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Radiation from Sources Near and on Solid Surfaces

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Acoustics

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Abstract

The present chapter begins with a discussion of the effects of nearby solid surfaces on the radiation of sound and then continues with the closely related topic of radiation from a planar surface when a portion of it is vibrating.

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Notes

  1. 1.

    F. A. Fischer, “Directionality and radiation intensity of acoustic ray groups in the vicinity of a reflecting plane surface,” Elektr. Nachrichtentech. 10:19–24 (1933).

  2. 2.

    U. Ingard and G. Lamb, Jr., “Effect of a reflecting plane on the power output of sound sources,” J. Acoust. Soc. Am. 29:743–744 (1957).

  3. 3.

    J. W. S. Rayleigh, The Theory of Sound, vol. 2, 2d ed., 1896, reprinted by Dover, New York, 1945, sec. 278.

  4. 4.

    A. Sommerfeld, “The freely vibrating piston membrane,” Ann. Phys. (5)42:389–420 (1943).

  5. 5.

    M. Born and E. Wolf, Principles of Optics, 4th ed., Pergamon, Oxford, 1970, pp. 378–381. Pertinent original references are A. Fresnel, “On the diffraction of light; examination of the colored fringes existing in the shadow of an illuminated body,” Ann. Chim. Phys. (2) 1:239–281 (1816); G. G. Stokes, “On the dynamical theory of diffraction,” Trans. Camb. Phil. Soc. 9:1 (1849), reprinted in Stokes, Mathematical and Physical Papers, vol. 2, Cambridge University Press, Cambridge, 1883, pp. 243–328; G. Kirchhoff, “On the theory of light rays,” Ann. Phys. Chem. 18:663–695 (1883).

  6. 6.

    H. A. Bethe, “Theory of diffraction by small holes,” Phys. Rev. 66:163–182 (1944); R. D. Spence, “A note on the Kirchhoff approximation in diffraction theory,” J. Acoust. Soc. Am. 21:98–100 (1949).

  7. 7.

    H. Lamb, “On the vibrations of an elastic plate in contact with water,” Proc. R. Soc. Lond. A98:205–216 (1920). A general result holding for arbitrary ka was later derived by N. W. McLachlan, “The acoustic and inertia pressure at any point on a vibrating circular disk,” Phil. Mag. (7)14:1012–1025 (1932).

  8. 8.

    L. M. Milne-Thomson, “Elliptical Integrals,” in M. Abramowitz and I. Stegun (eds.), Handbook of Mathematical Functions, Dover, New York, 1965, pp. 590–592, 608–611.

  9. 9.

    J. W. S. Rayleigh, “On the theory of resonance,” Phil. Trans. R. Soc. Lond. 161:77–118 (1870).

  10. 10.

    Rayleigh, The Theory of Sound, vol. 2, sec. 302.

  11. 11.

    The Bessel function J n(η) and the Struve function H n(η) for positive integer order n can be considered to be defined by the integrals

    $$\displaystyle \begin{aligned} \left\{\begin{array}{l} J_{n}(\eta)\\ {\mathbf{H}}_{n}(\eta) \end{array}\right\}=\frac{2(2n+1)\eta^{n}}{[(2n+1)(2n-1)\cdots 3\cdot 1]\pi} \int\nolimits_{0}^{\pi/2}\left\{\begin{array}{ll} \cos & \\ & (\eta\,\cos \phi)\\ \sin & \end{array}\right\}(\sin \phi)^{2n}\,d\phi \end{aligned}$$

    For a full discussion, see G. N. Watson, A Treatise on the Theory of Bessel Functions, 2d ed., Cambridge University Press, London, 1966, pp. 24–25, 328–338. The expression for J n(η) is known as Poisson’s integral for the Bessel function. The boldface symbol H n(η) for the Struve function is traditional and should not be construed as denoting a vector.

  12. 12.

    For the Struve functions, the identity (7b) follows from

    $$\displaystyle \begin{aligned} 1 &= \int\nolimits_{0}^{\pi/2} \frac{\partial}{\partial\phi}[\sin \phi\, \cos (\eta\,\cos \phi)]\,d\phi\\ &=\int\nolimits_{0}^{\pi/2}\left\{\frac{\partial}{\partial\eta}[\sin (\eta\,\cos \phi)]+\eta\,\sin^{2}\,\phi\,\sin (\eta\, \cos \phi)\right\}\,d\phi \end{aligned} $$
    (i)
    $$\displaystyle \begin{aligned} &=\int\nolimits_{0}^{\pi/2}\left\{\frac{\partial}{\partial\eta}\left[\eta\frac{\partial}{\partial\eta}\sin (\eta\,\cos \phi)\right]+\eta\,\sin (\eta\,\cos \phi)\right\}\,d\phi \end{aligned} $$
    (ii)

    Equation (i) leads to 1 = (π∕2)(d H 0∕dη + H 1), while (ii) leads to 1 = (π∕2)[(d∕dη)(η d H 0∕dη) + η H 0]. Since η d H 0∕dη = 0 at η = 0, the integral from 0 to η of the latter yields η = (π∕2)(η d H 0∕dη + L), where L is the left side of (7b). The derivation of (7a) for the Bessel functions proceeds in an analogous manner from

    $$\displaystyle \begin{aligned} 0=\eta\int\nolimits_{0}^{\pi/2}\frac{\partial}{\partial\phi}[\sin \phi\,\cos \phi\, \cos (\eta\, \cos \phi)]\,d\phi \end{aligned}$$
  13. 13.

    To derive the asymptotic expression for H 1(η), we write the integrand in Eq. (6) for H 0(η) as the real part of \(i\,\exp \,(-i\eta \,\cos \phi )\) and interchange the order of taking the real part and of integrating. The integration path is then deformed to one going from 0 to π∕2 + i ∞ plus one going from π∕2 + i ∞ to π∕2. For the first segment, the variable of integration is changed to s, so that \(\cos \phi = 1 - is^2\) and s goes from 0 to + ∞ along the path. In the second segment, one lets ξ = Im ϕ be the integration variable. Doing all this yields

    $$\displaystyle \begin{aligned} {\mathbf{H}}_{0}(\eta)=\left(\frac{2}{\pi}\right)2^{1/2}\,\mathrm{Re}\, \left[e^{-i(\eta - 3\pi/4)} \int\nolimits_{0}^{\infty}\frac{e^{-\eta s^2}\,ds}{(1-is^{2}/2)^{1/2}}\right]+\frac{2}{\pi}\int\nolimits_{0}^{\infty}e^{-\eta\,\sinh\,\xi}\,d\xi , \end{aligned}$$

    where the phase of the radical is understood to be between 0 and − π∕4. For large η one can approximate (1 − is 2∕2)1∕2 by 1 and \(\sinh \, \xi \) by ξ without appreciably changing the value of either integral, the resulting approximate integrals being then readily performed, so one obtains

    $$\displaystyle \begin{aligned} {\mathbf{H}}_{0}(\eta)\rightarrow\frac{2}{\pi\eta}+\left(\frac{2}{\pi\eta}\right)^{1/2}\,\cos\left(\eta-\frac{3\pi}{4}\right). \end{aligned}$$

    From (7b), one has H 1(η) = (2∕π) − d H 0∕dη; using the above and keeping only terms of order η −1∕2, we obtain (12b). The derivation of (12a) proceeds in an analogous manner from Eq. (6) except that one takes the imaginary part of \(i\,\exp \, (-i\eta \, \cos \phi )\). The asymptotic expression for J 1(η) is obtained from that of J 0(η) with the identity J 1(η) = −dJ 0(η)∕dη.

  14. 14.

    In the analogous Fresnel–Kirchhoff theory of diffraction by an aperture (Sect. 5.2), the diffraction is said to be Fraunhofer diffraction when the R in e ikR can be replaced by r −x S ⋅ e r. Points at which this approximation is satisfactory are said to lie in the Fraunhofer region. Similarly the terms Fresnel diffraction and Fresnel region are used when the quadratic terms (but not the higher-order terms) in the expression \(R\approx r -{\mathbf {x}}_{S}\boldsymbol {\cdot } {\mathbf {e}}_{r} + \tfrac {1}{2}[x_{S}^{2}+y_{S}^{2}-({\mathbf {x}}_{S}\boldsymbol {\cdot } {\mathbf {e}}_{r})^{2}]/r\) affect the value of the integral. See Born and Wolf, Principles of Optics, p. 383.

  15. 15.

    R. C. Jones, “On the Theory of the Directional Patterns of Continuous Source Distributions on a Plane Surface,” J. Acoust. Soc. Am., 16:147–171 (1945).

  16. 16.

    N. W. McLachlan, “Pressure distribution in a fluid due to the axial vibration of a rigid disc,” Proc. R. Soc. Lond. A122:604–609 (1928). For Fraunhofer diffraction by a circular aperture, the formula was first derived, although in a somewhat different form, by G. B. Airy, Trans. Camb. Phil. Soc., 5:283 (1835).

  17. 17.

    J. W. Miles, “Transient loading of a baffled piston,” J. Acoust. Soc. Am. 25:200–203 (1953); F. Oberhettinger, “Transient solutions of the baffled piston problem,” J. Res. Nat. Bur. Stand. 65B:1–6 (1961). The derivation in the text is similar to that of P. R. Stepanishen, “Transient radiation from pistons in an infinite planar baffle,” J. Acoust. Soc. Am. 49:1628–1638 (1971).

  18. 18.

    A. Schoch, “Considerations in regard to the sound field of a piston diaphragm,” Akust. Z. 6:318–326 (1941).

  19. 19.

    H. Backhaus and F. Trendelenberg, “On the unidirectional beaming of piston diaphragms,” Z. Tech. Phys. 7:630–635 (1926). The analogous result for diffraction by a circular aperture dates back to Fresnel, “On the diffraction of light …,” 1816, and to A. Schuster, “Elementary treatment of problems on the diffraction of light,” Phil. Mag. (5)31:77–86 (1891). The result is related to Poisson’s famous prediction (originally intended to debunk Fresnel’s theory of diffraction but shortly thereafter experimentally confirmed by Arago) that there should be a bright spot in the shadow of a circular disk along the axis of the disk. If the Fresnel–Kirchhoff integral with e R ⋅e z = n i ⋅ e z = 1 in Eq. (5.2.8) is used with \(\hat {\mathbf {v}}_{i}\cdot {\mathbf {e}}_{z} = \hat {\mathrm {v}}_{n}\) for w S > a, 0 for w S < a, and with a small attenuation factor inserted to make the integral convergent, one obtains (Babinet’s principle) an expression equal to the original incident plane wave minus what would be predicted for the problem of diffraction by a circular aperture of the same size. This difference for points on the symmetry axis, according to Eq. (2), is ρcv n(t − (z 2 + a 2)1∕2∕c), which has exactly the same amplitude as that of the incident acoustic-pressure wave. For a historical account, see E. Mach, The Principles of Physical Optics: An Historical and Philosophical Treatment, 1926, reprinted by Dover, New York, 1954, pp. 285–286.

  20. 20.

    Schoch, “Consideration … ,” 1941.

  21. 21.

    The derivation of this asymptotic expression proceeds as outlined on p. 225n; the result is due to Poisson (1823). For a general derivation that includes higher-order terms, see Watson, Treatise on the Theory of Bessel Functions, pp. 196–198.

  22. 22.

    The limiting case of a →∞, w − a finite and abbreviated by x, corresponds to the case when the x < 0 portion of the plane z = 0 is vibrating with constant amplitude and phase and the x > 0 portion is motionless. This limit applied to (8) gives

    $$\displaystyle \begin{aligned} \frac{\hat{p}}{\rho c\hat{v}_{n}}=H(-x)e^{ikz}-2^{-1/2}A_{D}(X)\exp\ \left\{i\left[k(x^{2}+z^{2})^{1/2}+\frac{\pi}{4}\right]\right\} , \end{aligned}$$
    (i)

    with X = −{k∕[π(x 2 + z 2)1∕2]}1∕2 x. This, with z ≫|x|, reduces to

    $$\displaystyle \begin{aligned} \frac{\hat{p}}{\rho c\hat{v}_{n}}=e^{ikz}[H(-x)-2^{-1/2}e^{i\pi/4}A_{D}(X)e^{i(\pi /2)X^{2}}]=e^{ikz}2^{-1/2}e^{-i\pi/4}\int\nolimits_{-X}^{\infty}e^{i(\pi/2)t^2}\,dt . \end{aligned}$$
    (ii)

    The mathematical steps leading to (ii) are explained later in the present section. This in the limit considered is the same as the classical result for Fresnel diffraction of a plane wave by a straight edge in the Fresnel–Kirchhoff theory. See Born and Wolf, Principles of Optics, pp. 433–434.

  23. 23.

    So called here because it is a ubiquitous feature of any asymptotic solution of the wave equation when the boundary involves a sharp edge. Born and Wolf, Principles of Optics, p. 428, use the term to refer, with some multiplicative factors, to the integral of e ikR over the aperture.

  24. 24.

    W. Gautschi, “Error function and Fresnel integrals,” in Abramowitz and Stegun (eds.), Handbook of Mathematical Functions, pp. 297–302, 323–324. Note that our A D(X) is (1 − i)∕2 times the w(z) in Gautschi’s eq. (7.1.4) with z = (π∕2)1∕2 Xe iπ∕4, so our (9a), giving iA D(X) = [(1 + i)∕2]w(z) as g(X) = if(X), is consistent with Gautschi’s (7.3.23) and (7.3.24).

  25. 25.

    A. G. Warren, “A note on the acoustic pressure and velocity relations on a circular disc and in a circular orifice,” Proc. Phys. Soc. (Lond.) 40:296–299 (1928). Warren omits all details; an explicit derivation is given by McLachlan, “The acoustic and inertia pressure … ,” Phil. Mag., (7)14:1012–1025 (1932).

  26. 26.

    Photographs resulting from exposure of a photographic plate to an ultrasonic beam radiating from a baffled piston exhibit such interference rings in a vivid manner. [J. T. Dehn, “Interference patterns in the near field of a circular piston,” J. Acoust. Soc. Am. 32:1692–1696 (1960).]

  27. 27.

    A. Sommerfeld, Optics, Academic, New York, 1950, pp. 218–220; Born and Wolf, Principles of Optics, pp. 371–375; F. W. Sears, Optics, 3rd ed., Addison-Wesley, Reading, Mass., 1949, pp. 245–251.

  28. 28.

    The analysis in the present section is largely due to Schoch, “Considerations … ,” 1941. For a comparable but mathematically dissimilar discussion of the field of a circular plane piston in the ka ≫ 1 limit, see P. H. Rogers and A. O. Williams, Jr., “Acoustic Field of a Circular Plane Piston in Limits of Short Wavelength or Large Radius,” J. Acoust. Soc. Am., 52:865–870 (1972). Some detailed computational results for the intermediate range of ka are displayed by H. Stenzel, Leitfaden zur Berechnung von Schallvorgängen, Springer, Berlin, 1939, pp. 75–79; they are also given by S. N. Rschevkin, A Course of Lectures on the Theory of Sound, Pergamon, Oxford, 1963, pp. 441–443.

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    Allan D. Pierce

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Pierce, A.D. (2019). Radiation from Sources Near and on Solid Surfaces. In: Acoustics. Springer, Cham. https://doi.org/10.1007/978-3-030-11214-1_5

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