Acceleration of the modal series in the Neumann scattering problem for a hemispherical shell

A mixed boundary value problem for the scalar acoustic field scattered by an axisymmetric plane wave incident upon a hard hemispherical shell is formulated. The resulting discontinuity in surface pressure is expressed in terms of a complete set of weighted Chebyshev polynomials that satisfy the correct asymptotic edge condition. In this way, the extremely slowly converging modal series of spherical wave functions is transformed to a convergent sum of physically motivated basis functions. Truncation to a finite number of unknown coefficients, together with Galerkin projection, yields a set of linear algebraic equations.