Scattering by a perforated sandwich membrane: method of Riemann surfaces

The model problem of scattering of a sound wave by an infinite plane structure formed by a semi-infinite acoustically hard screen and a semi-infinite sandwich panel perforated from one side and covered by a membrane from the other is exactly solved. The model is governed by two Helmholtz equations for the velocity potentials in the upper and lower half-planes coupled by the Leppington effective boundary condition and the equation of vibration of a membrane in a fluid. Two methods of solution are proposed and discussed. Both methods reduce the problem to an order-2 vector Riemann-Hilbert problem. The matrix coefficients have different entries, have the Chebotarev-Khrapkov structure and share the same order-4 characteristic polynomial. Exact Wiener-Hopf matrix factorization requires solving a scalar Riemann-Hilbert on an elliptic surface and the associated genus-1 Jacobi inversion problem solved in terms of the associated Riemann $\theta$-function. Numerical results for the absolute value of the total velocity potentials are reported and discussed.