On Transient Solutions of the "Baffled Piston" Problem

The case of the time harmoni c movement of a piston membrane in an infinite rigid wall (" baffle" ) can easily be generalized to the case ,v"hen the motion of the pis ton is not periodic but an arbi trary function of time. Such trans ien t solution ha ve become of considerable i nterest in recent times (for a detailed treatment of the propagation of such sound pulses, see [5)2). The procedure for the case treated here is the same as used elsewhere [7], i .e., t he Green's funct ion for the exponential decay case (modified wave equa tion /::" U-,,/2U= O, "'/ = ilc) is uscd to obtain the solu tion for the pulse problem . The aceustic field (velocit? po ten t ial) for the time harmonic movement of the piston includ e representations given by Bouwkamp [1], Kin g [6], and Wells and Leitne r [9J . The first of t hese contributions gives the solu tion in the form of a series expression while the second and third in volve in tegral reprcsentations that are obtained using integral transform methods (Hankcl transform [4 , p . 73J and Lebedev transform [4, p. 75J respectively). These l'eprescn taLions can be used Lo t l'Cllt Lhe ge ll eral case of an arbitrary movement of the p iston . In view of the method to be cmployed here, sueh J'cprcsen tations should be used for which the inverse Laplace transform of Lhe veloci ty poten tial with respect to the purely imaginary wave parameter "/ = ilc can be given. Such an expression can be obtained in a direct way by regardin g each point of the moving disk as an aecustic point source and in tegrating over all points of the disk .