Usually, the method of drawing front lines of traveling tsunami waves on a map uses the so-called Huygens' principle, that is, assuming the velocity of tsunami to be √
gh, where
g, the gravity acceleration,
h, the sea depth, and drawing sufficiently many circles of radius √
gh·
t (
t: a suitable short time interval) with their centers at as much points distributed adequately on a front line, we have the envelope of these circles as the next front line to be taken by the wave after
t from the preceding.
But due to the inevitable discrepancy of an employed map from a spherical surface, the position of any front line thus drawn must diverge much from that drawn on a sphere, especially when the tranaoceanic wave is considered (See Table 1).
Nevertheless, the effect of this destortion of map can be eliminated comparatively easily if we adopt a stereographic map and draw individual circle with radius
r=sin√
gh·
t/(sinφ+cos√
gh·
t) whose center deviates to the marginal direction of map from the original pasition on a front line by
d={cosφ/(cos√
gh·
t+sinφ)}-{cosφ/(1+sinφ)}, where φ is the latitude and √
gh·
t is measured by radian, while the map is assumed to represent a sphere of radius 1. By this method the wave front of tsunami due to the Sanriku Earthquake of 1933 is drawn successfully (See Table 1). Differences between the observed and the calculated travel times remain within 2%.
Moreover, since a stereographic projection is a conformal representation of a spherical surface, an energy contained between two adjacent orthogonal trajectories is conserved and this enable us to calculate the energy distribution of tsunami wave throughout the coasts around the ocean.
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