14 



the potential energy at any time t per wave length per unit breadth 

 is then: 



The kinetic energy per unit breadth per wave length is obtained 

 by integration of the equation: 



Ej,=^f' ^ r ^ {Uz'-\-Wz')dxdz 



In a deep-water wave of sinusoidal form 



wLlf 

 16 

 and the total energy is then: 



E,=^'^^^=E, 



E=E,+ E,='^ (11) 



and substituting for L from equation 2c 



E=OM wh?r (llo) 



Section 4. WAVES OF FINITE AMPLITUDE 



A. Irrotational theory. — The preceding equations were derived on the 

 assumptian that the wave height was very small as compared to 

 the wave length and water depth. However, it is known from experi- 

 ment that the equations continue to be valid for waves of small but 

 appreciable height. For waves of greater height, theory indicates 

 that certain corrections are necessary. Waves in water of unlimited 

 depth will be considered first. 



It has been shown by Stokes that waves of finite amplitude in deep 

 water are not sinusoidal in form and that there is an induced drift or 

 mass transport in the direction of wave motion which modifies the 

 wave velocity and orbital motion. Stokes' (2) conclusions are: 



I find that the expression for the velocity of propagation is independent of the 

 height of the waves to a second approximation. With respect to the form of the 

 waves, the elevations aie no longer similar to the depressions, as is the case to a 

 first approximation, but the elevations are narrower than the hollows, and the 

 height of the former exceeds the depth of the latter. 



There is one result of a second approximation which may possibly be of practical 

 importance. It appears that the forward motion of the particles is not altogether 

 compensated by their backward motion; so that, in addition to their motion of 

 oscillation, the particles have a progressive motion in the direction of propaga- 

 tion of the waves. 



The waves produced by the action of the wind on the surface of the sea do not 

 probably differ very widely from those which have just been considered, and which 

 may be regarded as the typical form of oscillatory waves. 



If the pressure of the air on the posterior parts of the waves is greater than on 

 the anterior parts, in consequence of the wind, as unquestionably it must be, it is 

 easy to see that some progressive motion must be produced. 



