Doppler effect of a planar vibrating piston: strong solution, series expansion and simulation
Résumé
This article addresses the Doppler effect of a planar vibrating piston
in a duct, as a plane wave radiation approximation generated by a
loudspeaker membrane.
This physical model corresponds to a nonlinear problem, because the
linear propagation is excited by a moving boundary condition at the
piston face: this introduces a varying propagation time between the piston
and a fixed receiver.
The existence of a regular function that solves the problem (a
so-called ``strong'' solution) is proven, under a well-posed
condition that guarantees that no shock occurs.
This function satisfies an implicit equation to be solved.
An algorithm based on the perturbation method is proposed, from which
an exact solution can be built using power series.
The convergence of the power series is numerically checked on
several examples.
Simulations derived from a truncated power series provide sound
examples with audible intermodulation and distortion effects for
realistic loudspeaker excursion and speed ranges.