Section I 

 SURFACE WAVES IN WATER 



General Discussion 



A wave is described by its length L, i. e., the 

 horizontal distance from crest to crest or trough 

 to trough (see fig. I.l) and by its height H, 

 i. e., the vertical distance from trough to crest. 

 A wave is further characterized by its period T, 

 i. e., the time interval in seconds between the 

 appearance of two consecutive crests at a given 

 position. 



B 



Figure I.l. — Surface Waves. A. Profile of wave. 

 B. Advance of wave, showing the wave profile at the 

 times t = O, i = r/4, and t = T/2. In the time 

 T/2 the wave has advanced one-half wave length, L/2. 



A wave may be standing or progressive, but 

 this discussion deals with progressive waves 

 only. In a progressive wave, if the length and 

 energy are constant, the wave height is the same 

 at all localities and the wave crest appears to 

 advance with a constant speed (fig. 1.2) . Dur- 

 ing one wave period T, the wave crest advances 

 one wave length L, and the speed of the wave 

 is therefore defined as 



C = L/T (I.l) 



The motion of the water particles depends 

 on the wave length and the depth of the water. 

 In general, it can be stated that the advance 

 of the wave form is associated with con- 

 vergences and divergences of the horizontal 

 motion. In front of the crest the motion is 

 converging and the surface is rising, but behind 

 the crest the motion is diverging and the sur- 

 face is sinking. 



By energy of the wave is always understood 

 the average energy over the wave length. The 

 energy is in part potential, Ep, associated with 



the displacement of the water particles above 

 or below the level of equilibrium, and in part it 

 is kinetic, E^-, associated with the motion of 

 the particles. In surface waves half the energy 

 is present as potential and half as kinetic. 

 The total average energy per unit area of the 

 sea surface is E =^ p gH"/8, where g is the ac- 

 celeration of gravity and p is the density of 

 the water. For a 10-foot high wave the total 

 average energy per unit area is 800 foot-pounds 

 per square foot.^ Since g and p can be con- 

 sidered constant the energy per unit area in a 

 wave depends only on the square of the wave 

 height. For the total energy per unit width 

 along a wave length it is necessary to multiply 

 the energy per unit area by the wave length. 



Waves of Very Small Height 



Waves of very small height are those for 

 which the ratio of height to length is 1/100 or 

 less. The simplest wave theory deals with such 

 waves, the form of which can be represented 

 by a sine curve (fig. 1.3) . In water of constant 

 depth d, such waves travel with the speed 



=V4 



tanh 2w -y- 



(1.2) 



where g is the acceleration of gravity. 



If d/L is large, that is, if the wave length is 



small compared to the depth, tanh 27r d/L ap- 

 proaches unity and one obtains 



= V^i 



(1.3) 



These waves are called deep-ivater waves. 



If d/L is small, that is, if the wave length is 

 large compared to the depth, tanh 2-Kd/L ap- 

 proaches 2Trd/L and one obtains 



C = V^ (1-4) 



These waves are called shalloiv-ivater waves. 

 In general, waves have the character of deep- 



^ p = 2 slugs/ft^ 

 g = 32 ft/sec^ 

 il = 10 ft 

 ^ _ 2 X 32 X 100 



800 ft-lbs/ft^ 



