DETERMINATION OF THE WAVE HEIGHT IN NATURE FROM MODEL TESTS

This paper deals with the problem of determining the wave characteristics in shallow water from those in deep water. In gene ral this can be done by means of a refraction calculation. If the sea bottom topography is too irregular the height of the waves can be determined by means of a small-scale refraction model. In both cases, however, some additional effects have to be taken into account, viz. the influence of the bottom friction and the influence of the wind.


1, INTRODUCTION
This paper deals with the problem of determining the wave characteristics in shallow water from those in deep water.In gene ral this can be done by means of a refraction calculation.If the sea bottom topography is too irregular the height of the waves can be determined by means of a small-scale refraction model.In both cases, however, some additional effects have to be taken into account, viz. the influence of the bottom friction and the influence of the wind.
Since a small-scale model does not correctly reproduce the breaking of the waves, this should be avoided by using such small waves, that no breaking in the model occurs.The influence of the breaking of the waves must then be studied in a separate model on a larger scale and the results of these separate tests must be taken into account as a correction factor by whicl the results of the refraction model have to be multiplied.
If a model is built in concrete, and the waves are long with respect to the water depth, the bottom friction in the model is no in accordance with that in the prototype.This can be compensated to a certain extent by a distortion of the model.In case of a ver; small scale of the model, however, this is not sufficient and additional measures have to be taken to compensate this scale effect, With regard to reproducing the influence of the wind on the height of the waves, it is often very difficult to generate wind in the model, in which case also this effect has to be taken into account as a correction factor.
For these reasons a small-scale refraction model cannot produce exact quantitative results and such a model will only give a correct representation of the refraction pattern.
The present paper describes a method for determining by meani of a snail-scale model, supplemented by calculations, the correct wave height in the prototype from the wave heights oeasared in the model.It is based upon the consideration of an energy balance for the prototype as well as for the model.This method nay also be used if the refraction is not determined by a model but by calcu-lations only.It can only be applied if the following conditions are fullfilled: 1) no breaking or surfing of the waves may occur 2) there must be only refraction of the waves and no reflection 3) the stretch of water, to which the method is to be applied, must be so short that the deviation of the wave height and the group velocity from the average values is not too great.

GENERAL PRINCIPLE
Let a b and c d be two approximately straight orthogonals and I and II two sections normal to these orthogonals, then the transport of wave energy through section II uust be equal to the transport of wave energy through section I, if between the sections I and Tl no energy is lost nor gained.However, neither in the model, nor m the prototype this is the case.In th« prototype, as well as in the model, energy is lost due to bottom friction, while in the prototype energy is gained from the wind.
The energy balances for prototype and nodel can be written as follows if losses due to internal friction are left out of consideration. Prototype In the above equations is: In the following it will be shown that the mnknown wave heighi H in section II of the prototype can bo calculated from the kncwn wave height II in section I by means of the above energy equations, if the corresponding wave heights in the model H' and Hi are measured and the losses in nodel and prototype, due to bottom friction are known.

TRANSPORT OF ENERGY IN SECTION I and II
The wave energy passing section I per second, may, with sufficient accuracy, be written as: 1 2 Ej = g pg H i B i u i watt and through section II: Hence, without wind and without bottom friction: After substituting u. and u_ by: L l L 2 U l = n l T and u 2 = Q 2 T respectively, the above equation becomes: The latter equation expresses that the wave energy present in section I on an area of n 1 B L is equal to that present in section II on an area of n B L .with the dimension of sinh -T~ ij The above formula for AE, is valid only for an impervious sea bottom, having a coefficient of bottom friction which is independent of the magnitude of the velocity at the bottom and of the water depth in the area under consideration, while the velocity must be sinusoidal.
From wave measurements in Lake Okeechobee, ref. 2., it appeared that k has a nearly constant value: k =• 0.01.
From the results of separate model tests on bottom friction, carried out in the Delft Hydraulics Laboratory, it appeared that the friction coefficient k for a model on a snail scale is dependent on the water depth D and the period T. With the help of the in this way obtained, values of k the correction for the bottom friction in the mode.lcould be determined.
The total loss of energy, due to bottom friction, in the stretch from se.ctionI to section II is then: 3, it appears that, after a certain period of time, for each water depth and each wind velocity, a state of equilibrium will be reached in which the waves no longer grow and the entire quantity of energy supplied by the wind is dissipated by the bottom friction.
By studying the amount of energy lost due to bottom fricti for various water depths and wave periods, an approximate constan value AE was found for the increase by wind of the wave energy p w sq.m over one metre.
For deep water, AE can directly be calculated from the dia gram showing the growth of wind-generated waves in deep w»»ter, ref. 4. For each value of the wind velocity, a value for AEj^is th found that is independent of the wave period and of the wave heig] This value does not differ much from the above mentioned value fo shallow water.This value is per metre displacement of the waver The total increase in energy, due to the wind, in the stret< from section I to section II is then: supplied, by the wind in the stretch I -II of the prototype 1^ » loss of energy due to bottom friction in the stretch I -II of 203 the prototype E' = loss of energy due to bottom friction in the stretch I -II of the nodel.
The time required by the energy on the area n..B, L-, to move to II amounts to velocity between I an,d II. 1 204 from I to II amounts to -sec, where u = the average energy u 4. LOSS OP ENERGY DUE TO BOTTOM FRICTION In the prototype, as well as in the model, there is a loss of energy due to bottom friction.Between section I and section II these losses amount to E, and EJ respectively.According to a theory on the dissipation of wave energy by bottom friction developed by Putnam and Johnson, ref. 1,. the loss of energy per unit of area, due to bottom friction, can be expressed by: AE b = kpH 3 f (D,T) watt/m 2 wheres H = wave height in m p = density of the water in kg/m f(D,T) = a function of water depth and period k = dimensionless coefficient of friction from the formula: v =V k ' v n> in which v = \/m/sec 1 = shear stress in N/m Vp= velocity at the bottom in m/sec.The function f(D,T) is: SUPPLIED BY THE WIND From the diagram showing the growth of wind-generated waves in shallow water, ref.