On the Propagation of Dispersine Waves (I)

[摘要] §. 1 Introduction. In the present paper, it is proved that the maximum amplitudes of a dispersive wave, the individual waves of which have maximum and minimum group velocities, are propagated with those group velocities. According to the late Professor K. Sezawa and Dr. K. Kanai(1), the maximum amplitude should always be propagated with the group velocity of the individua_wave of infinite wave length, being inconsistent with our conclusion. It is revealed that the examples treated by them are not suitable to examine the present problem, because such dispersion laws as discussed by them have no maximum or minimum group velocities except those corresponding to infinite wave length.§. 2 Fundamental Equations.Notations. V: Amplitude, C: Phase velocity, xπ/f: Wave length, t: Time, x: Space coordinate measured in the direction of propagation.The solution being responsible for the initial conditions, V=V0(x)and _??_ when t=0The relation between c and f is assumen to be given as where U is the group velocity. (Fig. 1). Then (2.4) may be reduced to and V0, V1 and V2 are certain constants.From physical point of view, the valucs of V2(±) when t is large will be most interesting; thus we will confine our attention only to the evaluation of V2(±).§. 3 Evaluation of V for a moderate valus of t under a Special Assumption of the Initial Conditions. (Omitted)§. 4 Evaluation of V for a Large Value of t under General Initial Conditions.To evaluate the right side of (2.10) for a large value of t, we must at first consider the integral under the condition _??_ To apply the method of steepest descent, let us suppose that f is, in general, a complex variable and putIt may be concluded, after some detailed examinations, that the positions of saddle points and the steepest paths in the (X, Y) plane are given as shown in Fig. 5. (In this figure, _??_, _??_ and _??_ indicate the terminal point of integral, the sadde point and triple point _??_espectively. A saddle point coincides with a terminal point at _??_.)Fig. 6 shows the relation between P/mg and x.The integrals near the terminal points (f=a and _??_) and the saddle points are carried out. Finally, comparing these values of integrals with each other, we may come to the conclusion that the most important terms are the integrals V2(+) älong the paths through the triple points, where P/mg=U1/β-x/βt=±1, and they are given by. where V1=A+(U0-U1)α+β/mcos(_??_) and A is an integral constant.The above values are O(g-1/3), Whereas the integrals through the other saddle points or the terminal points are O(g-1/2) or O(g-1) respectively and the largest values of V2(-) obtained are O(g-1/2) Thus, for a large value of t (viz_??_ -g_??_1), the maximum amplitudes in the wave train are found to appear at the points x=(U1±β)t, in which U1+β and U1-β mean the maximum and minimum group velocities respectively.§. 5 Some Notes on the Results obtained. (4.19) and (4.20) show that in general cases the amplitude of the wave group when U has its maximum value are greater than that when U has its minimum value. It agrees with the observational fact for Rayleigh waves of higer degrees along a surface of stratified ground as already discussed by Dr. K. Kanai(1) or the present author(2).If the extremum value of U corresponds to f=∞, V2(+) will not have such large magnitude of order as (4.19) or (4.20), because in this case _??_(f) and _??_(f) may be small quantities of O(f-1), as will be easily shown by Riemann-Lebesque's Lemma.

[学科分类] 大气科学