The solution of the Helmholtz equation which satisfies the energy conservation principle is the limit for t → ∞ of P ω( M t). Let us consider the unique solution P ω( Mt t t) of the wave equation which corresponds to a harmonic source with angular frequency ω starting at a time t = 0. For a real wavenumber k, the solution which satisfies the energy conservation principle is the limit, for ε → 0, of p ε. Let ε be a positive parameter and p ε be the unique bounded solution corresponding to a wavenumber k ε = k(1 + ɩε). To ensure the uniqueness of the solution for any real frequency when the domain Ω is unbounded, the principle of energy conservation must be respected: this is done by adding a Sommerfeld condition at infinity (which has already been given) or by using either the principle of limit absorption or the principle of limit amplitude. In what follows, it is assumed that α and β can be piecewise continuous functions: this allows the boundary σ to be made of different materials (for example, one part is rather hard, another one being highly absorbent). These remarks justify the necessity to use the symbol ‘Tr’ as a reminder of the mathematical requirements which must be respected to get a description of the physical phenomena. The consequences are: (a) the value on the boundary of the function which represents the sound pressure inside the propagation domain can be different from the value of the sound pressure (b) the value of the pressure field (or of its gradient) on the boundary of the propagation domain is obtained as a limit of a function defined inside the propagation domain (c) the limits on the boundary of the functions representing the pressure field and its gradient need to be (at least locally) square integrable functions. Furthermore, an energy flux density across any elementary surface of the propagation domain boundary must be defined. In acoustics, this implies that the sound pressure and its gradient are described by functions which are (at least locally) square integrable. The basic hypothesis which is always made in physics is that any system has a finite energy density and its behaviour is based on the Hamilton principle (energy conservation). Some remarks must be made on the mathematical requirements on which mathematical physics is based. The possible existence of a solution of this boundary value problem is beyond the scope of this book among the best textbooks dealing with boundary value problems of classical physics and in which such a proof is given, we can mention Methods of Mathematical Physics by R. But, as has been seen already, the sound field can be represented by surface integrals which are discontinuous or have a discontinuous normal gradient or involve, on σ, a finite part of the integral: in these cases, the symbol ‘Tr’ is absolutely necessary to indicate that the value on σ of the function is obtained as the limit of a Riemann integral. Very often, the symbol ‘Tr’ can be omitted.
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