Generation, Growth and Propagation of Waves 91 



We thus obtain from the equation (IV. 36): 



This relation applies to a train of conservative waves, but not to significant 

 waves, because experience shows that under the stated conditions the energy 

 of the significant waves is independent of x; therefore, dEJdx = 0, with this 

 condition the equation (IV. 37) takes the form 



dE E dc 



f + f£ = * r± *„. (iv.38) 



The integration of (IV.38) gives the change with time of the significant 

 waves at any locality in the storm area. Initially the significant waves will 

 have originated in the immediate neighbourhood of the locality under con- 

 sideration. As the time increases, the waves reaching this locality will have 

 travelled a longer time and originated at larger distances. In practice, the 

 distance from which waves can come is limited by the dimensions of the 

 storm area or by a shore line. This distance is called the fetch. The time 

 necessary for the waves to travel from the beginning of the fetch to the locality 

 under consideration is called the minimum duration / min . If the duration 

 of the wind exceeds / min , the character of the significant waves which are 

 present in the fetch remains constant in time and a steady state is established. 



To examine the steady state, consider a parallelepiped fixed in space of 

 unit width and of the length dx, but otherwise similar to the one shown in 

 Fig. 42. Since the parallelepiped is fixed in space, potential energy flows 

 into the volume at the rear edge at the rate chE and leaves at the forward 

 edge at the rate 



4+k(4) dx - 



The local change in energy must equal the sum of the amounts which enter 

 or leave the parallelepiped and one obtains 



cE c cE E dc „ „ ,„ T v 



p7+o p-+T - = Rt±Rn. (IV. 39) 



at 2 dx 2 ex 



This equation applies only to conservative waves. In order to apply it to 

 significant waves which are present over a limited fetch after a steady state 

 has been reached, we must write 8E/dt = and we obtain: 



ii+f !-*■=«- < IV - 40 > 



In the equations (IV.38) and (IV. 40) the + and — sign applies respectively, 



