Issue 
La Houille Blanche
Number 8, Décembre 1967



Page(s)  861  870  
DOI  https://doi.org/10.1051/lhb/1967060  
Published online  24 March 2010 
Évolution récente des modèles mathématiques d'agitation due à la houle. Calcul de la diffraction en profondeur non uniforme
New developments in mathematical models of sea waves : calculation of wave diffraction over an uneven bottom
SOGREAH
The present paper describes a method used on a digital computer for calculating the effects of wave refraction and diffraction simultaneously at harbour sites. It is derived from the method used by Biesel and Ranson [3] in the case of uniform depth by consideration of the radiation of wave energy in water of gradually varying depth. The wave height and wavephase are given in complex forme ζ by formulae (1) and (2) where the Green function G2 (OM) is defined by (19) (20) and (21). For practical purposes the area is divided into theoretical basins of convex shape and the integral (2) is applied to the boundaries of these basins. The derivative normal to the boundary Γ is given by formulae (4) (5) and (6) and the derivative of the Green fonction is given by (22) (23) and (24). For calculation, the integrals (2) and (6) are given the form (8) (9) with an increment ΔSR chosen small enough with respect to the local wave length The coefficients RJK, SJK defined by (25) (26) are determined in the following manner : (i) The paths of a given number of wave rays, starting from the points Oi and Mk, where ζ and ∂ζ/∂n are to be found, are computed by means of formulae (28) to (31) inclusive; (ii) The auxiliary functions I, φ, β, σ, are computed at the points where the wave rays reach the bounndary Γ. Their values at the given points Mp Mg, are then derived by interpolation along Γ. At this step, the differential equation (33) is integrated with the initial conditions (36) for MK and (37) for Oi. This mathematical model was compared to laboratory scale models and proved to be satisfactory. Two examples are given here. (i) The wave pattern in the harbour illustrated in Figure 2 for a wave period of 10 sec. Owing to the narrowness of the breakwaters compared to the wave length, it was possible to consider them as thin walls in the mathematical model. The results given by both models as shown in Figure 3. (ii) The wave pattern just inside the harbour entrance illustrated in Figure 4. In a previous study by Montaz [7] the breakwaters were considered as thin walls and the mathematical model gave the results of Figures 5 and 7.2, which depart considerably from the results of Figures 5, 6 and 7.1 obtained with a physical model. An accurate reproduction of the sloping breakwater head on the mathematical model leads to the consistent results of Figures 6 and 7.3.
© Société Hydrotechnique de France, 1967