Generation, Growth and Propagation of Waves 71 



process, it appears to an observer at a given distance from the wave source 

 that the first waves of a long wave train are of a very small height. This 

 height can be so small, that these waves often escape the observer's attention. 

 Then the wave height increases with time and, when the wave train is only 

 composed of a finite number of waves, i. e. when the wave source only sends 

 out a finite number of waves, the maximum wave height is found in the 

 centre of the group, whereafter it decreases again. The rate of travel of the 

 place in the group where there is maximum wave height always equals the 

 group velocity, although each wave travels with its own wave velocity 

 (see p. 12). On the contrary, when there is a continuous generation of waves 

 of constant height at the source, one will find that after a transient stage 

 of wave growth, a steady state condition involving constant wave height 

 will be approached. 



Sverdrup and Munk (1947) based their study of the propagation of 

 a disturbance into an area of calm, on considerations of energy. 



It has been previously shown, equation (11.146) that the energy of a wave, 

 and also of a wave group, is propagated with half the wave velocity vE = {\E)c; 

 inasmuch as half of the wave velocity is equal to the velocity C of the wave 

 group, we can also say that all the energy advances with the group velocity C, 

 vE = EC so that v = C. Here is again the question (1) does all the energy 

 travel with the group velocity C; or (2) does half of the energy travel with 

 the wave velocity c? In order to decide between these two possibilities, we 

 consider the flow of energy through a parallelepiped of unit width, length dx, 

 which extends from the surface to a depth where wave motion is negligible. 

 The time rate of change of energy within this parallelepiped must equal 

 ~8vE/dx, the net inflow in the direction of the x-axis, and therefore 



The first of the two interpretations gives 



SF SF 

 — + C— = 

 9t + dx ' 



with the solution 



E=f{x-Ct). 



It does not explain in which manner E varies with x. 



The second interpretation of equation (IV.7) gives the differential equation 



SE 31E 



As the energy of a wave is proportional to the square of the wave height, 

 equation (IV.8), taking into consideration the necessary physical boundary 



