Acoustic Waves

Abstract

Section 1.4 introduced the basics of sound propagation in isotropic media, and some elementary properties of compressional wave reflection and transmission. An important aspect noted there was the possibility of zero reflection at a sharp boundary between two media at the Green’s angle, the acoustic analogue of the Brewster angle.

Appendix: Universal Properties of Acoustic Pulses and Beams

The results of this Appendix are restricted to pulse and beam propagation within isotropic homogeneous media, within which the acoustic pressure satisfies the wave equation ((17.1) of this chapter with $\nabla \rho = 0$). We also neglect dissipation of energy or momentum, due either to viscous damping, or to scattering by impurities or bubbles.

We begin with a summary of the existing known universal properties of localized acoustic pulses , namely (i) the time invariance of the total energy $E$, momentum $P_{z}$ and angular momentum $J_{z}$ of the pulse, and (ii) the inequality $cP_{z} \, < \,E$. (In this Appendix pulse propagation is along the $z$-direction, and the speed of sound is $c$, a constant.)

The conservation of energy, momentum and angular momentum in the absence of dissipation is as expected, but the inequality $cP_{z} \, < \,E$ is in contrast to the sound quantum ‘phonon’, for which the momentum is unidirectional, and $cP = E$. The implication of the inequality $cP_{z} \, < \,E$ seems to be that we cannot model the phonon by any localized pulse wave-function satisfying the wave equation.

Peierls (1983) considered the energy and momentum of localized sound pulses. However, in calculating the energy and momentum, Peierls made approximations that removed the transverse localization, and in the long-wave limit his equation (3.12) gives equality of energy and $c$ times momentum. Lekner (2006a) examined the energy and momentum densities $e\left( {\mathbf{r},t} \right), \mathbf{p}(\mathbf{r},t)$ of a three-dimensionally localized sound pulse, and the total energy and total momentum

$$ E = \int\limits {\text{d}}^{3} r\;{\text{e}}\left( {\mathbf{r},t} \right),\quad \mathbf{P} = \int\limits {\text{d}}^{3} r\; \mathbf{p}\left( {\mathbf{r},t} \right) . $$

(17.146)

He showed that $E$ and $\mathbf{P}$ are independent of time (are conserved, as expected), and further, that $cP_{z} \, < \,E$ for predominant propagation in the $z$-direction. Localized pulses are always converging to or diverging from their focal region, hence there is a transverse momentum density, integrating to zero. This is the reason for the inequality, and the prime distinction between pulses and phonons. A consequence of this universal convergence toward or divergence from the focal region is that the pulse pressure gradient, density gradient and particle velocity are not purely longitudinal, as they are for pulses localized in only one dimension. Exact solutions of the wave equation are used to construct specific examples of localized pulses in Lekner (2006b).

We may expect that the total angular momentum of a localized sound pulse should also be constant in time, again in the absence of dissipation. This is indeed true: the angular momentum density $\mathbf{j} = \mathbf{r} \times \mathbf{p}$ integrates to give the total angular momentum

$$ \mathbf{J} = \int\limits {\text{d}}^{3} r\; \mathbf{j}\left( {\mathbf{r},t} \right) = \int\limits {\text{d}}^{3} r\; \mathbf{r} \times \mathbf{p} \left( {\mathbf{r},t} \right) . $$

(17.147)

In the translation of the coordinate system, $\mathbf{r} \to \mathbf{r} - \mathbf{a}$, the total angular momentum transforms to $\mathbf{J} \to \mathbf{J} - \mathbf{a} \times \mathbf{P}$. Thus the component of $\mathbf{J}$ parallel to $\mathbf{P}$ is invariant to the choice of origin. As before, we take direction of $\mathbf{P}$ to define the $z$-axis; then $J_{z}$ is the invariant component of interest. Lekner (2006c) has shown that the time derivative of $J_{z}$ is zero, and gives examples of exact localized solutions of the wave equation, with analytic expressions for the energy, momentum, and angular momentum.

There are analogous conserved quantities for acoustic beams , again in the absence of dissipation. These follow from the conservation of matter, momentum and angular momentum:

$$ \partial_{t} \rho + \nabla \cdot \left( {\rho \mathbf{v}} \right) = 0 , $$

(17.148)

$$ \partial_{t} \left( {\rho v_{i} } \right) + \mathop \sum \limits_{k} \partial_{k}\Pi _{ki} = 0 \quad (i,k = x,y,z) , $$

(17.149)

$$ \partial_{t} j_{i} + \mathop \sum \limits_{k} \partial_{k}\Lambda _{ki} = 0\quad(i,k = x,y,z) . $$

(17.150)

The tensors $\Pi ,\Lambda$ are the momentum flux density and angular momentum flux density tensors (Lekner 2007; Zhang and Marston 2011). Equation (17.148) may be written as $\partial_{t} \rho + \nabla \cdot \mathbf{p} = 0$, with $\rho$ the mass density and $\mathbf{p} = \rho \mathbf{v}$ the momentum density.

For monochromatic acoustic beams of angular frequency $\omega$ the motion everywhere within the sound beam is periodic with period $T = 2\pi /\omega$, and the cycle average of (17.148) gives $\nabla \cdot \overline{\mathbf{p}} = 0$, where the bar denotes average over one or any number of periods, at a fixed point in space. Suppose that the acoustic beam is propagating in the $z$ direction. Integration of $\nabla \cdot \overline{\mathbf{p}} = 0$ over the transverse directions $x$ and $y$ then gives (Lekner 2007)

$$ \partial_{z} \int\limits {\text{d}}^{2} r\; \overline{{p_{z} }} = \partial_{z} P_{z}^{\prime} = 0\quad ({\text{d}}^{2} r = \text{d}x\text{d}y) . $$

(17.151)

The meaning of (17.151) is that the longitudinal cycle-averaged momentum content within a transverse slice of the beam is constant along the beam: $P_{z}^{\prime} = \int\limits {\text{d}}^{2} r\;\; \overline{{p_{z} }}$ is an invariant. Note that it was conservation of matter which led to the momentum content beam invariant, not conservation of momentum.

Each component of the conservation of momentum, equation (17.149), leads to another invariant on cycle averaging. The conservation of angular momentum leads to three more. Thus the conservation laws give seven universal beam invariants , just as in the electromagnetic case (Sect. 20.1). Perhaps surprisingly, the cycle-averaged energy content in a transverse slice of the beam, $E^{\prime} = \int\limits {\text{d}}^{2} r\; \overline{\text{e}}$, is not an invariant in general. Neither is the angular momentum content, but both are constant along the length of the beam for a special class of generalized Bessel acoustic beams (Lekner 2006d). For generalized Bessel beams one finds $cP_{z}^{\prime} \le E^{\prime}$, and, for beams with azimuthal dependence ${\text{e}}^{{{\textit{im}}\phi }}$, a proportionality between the energy and angular momentum contents per unit length of the beam. Beams with azimuthal dependence ${\text{e}}^{{{\textit{im}}\phi }}$ are termed acoustic vortex beams by Zhang and Marston (2011).