Les phénomènes du second ordre rayonnants dans les ondes de gravité

Abstract

From the mathematical point of view, an irrotational gravity wave problem (in a homogeneous inviscid fluid) consists of finding a velocity potential φ as the solution of a system of differential equations and boundary conditions. The latter include the "free surface" condition which, as a rule, states that the pressure at the free surface is constant (atmospheric pressure). They also include what could be called "generating boundary conditions" specifying, for instance, the motion of some solid boundaries and/or the existence of sources at infinity. These"generating conditions" (which, broadly speaking, are the non-zero "second members" of the equation system) exclude the solution φ = 0 [or its equivalent φ - f (t) ] in other words they do cause at least some sort of motion to occur. Moreover if, as we shall assume, "parasite" forms of motion are eliminated by appropriate precautions (e.g. vanishing dissipative forces) the solution will be uniquely determined by the generating conditions. In particular, if there are no generating conditions, no motion will occur. Since the set of wave equations is non-linear-principally because of the free surface condition-its rigorous solution is a rather formidable mathematical undertaking and, in fact, has only ever been successfully achieved in a few very special cases. One therefore has no alternative but to make do with approximate solutions, the simplest of which are obtained by solving the linearised system, and are generally referred to as "first order (of approximation) solutions". Because of their linearity, these solutions are very practical tools in wave analysis. For instance, the velocity potential satisfying several generating conditions is the sum of the velocity potentials referring to each generating condition taken individually (all other "second members" being zero). These individual polentials are themselves merely proportional to the corresponding second member, and so on. Because of the linear character of these solutions, several powerful mathematical methods can be used to deal, with the gravity wave problem. In fact, it can be claimed that, with these methods and the use of modern electronic COIllputing facilities, it is now possible to at least find numerical solutions 10 the first order of approximation for most practical gravity wave problems. Some of the equations are obviously not rigorously satisfied by first-order solutions; errorsremain, the principal parts of which are proportional to the squares and binary products of the "second members". These errors, or rather their "quadratic" principal parts, can be corrected by modifying the first-order solution so as to convert it into a second-order solution. It can be shown that this second-order solution can be constructed in the following way : 1. A first order solution φl is found, regardless of whether there is any need to find a second-order solution. This, as we said above, is a problem which can very often he solved satisfactorily ; 2. A second-order correction φ2 is added to φl and must satisfy the following conditions : φ2 must be a solution of the same linearised equations that are safisfied by φl, but with different "second members", which are quadratic functions of φ2 and its derivatives. Thus, the first-order solution can be said to create what really amounts to generating boundary conditions for the second-order solution. When Euler variables are used, the most important of these generating conditions for practical purposes are those created at the free surface, where they will have the same generating effect as a pressure fluctuation. This shows that the second-order terms are almost as "easily" found as the first-order ones. Both obey the same set of equations, but the surface condition has a second member for the second-order terms, while there is usually none for the first order terms. This does not introduce many new difficulties into the mathematical problem, however, and the second order equations for gravity waves can just as well be solved with electronic machines as the first-order equations, except that they require a much greater expense of machine time. We shall not go into computing details, but rather concentrate on the physical implications of the above. The second-order phenomenon can be considered as a "wave" system obeying the usual (linearised) wave equation but generated by a fluctuating pressure distribution on the free surface (and of course satisfying the other boundary conditions). This pressure q is given by the following relation : q (x, y, t) = - ρ/g t δφ1/δt δ/δz (δ2φ1/δt2 + g δφ1/δz) dt + ρV12 where φ1 is the first-order potential, x, y, z are tri-rectangular coordinates (Oz vertical upwards), and V1 is the velocity (grad φ1), all values being at z = 0 (free surface). This formula is absolutely general. If, to be more specific, we consider a "monochromatic" first-order wave, i. e., φ1 (x, y, z, t) = φ1 (x, y, z) cos kt + φ1 (x, y, z) sin kt it follows from the above that the frequency of the pressure fluctuations (the non-fluctuating part is of no importance) is twice that of the first-order wave. It is easy to prove that any local surface pressure fluctuations of a given frequency will give rise to a circular wave of the same frequency radiating energy in all directions. Second-order phenomena can therefore be considered as resulting from waves emitted by practically all the points of the free surface at which some first-order motion occurs. They should therefore generally have a "radiating" character, Le second-order waves should be generated "by" the first order phenomena and then radiate independently, as free waves, and in particular, with the velocity corresponding to their own wave length and frequency. The second-order solutions which have been worked out so far show, on the contrary, no apparent generation of such second-order waves. As a rule, it has only been found that individual waves are subject to local distortion (e.g. crests sharper than troughs), that special "accompanying waves" travel with irregular wave trains and so on, but these phenomena remain "linked" with the underlying first-order phenomena, and in particular, they propagate at the same velocity and not at the velocity corresponcling to "free" waves of the same length and period. The explanation of this apparent contradiction is that the known second-order solution in fact only concerns very simple cases, in all of which the second-order "radiation" is cancelled out by interference. How this happens is clearly illustrated in a two- dimensional monochromatic wave, for instance, where the second order emission is distributed along the entire course of the first-order waves, the phase distribution being uniform through all possible angles. Finally, for second-order radintion to occure "something" must happen to the waves to disrupt their orderly array and upset the finely-balanced mutual cancelling out of their second-order emissions. In fact, "things" do happen to real waves which never conform to the rigid patterns of the available second-order theories (at least at most points of interest such as harbours, beaches, etc.). Beal waves refract, diffract, break and ... are generated. In all these cases, the only outlined here indicates that some second-order radiation should occur, and allows to answer the question of how important this radiation is liable to be in practice. Experience has shown that this kind of question should not he brushed aside because, although the first-order phenomena are by far the largest, many second order phenomena are nevertheless also of engineering significance. A mathematical model of wave penetration in a harbour must therefore also be able to allow for such effects, just in case. In conclusion, it should be emphasized that the existence and occasional importance of second-order radiating phenomena has been confirmed by experience. Figure 2, for instance, shows clearly how second-order wave emission results from diffraction around the end of a pier. Although not very high, these waves penetrate into a region in which practically no first-order phenomena occur, 50 that they are the main wave action to be considered in these areas. Figure 3 shows the sudden appearance of second-order radiation in a complicated refraction wave pattern, in which the first-order refraction pattern is shown by the white lines. It is clear from this picture that "mild" refraction does not cause any appreciable second-order radiation ; it mainly seems to originate from the most "disorderly" part of the wave pattern.