On the transformation of a continuous spectrum by refraction

When waves are propagated through a medium whose velocity of propagation varies gradually from place to place, the wave direction and intensity vary according to the laws of refraction. Although the geometry of ray-paths has been well explored, and so also the laws governing the intensity of a coherent train of waves, little attention has apparently been given to the variation in intensity of an incoherent beam having a broad spectrum. The transformation of the energy spectrum is of practical importance in branches of geophysics, for example, in the study of sea waves entering shallow water, or of microseismic waves propagated through inhomogeneous regions of the earth's crust. Accordingly, it seems worth while to state and prove the rule governing the transformation of the two-dimensional spectrum function of a wave disturbance undergoing refraction.