Chapter Thirteen

Radiation and scattering of sound by the boundary integral method

Abstract

This chapter starts with definitions of the Huygens–Fresnel Principle; Rayleigh integrals, Green's function; Kirchhoff–Helmholtz boundary integral; and Green's function in different coordinate systems. Then comes a succession of applications: radiation from a pulsating cap in a rigid sphere; reflection of a point source near a plane; radiation of a rigid circular piston in an infinite baffle; radiation from a resilient circular disk without a baffle and with an infinite baffle; radiation from a rigid circular piston in a finite circular open baffle and a finite circular closed baffle. This is followed by a definition of the Babinet–Bouwkamp principle, with applications to several cases. Continuing, we have radiation from an infinitely long oscillating ribbon in an infinite baffle and the far-field pressure distribution as a spatial frequency spectrum of the source velocity distribution. The chapter concludes with a definition of the bridge product theorem, with applications to radiation from a rigid rectangular piston in an infinite baffle and to mutual radiation impedance between circular pistons in an infinite baffle.

Keywords

Bouwkamp impedance theorem; Fourier transform; Kirchhoff–Helmholtz boundary integral; Radiation impedance; Sound field; Spherical coordinates

Part XXXV: Boundary integrals and the Green's function

13.1. The Huygens–Fresnel principles

The Huygens–Fresnel principle states that each point on the wave front of a propagating wave can be replaced with a point source as illustrated in Fig. 13.1, thus creating an array of wavelets whereby each wavelet is unaffected by the presence of all the other wavelets. Some time later, the wave front is equivalent to the envelope of these wavelets. In other words, the resultant field is due to the sum of the point sources, using the principle of superposition. The point sources can be monopoles or dipoles. Although the forward propagating wave remains unaltered, the principle does not explain the creation of a backward propagating wave that was not present in the original. However, if the surface over which the point sources are distributed encloses the original source(s) fully, we can use the principle to analyze the internal or external fields separately. It can also be used to analyze sound radiation from vibrating surfaces, which may or may not be closed, provided that the boundary conditions are correct.
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Figure 13.1 Huygens–Fresnel principle.
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