QUASI-TRANSLATIONAL ADDITION EXPRESSIONS OF PROLATE SCALAR WAVE-FUNCTIONS FOR THIN PROLATE SPHEROIDS

The conventional, translational addition theorems for both spherical a nd prolate spheroidal wave functions applied to incoming or outgoing w aves possess a well-known convergence sphere near which the addition s eries fail to converge rapidly. To overcome this deficiency the quasi- translational addition expressions for prolate wave functions are deve loped by using a physical-geometrical transformation which allows the separation in the azimuthal direction in the translated prolate sphero idal coordinate system. These quasi-translational addition expressions are valid everywhere and, in particular, exhibit a fast convergence c haracteristics for thin spheroids which are commonly used in physical and engineering applications.