Sur l'amplitude de la houle émise par une source ponctuelle isotrope dans un domaine de profondeur variable

On the amplitude of waves generated by an isotropic point source in a region of varying depth

Ingénieur à la SO.GR.E.A.H., Grenoble.

Résumé

Waves generated by a hypothetical point source are investigated with a view to the application of Huygens's principle to computer calculations of waves in harbour areas in which the depth of water varies. Two useful reciprocity relationships for the calculation of wave amplitude at any point and its derivative along a given direction are demonstrated. It is known that the wave amplitude Aa at a point A can be deduced from the local distance σa between two adjoining wave radii originating from source B by relation (1), where K is the energy flux transmitted per unit angle, δ the specific gravity of water, Cga the group wave celerity at A, and θ the angle between the two wave radii at B. The quantity σa is the value at A of the solution σ2 (8) of differential equation (2) associated with boundary conditions (3) and (4). In this equation, s is the curvilinear abscissa along wave radius AB, n the ordinate along a line normal to the same wave radius at an arbitrary point, and c the corresponding phase celerity. By considering two isotropic fluctuating sources of equal intensity at points A and B successively, it is shown that : __ 1. The quantities σa with respect to source B considered by itself and σb with respect to source A are connected by relationship (15), in which ca and cb are the respective phase celebrities at these two points ; 2. The values of the derivatives with respect to s, σ'a and σ'b do not generally satisfy any analogous relationship. By means of relationship (25), however, σ'a can be found from the previous quantities and the solution ω1 (s) of equation (2) subject to conditions (22). The normal procedure for ca1culating the wave amplitude at point A from the known characteristics of fluctuating sources such as B distributed along an arbitrary line (r) surrounding this point would require the following : __ a) A first phase involving the determination of the wave radii from point A, knowing their curvature due to bed variations, and the calculation and memory-storage of coefficients P(s) and q (s); b) A second phase involving the calculation of σa by determining σ2 (s) in a direction opposite to the direction of progression. With relationship (15), the calculation can be carried out by simultaneously determining the wave radii and finding the integral σ1 (s) of equation (2) associated with boundary conditions (3) and (4). σ'2 (s) has to be calculated where waves are likely to reflect back from structures in the harbour ; if σ1 (s) and ω1 (s) are calculated instead, a similar advantage is gained by using relationship (25).