Impulsively Generated Waves Propagating into Shallow Water 



for this application. When the modulus, k, assumes its minimum 

 value, zero, en reduces to cosine. When it assumes its maximum 

 vctlue, unity, it reduces to the hyperbolic secant, sech. Since cos^ 

 reduces to cos by the double angle formula and since the form of a 

 solitary wave is given by a sech function, the en function by 

 appropriate choices of k describe the complete transition from 

 sinusoidal waves to solitary waves. As the value of the modulus 

 increases from zero toward unity the wave crests become more 

 sharply peaked and the troughs longer and flatter; the height of the 

 crest above the still water line increases and the depth of the trough 

 decreases. The wave form is symmetrical about a vertical through 

 the wave crest so that no wave slope asymmetry is reflected. Thus 

 the cnoidal feature may be used to improve the realism of the phase 

 waves in several ways: 



(a) to give the non-breaking waves a more realistic profile 



(b) to introduce asymmetry of the waves about the still water 

 line 



(c) to increase the velocity of the waves in very shallow water. 

 This feature has not been utilized in this application. 



Computationally, the frequency, co, and the water depth, h, 

 are defined in the given frequency and spatial array. The wave 

 height, H, is obtained from the linear theory of the previous section. 

 The elliptic modulus, k, is computed from an iterative solution of 

 Eq. (18), K(k) and E(k) are computed from a series expansion In 

 terms of k. Further details appear In the appendix on computational 

 procedure. 



The application, then, Involves the use of a theory developed 

 by Irrotatlonal, periodic, non-dlsperslve waves of permanent form 

 In water of uniform depth to represent a wave system which Is dis- 

 persive and not periodic passing over a bottom of variable depth and 

 In which vortlclty due to bottom friction Is present at least to some 

 extent. The assumptions Implicit In this extension are that the phas e 

 waves assume a form appropriate to their local frequency wltliln the 

 group and that this frequency content changes only slowly, and further 

 that rotationality effects are small . The latter assumption appears to 

 be the most vulnerable. 



V. WAVE BREAKING AND ENERGY DISSIPATION 



Despite abundant literature on the subject of wave breaking 

 there does not exist today a fully -ad equate mathematlccd description 

 of the process, and, In fact, much of the experimental evidence Is 

 contradictory and subject to wide scatter bands, Ih. the case of this 

 analysis the need is for a criterion for wave breaking which relates 

 the wave frequency, co, the linearly computed wave height, H, and 



251 



