Chapter II 



Theory of Short and Long Waves 



1. Waves with Harmonic Wave Profile. (Stokes' Waves) 



The basic prerequisite for a wave theory is that the motion of the water 

 mass obeys the hydrodynamic equations and the equation of continuity. 

 If this is the case, it only means that this motion is possible, and not whether 

 and under which conditions it occurs. This theory can first be considered 

 from the viewpoint that the motion of the individual water particles is 

 stationary and irrotational, i.e. that the motion can be generated from rest 

 by the action of ordinary forces. Then there exists a velocity potential ip which 

 will satisfy the equation of continuity in the form 



S+S- - <™ 



Instead of the three equations of motion, we have the equation of Bernoulli 



|+f +*+*-*». CH.2) 



in which we substitute for the gravity potential gz. Let us consider oscillations 

 of a horizontal sheet of water. We will confine the problem first to cases 

 where the motion is in two dimensions, of which one (x) is horizontal and 

 the other (z) vertical, counted positive upwards; water surface at rest z = 0, 

 water depth z = — /?. The waves then present the appearance of a series of 

 parallel straight ridges and furrows perpendicular to the plane xz. The 

 amplitude of the waves shall be small at first and we will neglect factional 

 influences. 



One of the kinematic boundary conditions to be fulfilled by the motion 

 of the water masses is that at the bottom (z = —h) the vertical velocity 

 component w = dcp\dz must disappear; the other boundary condition requires 

 that at the free water surface the normal component of the fluid velocity 

 be equal to the normal component of the surface itself. If I and r\ are the 

 horizontal and vertical displacements of the surface, then with sufficient 

 approximation : 

 for 



-O:*-"-*. (IL3) 



