Numerical evaluation of the Kirchhoff--Helmholtz integral outside a sphere

A method is presented for the fast evaluation of the transient acoustic field generated outside a spherical surface using surface data on the sphere. The method employs Lebedev quadratures, which are optimal integration on the sphere, and Lagrange interpolation and differentiation in an advanced time algorithm for the evaluation of the transient field. Numerical testing demonstrates that the approach gives near machine-precision accuracy and a speed-up in evaluation time, which depends on the order of quadrature rule employed but breaks even with direct evaluation at a number of field points about 1.15 times the number of surface quadrature nodes, thus making the method an efficient means of evaluating the field generated by a large number of sources.