Gerstner Wave Research Articles

THE DEVELOPMENT OF A SAND BEACH BY DEEP-WATER WAVES

In a previous paper**, it was shown that the mechanism of the trochoidal waves can be used to determine the equilibrium slope of a sand beach under any wave conditions. As a start it was assumed that the beach material was of uniform grain size, and that the waves approached the beach directly with all motion in planes at right angles to the shore line.
 In the present paper, the application of the theory is shown in the development of various sand and gravel beaches. The equilibrium theory is studied in the light of the fact that there is usually considerable transportation of material along the shore. In particular, attention is called to the characteristics of beaches with rounded or pointed contours, of beaches whose ends are closed off by rocks or cliffs, or whose ends are open and extend into deep water without barriers of any kind. 
 A method of study and analysis is demonstrated which can be applied to all beaches. Finally, it is shown that an accurate forecast of the natural development of a beach can be made on the basis of the equilibrium slope equation, as well as a forecast of the effect of any structure placed in a naturally developing beach.

THE HYDRAULIC DREDGING AND PUMPING OF LAKE AREA DEPOSITS

First let me say that it is a sincere pleasure and an honor to address such a distinguished audience. More people should be made aware of the work that you are doing in the protection of our shores and coastline. It is a great satisfaction to me to he associated with an industry that is instrumental in rebuilding our shores along the lines of your extensive studies and recommendations. I recall laboring over the trochoidal wave theory in my study of naval architecture quite a few years ago. Frankly, it is not used very much in dredge design except in the larger molded form sea going hopper dredges. There is no doubt that you have made great strides in the theory of wave motion and transmitted energy since that time.

ON THE THEORETICAL FORM OF OCEAN SWELL. ON A ROTATING EARTH

In the discussion of the form of ocean swell the waves studied by Gerstner’ (1802) and Stokes (1847) have received particular attention. In Gerstner waves each fluid particle describes a circle about a fixed point. Their mathematical properties are very simple, but they cannot be generated from rest by conservative forces. On the other hand, Stokes waves can be generated from rest by conservative forces. There is a surface drift associated with Stokes-waves; but the mathematical difficulties are considerable. It has been customary to consider swell as a train of Stokes waves. One of the fundamental assumptions underlying these theories is that the waves are moving on a non-rotating Earth. It is now shown that if the curvature of the Earth can be neglected, the effect of the Earth's rotation makes swell waves in their stationary state differ very little from Gerstner waves. The work suggests that in the general case each particle moves approximately in a horizontal circle of inertia as well as in the nearly vertical Gerstner motion;. the diameter of the inertia circles is greatest at the surface, where it may be several hundred metres, and the period of revolution is half a pendulum day (equal to 12 cosec ø hours, where ø is the latitude). Except for this circular movement there is no mass transport, and thus. there is no ground for supposing that ocean drift currents are due to fluid transport with the waves. On the other hand, waves appear to provide a reasonable mechanism for the generation of inertia currents which have been frequently observed.

The effect of shallow water on wave resistance

1. The general character of experimental results dealing with the effect of shallow water on ship resistance may be stated briefly as follows:—At low velocitie the resistance in shallow water is greater than in deep water, the speed at which the excess is first appreciable varying with the type of vessel. As the speed increases, the excess resistance increases up to a maximum at a certain critical velocity, and then diminishes. With still further increase of speed, the resistance in shallow water ultimately becomes, and remains, less than that in deep water at the same speed. The maximum effect is the more pronounced the shallower the water. For further details and references one may refer to standard treatises, but one quotation may be made in regard to the critical velocity: "This maximum appears to be at about a speed such that a trochoidal wave travelling at this speed in water of the same depth is about times as long as the vessel... It was at one time supposed that the speed for maximum increase in resistance was that of the wave of translation. This, however, holds only for water whose depth is less than 0.2 times the length of the vessel. For greater depths the speed of the wave of translation rapidly becomes greater than the speed of maximum increase of resistance." In a recent analysis of the data, H. M. Weitbrecht expresses a similar conclusion by stating that for each depth of water there is a critical velocity, but that the critical velocity does not vary as the square root of the corresponding depth. It should be noted that experimental results are for the total resistance. If we assume that this can be separated into three terms, which are simply additive, namely, eddy, frictional, and wave-making resistance, it must be admitted that probably all are affected by limited depth of water. However, the main differences are due to the altered wave-making, and the general explanation is to be found in the fact that there is a limiting velocity, √( gh ), for simple straight-crested waves on water of depth h .

Societies and Academies

Societies and Academies