Abstract
Abstract In this chapter, solutions to the wave equation that satisfies the boundary conditions within three-dimensional enclosures of different shapes are derived. This treatment is very similar to the two-dimensional solutions for waves on a membrane of Chap. 10.1007/978-3-030-44787-8_6. Many of the concepts introduced in Sect. 10.1007/978-3-030-44787-8_6#Sec1 for rectangular membranes and Sect. 10.1007/978-3-030-44787-8_6#Sec5 for circular membranes are repeated here with only slight modifications. These concepts include separation of variables, normal modes, modal degeneracy, and density of modes, as well as adiabatic invariance and the splitting of degenerate modes by perturbations. Throughout this chapter, familiarity with the results of Chap. 10.1007/978-3-030-44787-8_6 will be assumed. The similarities between the standing-wave solutions within enclosures of different shapes are stressed. At high enough frequencies, where the individual modes overlap, statistical energy analysis will be introduced to describe the diffuse (reverberant) sound field.
Highlights
The formalism developed for three-dimensional enclosures provides the description for sound propagation in waveguides, since a waveguide can be treated as a three-dimensional enclosure where one of the dimensions is extended to infinity
The linearized wave equation for the acoustic pressure, p, can be written in a vector form that is independent of any particular coordinate system
Since k 1⁄4 ω /c, Eq (13.1) can be written in the time-independent form known as the Helmholtz equation
Summary
The linearized wave equation for the acoustic pressure, p, can be written in a vector form that is independent of any particular coordinate system. Since each term in the separated Helmholtz equation (13.5) depends upon a different coordinate, and their sum is equal to a constant, Àk2, each term must be separately equal to a constant This is the same as the “separation condition” imposed in the two-dimensional case in Eq (6.8). Each term generates a simple harmonic oscillator equation By this time, we are quite familiar with the solutions to the above ordinary, second-order, homogeneous differential equation. P1ðx, y, z, tÞ 1⁄4 ReÂbp cos ðkxx þ φxÞ cos Àkyy þ φyÁ cos Àkzz þ φzÁejω tà ð13:8Þ To emphasize that this is could be a traveling plane wave (before imposition of boundary conditions), the solution can be written as a product of complex exponentials
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