shown in Fig. 15.31 and Fig. 15.32 (center). The waveguide is driven by two compression drivers, shown in Fig. 15.32 (left), which can produce large amplitude sound waves.^{Footnote 14} The waveguide is terminated by a porous anechoic cone.
Three ¼” microphones are flushmounted at three locations using the fixture that joins smoothly to the PVC pipe to eliminate reflections, shown in Fig. 15.32 (right). One microphone is located very close to the drivers at a position designated x = 0. The second microphone is located at x = 3.17 m, and the third is located at x = 17.9 m.
Assume that the waveguide contains dry air at p_{m} = 100 kPa with a sound speed, c_{o} = 345 m/s, and it is driven sinusoidally at f_{1} = 880 Hz.

(a)
Attenuation length. Using the expression in Eq. (13.78), determine the exponential thermoviscous attenuation length,\( \mathrm{\ell}={\alpha}_{TV}^{1} \), due to boundary layer dissipation at the fundamental frequency, f_{1}. Is that attenuation length shorter or longer for the higher harmonics?

(b)
Shock inception distance and Gol’dberg number. Determine the shock inception distance, D_{S}, and using the result of part (a), determine the Gol’dberg number, G, if p_{1}(0) = 100 Pa (131 dB re: 20 μPa_{rms}), p_{1}(0) = 300 Pa (140.5 dB re: 20 μPa_{rms}), and p_{1}(0) = 1000 Pa (151 dB re: 20 μPa_{rms}).

(c)
Harmonic distortion. Using Eq. (15.43) and neglecting attenuation, determine the amplitude of the fundamental, f_{1} = 880 Hz, second harmonic, f_{2} = 1.76 kHz, third harmonic, f_{3} = 2.64 kHz, and forth harmonic, f_{4} = 3.52 kHz, at x = 3.17 m and at x = 17.9 m, assuming p_{1}(0) = 100 Pa.

(d)
More harmonic distortion. Repeat part (c) assuming p_{1}(0) = 300 Pa.

3.
Repeated shock. Determine the ratio of the amplitudes of the harmonics to the amplitude of the fundamental, C_{n}/C_{1}, for a fully developed shock wavelike that shown in Fig. 15.8.

4.
Levitation demonstration resonator. A groundbased levitator (i.e., g = 9.8 m/s^{2}) is being designed to demonstrate acoustic levitation by levitating the bottoms of Styrofoam coffee cups. Those disks have a diameter of 5.0 cm and a thickness of 1.5 mm, and each has a mass, m_{disk} = 0.15 gm. Assume that resonator will be constructed from a 1.5 m long, 6″ (nominal) diameter, optically clear cast acrylic tube with inside diameter, D_{tube} = 14.0 cm.

(a)
Levitation force. If the resonator is operated in its n = 3 standing wave mode, f_{3} = 350 Hz. Determine the pressure amplitude of the standing wave at the rigid end of the resonator so that the levitation force on the disk is three times its weight.

(b)
Equilibrium location. If the tube is oriented so that the speaker is at the bottom and the rigid end is at the top (like those in Fig. 15.29), how far from the top end of the resonator will the disk be levitated at its highest stable location if the standing wave amplitude is that calculated in part (a)?

(c)
DeltaEC model. Make a DeltaEC model of the resonator (without the levitated disk) to determine the volume velocity of a piston that has the same diameter as the tube, D_{tube}, which would be required to produce the standing wave pressure amplitude at the rigid end calculated in part (a) for the n = 3 mode. You may make a slight adjustment of the tube’s length to force f_{3} = 350 Hz. What are the frequencies of the f_{1}, f_{2}, and f_{4} modes?

(d)
Adiabatic invariance. Use your DeltaEC model in part (a) to estimate the frequency as a function of disk position by moving a constricted DUCT segment that is the same length as the disk (1.5 mm) and has a crosssectional area equal to that of the empty tube minus the crosssectional area of the disk. Move that constricted section from the rigid end to about 0.3 m from the driven end of the resonator. Plot f_{3} vs. position to produce a graph similar to Fig. 15.27. Repeat for f_{2} vs. position.

(e)
Advanced DeltaEC model. Repeat part (c) but explicitly includes the loudspeaker in Fig. 2.43 using the following speaker parameters: m_{o} = 12.0 gm, Κ = 1440 N/m, Bℓ = 7.1 N/A, R_{dc} = 5.2 Ω, L = 0.1 mH, R_{m} = 1.9 kg/s, and A_{pist} = 98.5 cm^{2}. The rear of the speaker is enclosed (to protect your hearing!) in a cylindrical enclosure that has an inside diameter of 6″ (15.2 cm) and a length of 8″ (20 cm). What is the electrical current that must be supplied to the voice coil to produce the n = 3 standing wave amplitude calculated in part (a) at f_{3} = 350 Hz? What are the frequencies of the f_{1}, f_{2}, and f_{4} modes of the coupled speakerresonator system (see Sect. 10.7.5)?

(a)

Hints: The DeltaEC model of the bassreflex loudspeaker enclosure in Fig. 8.41 might provide a helpful starting point. An “enclosed current driven speaker” segment, IESPEAKER, will provide a way to incorporate the rear enclosure with the electrodynamic speaker’s excitation of standing waves in the tube.

(5)
Rayleigh disk. The apparatus in Fig. 15.33 shows a rigid disk (e) suspended at the midplane of a cylindrical resonator from a torsion fiber (b). The resonator has an electrodynamic dome tweeter (g) at one end and an electret microphone (see Sect. 6.3.3) providing a rigid termination at the other end (f). The disk’s angular orientation is detected with the coils surrounding the resonator that incorporates a splitsecondary astatic transformer [60]. A gearing system (a) and a coil (d) and magnet structure (n and s) from an analog meter movement can be used to adjust the equilibrium orientation, θ_{o}, of the disk or excite a freedecay oscillation. The maximum occlusion of the resonator occurs when θ_{o} = 0°.
The resonator’s inside diameter is 3.0 cm and its length, L = 12.0 cm. The diameter of the disk is D_{disk} = 1.2 cm. The disk has a mass, m_{1} = 0.80 gm and a moment of inertia of about its diameter of I_{disk} = 2.0 x 10^{−8} kgm^{2}

(5)
Assume the resonator contains dry air at 300 K with p_{m} = 100 kPa.

(a)
Fundamental resonance frequency. What is the frequency of the fundamental halfwavelength mode of the resonator?

(b)
Torsional stiffness. If the frequency of disk oscillations is 1.1 Hz, what is the torsional stiffness of the disk’s suspension?

(c)
Standing wave pressure amplitude. The disk’s equilibrium position is adjusted so that θ_{o} = 45°. What is the acoustic pressure amplitude, p_{1}, at the surface of the electret microphone if the standing wave causes the disk equilibrium orientation to be θ = 35° and the corrections in the curly brackets of Eq. (15.79) are ignored?

(d)
Scattering corrections. How large is the correction provided by Eq. (15.79) relative to the simpler expression for the torque in Eq. (15.78)? Express your result as N_{antinode} (35°)/N (35°).

(e)
Electret microphone sensitivity. If the opencircuit output voltage of the electret microphone is 285 mV_{ac} under the conditions of part (b), what is the microphone’s opencircuit sensitivity?
Notes
 1.
Although instability requires nonlinearity, nonlinearity does not necessarily always result in instability. Solitons are waveforms that remain stable due to the compensatory influences of nonlinearity and dispersion.
 2.
The exact result for the propagation speed at all depths reduces to c_{grav} in Eq. (15.4) in the limit that kh_{o} → 0. Since this result depends upon k, it is dispersive, so the phase speed, c_{grav}, will not be equal to the group speed except in the “shallow water” kh_{o} → 0 limit.
$$ {c}_{grav}=\frac{\omega }{k}=\sqrt{\frac{g}{k}\tanh {kh}_o} $$  3.
An excellent history of the early development of nonlinear acoustics is provided by D. T. Blackstock, “History of Nonlinear Acoustics: 1750s–1930s,” as Chap. 1 in Nonlinear Acoustics, 2nd ed. (Acoust. Soc. Am., 2008), M. F. Hamilton and D. T. Blackstock, editors; ISBN 0–9,744,067–59.
 4.
If the elastic potential of a solid depended on only the parabolic potential energy contribution (see Sect. 1.2.1), then as a solid heated up, the amplitude of the motion of the point particles (molecules) would increase, but their equilibrium position would remain unchanged. If there is a cubic contribution to the interparticle potential energy, then as the amplitude of the molecular vibrations increased (with increasing temperature), the equilibrium position would shift causing thermal expansion or contraction of the solid.
 5.
In 1962, Lev Landau won the Nobel Prize in Physics for his prediction of the temperature dependence of second sound using his twofluid theory of superfluid hydrodynamics after the speed of second sound was first measured by Pyotr Kapitza. Kapitza won the Nobel Prize in Physics in 1978 for his measurement of the speed of second sound in superfluid helium.
 6.
Prof. I. Rudnick has pointed out that superfluids are interesting because they obey the laws of hydrodynamics on the microscopic scale and obey the laws of quantum mechanics on the macroscopic scale.
 7.
Do not confuse the wave equation for the secondorder deviations from equilibrium with the fact that both the first and secondorder wave equations are both secondorder partial differential equations. For the classification of differential equations, secondorder refers to the highestorder derivative that appears in the equation.
 8.
Westervelt first presented his parametric array at a meeting of the Acoustical Society of America in Providence, RI, in 1960, J. Acoust. Soc. Am. 32, 934 (1960). The abstract for that presentation included an expression for the radiated intensity of the differencefrequency beam.
 9.
Daniel Bernoulli (1700–1782) was a Dutch physicist and mathematician who published Hydrodynamica in 1738 that provided the basis of the kinetic theory of gases which he applied to explain Boyle’s law. He was also well known for early development of elasticity theory with Leonard Euler, an effort recognized to this day by the fact that Eq. (5.36) is called EulerBernoulli beam equation.
 10.
The motion of a small particle in a sound field will depend upon the competition between the particle’s inertia (mass), which tends to make it remain stationary in the laboratory frame of reference, and the Stokes drag due to the viscosity of the medium which tends to force the particle to move along with the acoustically oscillating fluid. The inertial force is given by Newton’s Second Law, F_{inertia} = m (dv_{1}/dt), and the Stokes drag force on a spherical particle of radius, a, (at sufficiently low Reynolds number) is F_{drag} = 6πμav_{1}. Their dimensionless ratio will determine if the particle moves with the fluid or if the fluid moves around the particle. That ratio can be written for a spherical particle with mass density, ρ, and sound with frequency, f.
$$ \frac{F_{inertia}}{F_{drag}}=\frac{4\pi }{9}\frac{a^2 f\rho}{\mu } $$For a particle with the density of water (ρ = 10^{3} kg/m^{3}), in air with μ ≅ 1.8 x 10^{−5} Pas, and then at 100 Hz, that ratio is one for a spherical particle with a radius of about 10 microns. A larger radius particle, like cork dust, coffee whitener, or a seed, will remain nearly stationary in the laboratory frame, and the fluid will oscillate around it, while a much smaller particle, like smoke, will move with the fluid.
 11.
Video demonstrations of several of the nonzero, timeaveraged effects in this section were recorded at the 100^{th} meeting of the Acoustical Society of America held in Los Angeles, CA, in 1988. This video is included in the second disk of the Collected Works of Distinguished Acousticians—Isadore Rudnick, compiled by J. D. Maynard and S. L. Garrett (Acoust. Soc. Am., 2011); https://www.abdiecommerce10.com/ASA/p230collectedworksofdistinguishedacousticians.aspx.
 12.
This assumption is not as bad as it seems since the Bernoulli pressure, as described in Eq. (15.76), is only valid along a streamline. The streamlines in Fig. 15.20 (right) will follow the contours of the disk accounting for the fact that simple results of Eqs. (15.77) and (15.78) are very nearly the correct result.
 13.
 14.
The use of two drivers not only increases the achievable amplitudes but also facilitates measurements of the interaction of two waves of different frequencies.
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Garrett, S.L. (2020). Nonlinear Acoustics. In: Understanding Acoustics. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/9783030447878_15
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