The bottom slopes encountered off sandy coasts are usually gradual 

 and, when plotted to true scale, seem almost horizontal. Whether 

 or not the theories applicable to a horizontal bottom may be applied 

 to these very gradual slopes remains to be determined by experiment 

 or direct observations. Though identity of individual wave crests 

 seems to be maintained over a gradually sloping bottom even this 

 feature should be investigated. 



The relationship between period, length, and velocity characteristic 

 of all wave phenomena is 



L=CT (1) 



This formula is essentially a definition. Independent measurements 

 of length, velocity, and period should agree with equation 1 exactly; 

 and deviations therefrom should be considered as observational 

 errors. Agreement with this equation, however, is not a confirma- 

 tion of the dynamical theories of oscillatory waves. 



Oscillatory waves on the surface of water are characterized by 

 periodic variations in the motion of the water particles, pressure, 

 and surface elevation, each quantity passing through a series of 

 values and returning to its initial value at regular intervals of time. 

 At intervals equal to the wave length measured in the direction of 

 motion, each of these quantities will be equal to and in phase with 

 its former value. From the standpoint of the mathematical analysis, 

 the approximately constant atmospheric pressure on the water surface 

 and the existence of appreciable vertical accelerations are of basic 

 importance. The ampHtude of orbital motion decreases exponen- 

 tially below the surface and becomes very small at a depth equal to 

 half the wave length and for most purposes the orbital velocity is 

 inappreciable at even smaller ratios of depth to length. The effect 

 of decreasing depth is a continuous function. Although some authors 

 give a contrary impression, there is therefore no sharp line of demarca- 

 tion between shallow-water and deep-water waves. The dividing 

 point is purely arbitrary and is dependent upon the desired accuracy 

 of computation. No abrupt change occurs when the depth is equal 

 to half the wave length and the present arbitrary division of deep- 

 water and shallow-water waves on the basis of the depth-length ratio 

 is meaningless. 



Present-day wave theory deals with waves of stable form in which 

 all elements of the wave profile advance with the same velocity 

 relative to the undisturbed water. From a theoretical standpoint, 

 the wave motion is unchanged if the water is imagined to move 

 relative to a fixed point at a velocity equal and opposite to the wave 

 velocity, thus bringing the waves to rest relative to the same point 

 of observation. The analysis is thus changed to one of steady flow 

 in which the surface form, and distribution of horizontal and vertical 

 accelerations must be so distributed as to satisfy the equation of 



