Greenʹs function for a plane three-dimensional fluid layer at small horizontal distances from the source

The Greenʹs function for a plane three-dimensional layer in its common form is usually determined as an infinite series of the waveguideʹs normal modes. The slow convergence of the series close to the source, limits the applicability of Greenʹs function to solving sound generation and scattering problems in waveguides, especially if near-field effects are significant. In the present work, a more convenient form of Greenʹs function for such a waveguide is obtained as a sum of two quickly converging series. The first series is a difference between the Greenʹs functions for Helmholtz and Laplace equations, whereas the second series is Greenʹs function for the Laplace equation, determined as a sum of the mirror images of the source in the waveguide boundaries. The behaviour of the obtained function close to the source is investigated. Numerical experiments show significantly better convergence for the obtained function as compared with the Greenʹs function in its common form. It is also shown that the obtained function can be easily calculated at the points directly underneath or above the source, where the terms of the Greenʹs function series in the common form are singular.