Periodic solutions and multiharmonic expansions for the Westervelt equation

Abstract

In this paper we consider nonlinear time periodic sound propagation according to the Westervelt equation, which is a classical model of nonlinear acoustics and a second order quasilinear strongly damped wave equation exhibiting potential degeneracy. We prove existence, uniqueness and regularity of solutions with time periodic forcing and time periodic initial-end conditions, on a bounded domain with absorbing boundary conditions. In order to mathematically recover the physical phenomenon of higher harmonics, we expand the solution as a superposition of contributions at frequencies that are multiples of a fundamental excitation frequency. This multiharmonic expansion is proven to converge, in appropriate function spaces, to the periodic solution in time domain.

On the identification of the nonlinearity parameter in the Westervelt equation from boundary measurements

“…. Thus we have shown that T is a self-mapping, provided (23), (25), (28), (29), (30) hold. These can be achieved by making κ L ∞ (Ω) small:…”

“…We now proceed to derive contractivity of T by applying the estimates (20), (21) to the pde (19). In the above, we have already shown that for any p ∈ M , under conditions (23), (25), (28), (29), (30), the coefficients α = 1 − 2κp t , β ≡ b, γ ≡ c 2 , δ ≡ 0, µ ≡ 0 satisfy the assumptions of Lemma 2.1 with 1 2 ≤ α ≤ 3 2 ,γ ≤γ < 1 16 , so that the estimates (20), (21) with u(0) = 0, u t (0) = 0 apply to (19). Using them together with Gronwall's inequality yields existence of constants C,C > 0 (depending only on the constants R 0 , R 1 , R 2 ,γ in the definition of M as well as b, c 2 and T ), such that…”

“…f , so that we have (29), (30); then possibly decrease κ L ∞ (Ω) to achieve (23), (25), (28). These requirements also show that smallness of b can to some extent be compensated by smallness of T , cf.…”

“…and further conditions will be imposed on R 1 , R 2 , cf. (25), (28), (29), (30) below. Now fix p ∈ M .…”

On the inverse problem of vibro-acoustography

“…To achieve the separability (33) of m we supplement the boundary excitation ĝk (𝜔) by an interior one f g k ( ) , which we view as an approximation of a source g(𝜔) 𝛿 Σ k concentrated on Σ k , cf., e.g., [15]. The resulting equation for φk (𝜔) then has a solution of the form φk (x, 𝜔) = ã(𝜔) b(x) if, e.g., we choose b such that…”