Assume that the speed of sound in the water and the bottom of the ocean is a function of only the depth, and not the range. Also assume that the ocean and its bottom eventually interface with a rigid halfspace. This problem can be solved by the method of normal modes, involving the eigenvalues and eigenfunctions of depth dependent ordinary differential equation. Since the sound speed in this problem varies only a little from its average value, the eigenfunctions and eigenvalues are known when the sound speed is constant. The changes in these eigenvalues and eigenfunctions that result from changes in the depth dependent sound speed within the ocean and its bottom, using a algebric formulation of the effect of the perturbation. Another more recent approach to finding the changes in the eigenvalues and eigenfunctions is a transmutation approach. We show a method of approximating the kernal of an integral transform and use it to find the first order corrections to the eigenvalues and eigenfunctions. Finally we compare the results of these two approaches with the results of classical perturbation theory for the same problem.