A Transparent Boundary Condition for an Elastic Bottom in Underwater Acoustics

Abstract

This work deals with the derivation of a novel transparent boundary condition (TBC) for the coupling of the standard “parabolic” equation (SPE) in underwater acoustics (assuming cylindrical symmetry) with an elastic parabolic equation (EPE) for modelling the sea bottom extending hereby the existing TBCs for a fluid model of the seabed.

Appendix: Laplace–Transformations

$$\begin{aligned} \mathcal {L}^{-{\scriptscriptstyle 1}}\left\{ \sqrt{s-\gamma }-\sqrt{s}\right\}&=\frac{1}{2\sqrt{\pi }}(1-e^{\gamma t})\,t^{-\frac{3}{2}},\end{aligned}$$

(L.1)

$$\begin{aligned} \mathcal {L}^{-{\scriptscriptstyle 1}}\left\{ \frac{1}{\sqrt{s}}\right\}&=\frac{1}{\sqrt{\pi }}\,t^{-\frac{1}{2}}, \end{aligned}$$

(L.2)

$$\begin{aligned} \mathcal {L}^{-{\scriptscriptstyle 1}}\left\{ \hat{\psi }(s+\sigma )\right\}&=\psi (t)\,e^{-\sigma t}, \end{aligned}$$

(L.3)

$$\begin{aligned} \mathcal {L}^{-{\scriptscriptstyle 1}}\left\{ s\,\hat{\psi }(s+\sigma )\right\}&=\,\dfrac{d}{d t}\,\left\{ \psi (t)\,e^{-\sigma t}\right\} \quad \text {if}\; \psi (0)=0. \end{aligned}$$

(L.4)