Sound radiation by a moving line source above an impedance plane with frequency-dependent properties

An analytic solution for the problem of sound radiation by a harmonic line source moving at a uniform subsonic speed parallel to an impedance plane is proposed. The main originality of this work is that the variation of the impedance with the frequency is taken into account. Compared to the case of a constant impedance, the reflection coefficient and the location of its poles in the complex plane are modified. A uniform asymptotic expression is then developed for moderate Mach numbers and a closed-form expression, corresponding to a Weyl-Van der Pol formula, is proposed for a grazing incidence for hard grounds and for low Mach numbers. Unlike previous analytical solutions derived in the literature for a point-source, the impedance is evaluated at the Doppler frequency instead of at the source frequency. The analytical solution and asymptotic expressions are then compared satisfactorily to a numerical solution obtained from a time-domain solver of the linearized Euler equations. Finally, a parametric study is carried out showing that the assumption of a constant impedance is valid if the source Mach number remains small, typically less than 0.2, and if the source is not too close to the ground.