74 Generation, Growth and Propagation of Waves 



region is so small that they will pass unnoticed, so that, due to the continuous 

 transmission of energy into the undisturbed water in front of the wave train, 

 the speed of propagation of a disturbance will appear to be one half of the 

 wave velocity, even though each wave travels with the full wave velocity. 

 This will be especially important in predicting the spreading of a wave dis- 

 turbance into undisturbed water. 



These results by Sverdrup and Munk are in good agreement with those 

 found by Poisson and Cauchy; they have the advantage of being much 

 clearer than these latter. 



3. Theories of the Generation and the Growth of Waves 



(a) General Remarks 



The generation of waves by the action of an air current on a water surface 

 will be more easily understood when we return to the dynamic fundamentals 

 of wave processes based on Bernoulli's Theorem. Let W, in Fig. 38, be 



Fig. 38. Stationary waves and Bernoulli's theorem. 



a wave-shaped wall extending in a horizontal direction, which has as its 

 lower boundary a current moving to the right. The motion of the water- 

 masses exercises pressures upon this wall. At a point B, where the section 

 across the current is greater, the velocity must be smaller, whereas at a point T, 

 where the section is smaller, the velocity must be greater. In other words, 

 in steady motion the pressure is greatest where the velocity is least and vice 

 versa. When the motion is steady the velocity is constant in magnitude and 

 direction at every point. Consequently, the water motion, according to 

 Bernoulli's Theorem, will generate an excess pressure at the point B, and 

 a deficit in pressure at the points T. The consequence of this is that the water 

 motion will try to exaggerate the deformation of the wall. Between B and T, 

 consequently, there is a "dynamic" pressure gradient in the direction opposite 

 to- that of the current (to the left). The force of gravity acting on the water 

 mass in the direction S generates pressure forces which increase with depth. 

 Should gravity act alone, the pressure at all points below B would be greater 

 than at the points below T, and between B and T there would be a "hydro- 

 static" pressure gradient to the right (in the direction of the current). Current 

 and gravity, therefore, produce between B and T, pressure gradients acting 

 in opposite directions. The velocity of the current can now be selected in 

 such a way that the two equalize each other and that the differences in 



