Van Mater and Neat 



r = dimensionless radius to field point = r'/h 



r\ = dimensionless elevation of water surface = "n'/h 



h = water depth 



a- = dimensionless wave number = Kh 



K = wave number = root of equation: co = g/C tanh Kh. 

 fl = dimensionless frequency = (w h/g) 



CO = frequency, radians /sec 



t = dimensionless time = t'Vg/h 

 t' = time, sec. 



The integral In (2) is evaluated using the method of stationary 

 phase (cf. Stoker [ 1957]). After doing this and making the substitu- 

 tions the result is: 



il(r,t) = 12^ . [- ^^]'^ • J3(o-r^) * cos (err - Qt) (4) 



Here, v = group velocity = |[ (6//c) tanh cr] [ 1 + (2a/slnh Zcr)] . 

 Eq. (4) is valid at distances from the source that are large In com- 

 parison with the radius of the cavity and in water of uniform depth. 

 A point is selected which satisfies these conditions and the spectral 

 energies of the system evaluated as will be discussed in the next 

 section. 



III. WAVE PROPAGATION OVER A BOTTOM OF VARIABLE DEPTH 



The extension of the solution to regions of variable depth is 

 based on a conservation of energy approach originally presented by 

 Van Dorn and Montgomery [ 1963] , The equation presented therein 

 evaluated the spectral energy, that is the energy per unit frequency 

 of the system, for the special case of propagation along a ray normal 

 to parallel bottom contour lines. The derivation is presented here in 

 a slightly different way to permit its extension to Include refractive 

 propagation. The topography considered is represented in Fig, 1 

 which also shows the coordinate system and a typical refracted 

 wave ray. 



The following assumptions apply: 



(a) wave frequency remains constant throughout the region of 

 wave travel and is unaffected by refraction 



(b) energy is transported at group velocity in a direction 

 normal to the wave crest 



(c) energy per unit frequency is conserved between adjacent 

 wave orthogonals, 



244 



