Chapter V 



Shallow Water Waves and Their 



Transformation through External 



Factors; Surf 



Surface or short waves generated by wind on deep water travel eventually 

 on to shallow water, where their profile changes when the depth becomes 

 equal to the wave length. 



In order to understand the phenomena accompanying this transformation, 

 it is necessary to deal first with the theory of the shallow water waves, that 

 is with waves whose vertical motions, in shallow water, can no longer be 

 regarded as small in relation to their horizontal motions. 



1. Shallow Water Waves; Theory and Observations 



As shown in the second chapter (see p. 16) there is a gap between surface 

 waves and tidal waves which is filled by waves where the water depth varies 

 between T V to h of the wave length. The orbits of the single water particles 

 in such waves are flat ellipses, whose vertical axes become gradually smaller 

 until they vanish at the bottom, where the ellipses degenerate into straight 

 lines. Figure 8 shows the wave profile and the position of the line of particles 

 which is vertical at rest. The velocity of propagation of such waves is given 

 by equation (II. 10), and the middle section of Fig. 9 shows its relation to 

 the ratio /?:/. The extreme values in either direction are the well-known 

 equation for the velocity of surface or deep water waves c = ] (g/%) and 

 for the long waves of small amplitudes c = }/(g/h). In a first approximation 

 the wave profile, for small amplitudes in shallow water, is very similar to 

 a harmonic form, but with large amplitudes the profile changes differently 

 for small and great depths. A short time ago, Struik (1926, p. 595) was 

 able to prove, by using a similar analysis as used by Levi-Civita (1925) 

 for deep water waves, that waves of a permanent type are possible in shallow 

 water. However, it is not yet known how the waves are generated and 

 maintained. Korteweg and de Vries (1895, p. 422) developed a theory 

 of a system of oscillatory waves of finite height in a canal of limited depth. 

 It gives wave profiles with steep crests and wide troughs as they are regularly 

 encountered in rather high shallow water waves. Regarding the theory of 

 these waves, we refer to Lamb's Hydrodynamics (1932, para. 253, p. 426). 



