TABLE OF CONTENTS The limitation of the simple ray tracing is first of all that it disregards possible variations in amplitude along the wave fronts, disregards the effects of curvature of the wave fronts. Such variations become important, in the neighborhood of caustics, and around structures such as breakwaters that partly obstruct the wave propagation in the horizontal plane. These diffraction effects imply that wave energy propagates across wave number directions, which was specifically assumed not to happen in the simple refraction considered above. It also disregards reflections caused by variations in the borrom topography.
The classical theory for linear constant depth diffraction is based on the same basic assumption of long crested waves as the refraction methods described above. However it allows for relatively rapid variations of the wave field also in the direction perpendicular to the wave fronts. Under such conditions, the wave motion can be described by the so-called wave equation.
For the sinusoidal ("harmonic") motion this equation, which describes the wave motion in time and space, reduces to a Helmholtz equation which governs the space variation of the amplitude. It is worth already here to emphasize that the wave equation describes the propagation of the wave motion in space and time and therefore is a hyperbolic equation. Conversely, the Helmholtz equation describes the (time constant) distribution in space of the amplitude and phase of a sinusoidal wave motion, and hence is an elliptic equation.
Few analytical solutions are available for this equation, but one...
