This thesis is in two parts. In Part I the independentvariable θ in the trigonometric form of Legendre's equation isextended to the range ( -∞, ∞). The associated spectralrepresentation is an infinite integral transform whose kernelis the analytic continuation of the associated Legendre functionof the second kind into the complex θ-plane. This new transformis applied to the problems of waves on a spherical shell, heatflow on a spherical shell, and the gravitational potential of asphere. In each case the resulting alternative representation ofthe solution is more suited to direct physical interpretation thanthe standard forms.
In Part II separation of variables is applied to theinitial-value problem of the propagation of acoustic waves in anunderwater sound channel. The Epstein symmetric profile is takento describe the variation of sound with depth. The spectralrepresentation associated with the separated depth equation isfound to contain an integral and a series. A point source isassumed to be located in the channel. The nature of thedisturbance at a point in the vicinity of the channel far removedfrom the source is investigated.
