Shallow Water Wave Transformation through External Factors 123 



pletely that the differential equations were not linear. The solution is valid 

 irrespectively if one starts either with Euler's equations or the Lagrangian 

 equations of motion. 



The approximation by Airy (V.21) applies to a simple two-dimensional wave train. Jeffrey 

 analysed in the same manner a three-dimensional wave system with shorter crested waves, as already 

 discussed on p. 29. If we take as an undisturbed wave form 



r] = bcosx'(x—c't)cos,uy, 



we find here too that if b attains the order of magnitude of depth, the wave profile becomes asym- 

 metric. In a first approximation 



3 gx'b 2 c' 2 

 r\ = bco%x\x — c t)cos[iy J \ .\sin2x; (x—c i)cos2juy . (V.22) 



As found before, the waves will overturn at a distance of about 



8 c* 



x = - 



3 gx'bc" 2 



In order to determine which of the two wave systems can keep itself longest before breaking 

 we establish the ratio of the two distances where the waves break. This ratio is x c' 2 b\2xac 2 . Inas 

 much as c is larger than c, the value of c will be > x'bjlxa, as long as fi is not large in compari 

 son to x' . 



From observations made far out from the coast it was found that the amplitudes of the long- 

 crested waves are so small that they are barely noticeable in a cross-sea. Consequently, b is many 

 times larger than a. Furthermore, the short-crested waves of a cross-sea have a short length as 

 compared with the long waves of the swell ; we therefore have x' > x. For these two reasons, the 

 value of the above-mentioned ratio is very large. This means that when a cross-sea enters into 

 shallow water close to the beach, the short-crested waves are destroyed rather soon by collapsing 

 and overturning. They vanish from the oncoming waves, whereas the long-crested swell is main- 

 tained. Numerical examples show that, generally, a stretch of water 500—1000 m long consists 

 only of long-crested swell, and this is also confirmed by observations. 



4. Surf on Flat and Steep Coasts 



The further development of shallow water waves advancing towards the 

 coast depends upon whether they can expand on a gradually sloping beach 

 or whether they progress against a steep coast where the water remains 

 relatively deep. 

 (a) Coastal Surf 



If the coast is gently sloping, the surf is regular and constant provided 

 the height of the waves is not too large and can be followed up qualitatively, 

 although not by mathematical analysis. In a wave system travelling from 

 deep water towards a soft sloping shore, the orbits which are in deep water 

 perpendicular to the wave front and to the water surface at rest, become 

 inclined from this vertical plane. Finally, these orbits will be parallel to the 

 plane of the beach. The projection of the orbit on the horizontal plane is, 

 therefore, a straight line in front of the coast; on approaching the shore, 

 it becomes an ellipse which gradually changes its main direction. When the 



