Tunami Wave in Osaka Bay (I)

Abstract

(I) Theory on Propagation of Tunami Wave.
Assuming that the tunami wave is appropriately the so-called long wave, its motion may be represented by the solution of where ζ is the elevation, h the depth and u the velocity. Now, for the sake of simplicity, we treat the periodic wave with period of2π/v only Thus the solution of the equation becomes of the form of f cos 2π (vt-φ). Then substituting it in (1) and putting both coefficients of cosine and sine to be zero, we get
(2) is _??_ Hamiltonian equation for the long wave and shows that the wave propagates normally to the surface φ=const. (wave front) with the velocity of In this case the bay of variable depths may be a dispersive medium for the long wave. But if the variation of the depth with the horizontal distancemay be negligible within the range of a wave length, the equation (2) and the wave velocity becomes (∂φ/∂x)²+(∂φ/∂y)²=v²/gh and √gh respectively.
Next. (3) indicates that the energy contained within a wave length is conservative and the average direction of flow of energy is normal to the wave front.
Applying this theorem, the present author draw the curves showing the wave front by the method of Huygens' principle, and also the curves normal to the wave fronts. Then we calculated the flux of energy and found that the average of the wave is conservative and for wave height the Green's formulae of energy flow hold. The chart of the wave thus obtained indicates the energy distribution of the wave in the field.
When the long wave progresses along a canal, and when the breadth of the canal is sufficiently wide compared with the wave length, the features of wave motion may be inferred by drawing the wave surfaces with the velocity of√gh. Then we shall understand directly from it that the energy of the wave flows towards both coasts of the canal and the most part of energy scarcely passes tbrough the canal regularly. But this is not the case when the breadth of the canal is comparable with the wave length, and the longer is the wave length, the closer the circumstances approach to the case which is applicable to the theory of narrow sea. Consequently, in this case, the wave front becomes nearly straight lines normal to the median line of the canal and its energy is accumulated in the deeper parts of the canal and the wave reaches far inner parts of it with a small loss of energy.
Thus the fact that tunami which occured far off in the Pacific Ocean could arrive at Osaka through the long canal bearing violent damages, can be explained.