A theory of propagation of waves inside a sphere is developed by using a metric in which the coordinates of a point are defined by its position along the trace of a ray reflected singly, or multiply, from the surface. The coordinates in this metric are orthogonal. Application of the theory is illustrated for propagation of a compressional pulse in a liquid sphere. The wave equation in the new metric is solved asymptotically in the limit of large wave-numbers. The resulting initial amplitudes of the reflected pulses are found to be in agreement (except for a factor of 2) with the results derived by Jeffreys & Lapwood, who used the classical methods of spherical harmonic representation, the Watson transformation, ray expansion, and the method of steepest descent.