The sound field caused by a monopole source above an impedance plane can be calculated by using a superposition of equivalent point sources located along a line in the mirror space below the plane. Originally, such an approach for representing the half-space Green's function was described by Sommerfeld at the beginning of the last century, in order to treat half-space problems of heat conduction. However, the representation converges only for masslike impedances and cannot be used for the more important case of reflecting planes with springlike surface impedances. The singular part of the line integral can be transformed into a Hankel function, which shows that surface waves are contained in the whole solution. Unfortunately, this representation suffers from the lack of validity at certain receiver points and from restrictions on wave number and impedance range to ensure the necessary convergence. The main idea of the present method is to use also a superposition of equivalent point sources, but to allow that these sources can be located at complex source points. The corresponding form of the half-space Green's function is suitable for both masslike and springlike surface impedances, and can be used as a cornerstone for a boundary element method.

Fractional derivatives of arbitrary order are incorporated into a model nonlinear equation in order to simulate the absorption mechanism in a finite frequency domain. That makes it possible to take into account a power dependence of the absorption coefficient on the frequency at any real degree. The model gives a mathematical description of a large set of well-known absorption mechanisms. The method is applied to calculate weakly nonlinear oscillations in a nonrigid tube. As a result, the value of the absorption coefficient and its dependence on the frequency are found based on experimental data.

The application of the averaging method to the one-dimensional inhomogeneous, nonlinear acoustic wave equation with dissipative term makes it possible to give asymptotic solutions for any kind of external resonance excitation. It shows that the lowest-order solution consists of the superposition of two modulated counterpropagating waves, where the amplitude of each is a solution of Burgers equation. The method is extended to the treatment of oscillating boundaries; in that case it also leads to Burgers equations. Explicit stationary solutions are given for the particularly important forms of the external excitation, harmonic distributed forces, and harmonic oscillating boundaries. The application of several other computational methods to this problem leads to the same results.

The transient sound field caused by a Dirac delta impulse function above an infinite locally reacting plane can be calculated by applying the inverse Fourier transform of the corresponding half-space Green's function in frequency domain. As a starting point, the representation given by Ochmann [J. Acoust. Soc. Am. 116(6), 3304-3311 (2004)] is used, which consists of discrete and continuous superposition of point sources. For a locally reacting plane with masslike character and also with pure absorbing behavior, it is possible to express the resulting impulse response in closed form. Such a result is surprising, because corresponding formulations in the frequency domain are not available yet. Hence, the first main result is the closed form solution Eq. (22) for an impulse response over an infinite plane with a pure imaginary impedance. The second main result is the closed form solution Eq. (53) for an impulse response over an infinite plane with a pure real impedance. As a particular application of both main results, a convolution technique is used for deriving formulas for point sources with a general time dependency. For special signals like an exponentially decaying time signal or a triangular shaped impulse, the resulting sound field can be presented in terms of elementary functions.

No abstract