Numerical modeling of waves for a tsunami early warning system
Numerical modeling of waves for a tsunami early warning system
Numerical modeling of waves for a tsunami early warning system
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<strong>Numerical</strong> <strong>modeling</strong> <strong>of</strong> <strong>waves</strong> <strong>for</strong> a <strong>tsunami</strong> <strong>early</strong> <strong>warning</strong> <strong>system</strong><br />
these methods <strong>for</strong> use in quantum mechanics in the 1920ies. The essence <strong>of</strong><br />
the WKBJ expansion method is to suppose that the amplitude function varies<br />
much more slowly than the phase function in the horizontal space. Thus the<br />
WKBJ approximated solution <strong>of</strong> the MSE is taken, with the restriction <strong>of</strong><br />
considering <strong>waves</strong> propagating only in x direction:<br />
N (x, y) =A (x, y) e −i kdx + c.c. (3.52)<br />
Since the wave motion is primary in the x-direction, the variation <strong>of</strong> the<br />
amplitude in the y direction is an order <strong>of</strong> magnitude slower than in the<br />
x direction. This can be expressed <strong>for</strong>mally by introducing other variables<br />
named X and Y and given by<br />
X = ɛ2x Y = ɛy<br />
(3.53)<br />
where ɛ is a small parameter.<br />
horizontal space derivatives become<br />
The changed variables implies that the<br />
∂A<br />
∂x<br />
∂A<br />
∂y<br />
∂A ∂X = ∂X ∂x<br />
∂A ∂Y = ∂Y ∂y<br />
= ɛ2 ∂A<br />
∂X ;<br />
= ɛ ∂A<br />
∂Y ;<br />
∂2A ∂x2 = ɛ4 ∂2A ∂X2 ∂2A ∂y2 = ɛ2 ∂2A ∂Y 2<br />
The first term <strong>of</strong> the MSE (3.51) can be expanded, yielding to<br />
(3.54)<br />
ccgηxx +(ccg) x ηx + ccgηyy +(ccg) y ηy + k 2 ccgη = 0 (3.55)<br />
Then substituting equation (3.52) into equation (3.55) and using the new<br />
variables (X and Y ) it yields to<br />
ccg [ɛ 4 AXX +2ɛ 2 AXik + ɛ 2 ikX − k 2 A]+ɛ 2 (ccg) X [ɛ 2 AX + ikA]+<br />
ɛ 2 (ccgAY ) Y + k 2 ccgA =0<br />
(3.56)<br />
if the terms <strong>of</strong> order O (ɛ 4 ) are neglected, the equation (3.56), after some<br />
simplifications, at the order O (ɛ 2 ) becomes<br />
2ikccgAX + i (ccgk) X A +(ccgAY ) Y = 0 (3.57)<br />
Equation (3.57) is the parabolic approximation <strong>of</strong> the MSE, its solution<br />
provides the slowly varying amplitude A (x, y), which multiplyed by<br />
Università degli Studi di Roma Tre - DSIC 37