(wavelength, phase speed, or frequency) are given. In the absence of 

 current, some of the conditions investigated lead to explicit solutions 

 and others to implicit solutions. 



This paper also notes the problem of critical current velocities; 

 i.e., when wave phase speed is equal and opposite to current velocity. 

 Solutions then appeared to be unobtainable, but more recent work has 

 developed ways of dealing with this problem (see PEREGRINE, 1976, for 

 more on this topic). 



Coastal Engineering Significance. In addition to presenting the 

 mathematical solution, the author discusses physical points of interest 

 to engineers. He emphasizes how vertical shear of the current can have 

 practical importance in changing wave properties. Numerous more recent 

 papers have rediscovered or elaborated on Biesel's (1950) results. 



4. BOOIJ, N., "Gravity Waves on Water with Non-Uniform Depth and 

 Current," Communications on Hydraulics, Report No. 81-1, Department 

 of Civil Engineering, Delft University of Technology, The Nether- 

 lands, 1981. 



Keywords . Comparison of Theory and Measurement; Currents, Large-Scale; 

 Numerical Model; Refraction-Diffraction; Wave Dissipation; Wave Height. 



Discuss ion . A parabolic approximation of a new wave equation is 

 developed for the practical calculation of wave propagation in an area 

 with slowly varying depth and current. 



Using a variational principle, the author derives a water wave 

 equation which is probably the first to include the effect of variable 

 depth and current. The derivation assumes small-amplitude waves, mild 

 bottom slope, slowly varying current, no velocity variation with depth, 

 nearly periodic water motion, and no dissipation. The new equation is a 

 partial differential equation of the hyperbolic type. It has the 

 important restriction that the frequency observed from a fixed point 

 must vary slowly and within a narrow band. For purely periodic waves it 

 transforms to an elliptic-type wave equation which, in the absence of 

 currents, reduces to the mild-slope wave equation developed by Berkhoff 

 (1972). 



The new wave equation has two drawbacks: it does not include 

 dissipation, and for areas many wavelengths in size, computer time 

 becomes too long for practical application. Moreover, in irregularly 

 shaped regions, the refraction method tends to leave areas with hardly 

 any rays, where experience indicates appreciable wave heights may occur. 



The author addresses these problems by adopting a parabolic approx- 

 imation (developed mainly in acoustics) which allows for variation of 

 wave height along wave fronts. The parabolic approximation to the 

 steady-state wave equation obtained by the author is used to develop a 



