Directional Recording of Swell from Distant Storms - Department of ...

520 W. H. MUNK, G. R. MILLER, F. E. SNODGRASS AND N. F. BARBERFor very low frequencies V- V0(1--2u2), ji 2= f(h/g)fIt follows that t-to - [ + 7rhf0]where x/VO is the time of the earliest arrival. At high frequencies the solution approachestto = 4irxf/g,as previously. The general relationf(t) is shown in figure 6 for the case h -- 4 km, A == 1 IO'.The ridge does not deviate appreciably from a straight line for frequencies above 15 c/ks.6. GENERAL THEORY OF ARRAYS(a) Elementary wave trainsConsider a wave whose measurable value of q varies with the space co-ordinates x, yand the time co-ordinate t according tov = a exp'i 2nr(lx + my -ft + a). (6.1)This does not of course represent a real sea wave since the values of a are complex, but'it willbe apparent that a wave with real values can be created by adding two complex wavesthat differ only in the sign of i. A positive sign has been used beforeft for the sake of symmetry.It implies that a wave whose 'space frequencies' 1, m and 'time frequency 'fare allpositive is one that approaches the space origin from the positive region of x and y. For alocus of constant phase, a 'wavecrest', is a lineIx + my +ft - a = constant,and its points of intersection with the x and y axes move with velocities -fi, -flm.Wave analysis will be concerned with finding the coefficients in (6. 1), in particular I andm. When I and m have been found for a wave train such as that sketched in figure 7 (a), onemay use them as rectangular coordinates to plot a point P as shown in figure 7 (b). Thebearing of P from the 1-m origin is the same as the direction from which the wave isapproaching the site. The distance of P from the origin measures the wave number (thereciprocal of wave length). One may record the wave 'power' a2 at point P. Many wavetrains can be summarized by a diagram such as figure 7 (b). The wave number k and thedirection 0 are the polar coordinates corresponding to1z=kcosO,m-ksinO.So figure 7 (b) is sometimes called a plot in 'wave-number space', and I and m are 'componentsof wave number'.The coefficients I and m are the rates of change of phase (in cycles) with the space coordinatesx and y, just asf is the rate of change of phase with time t. They may be found byobserving how the phase depends on x, y, and t. In the complex wave of (6.1), the phase ofthe wave is the argument (in cycles) of the wave value q. If one knows wave values at twoplaces and instants, say (x, y, t) and (x +X, y -1- Y, tA-- T) then the amount by which the phaseof the wave at the second point exceeds that at the first is the argument of the conjugateproduct n* (x,y,t)n(x-l-X,y+rY,t-1-T) =a2expi2ir(lX-1-mY?4fT). (6.3)