3. Asymptotic Formulas for the Wave Profile in the Non-Uniform Depth Basin

3.1. Relationship Between Fast Oscillating and Localized Solutions

[19]  In this section we begin an asymptotic analysis of Cauchy problem (1, 2). We use here well known objects and their characteristics which one can find in books connected with the semiclassical asymptotic and ray method, geometrical optics and wave fronts, Hamiltonian mechanics, catastrophe theory etc. We try to collect here all necessary concepts and give their description in elementary form. A more complete presentation and details can be found in [Maslov, 1965; Maslov and Fedoiuk, 1981; Arnold, 2001; Babich and Buldyrev, 1991; Kravtsov and Orlov, 1990].

[20]  We introduce a parameter


expressing the relationship between the characteristic size of the source l and the characteristic length L of the interval of slow varying depth. Our asymptotic expansions are derived under the assumption that parameter mll 1 and C(x)=(gH(x))1/2 is a slowly varying function.

[21]  The problem now is to find asymptotic solutions h(x,t) to the wave equation with variable coefficient. They can be expressed by means of the wavefront formed by rays (an accurate definition is given in the next subsection). One has to introduce curved rays and characteristics given by 1-D family of trajectories P(t,y),X(t,y) of an appropriate Hamiltonian system. This Hamiltonian system can be found using a WKB expression for h= A(x,t) exp(iS(x,t)/h) (with some small artificial parameter h ) inserting it in the equation and considering the equations of zero and first orders. The first order equation is the Hamilton-Jacobi equation similar to (15) but with coefficient C(x) instead of C0.

[22]  The solutions of the corresponding Hamiltonian system define trajectories which are not straight lines as in the case with the constant coefficient C0, but are curves. We mentioned before that WKB solutions do not describe the localized solutions. In order to pass from oscillating solutions to localized ones we introduce a new variable r, put h=l/r, multiply WKB solutions by some decaying (as rto infty ) function g(r, y) and integrate this product over r from 0 to infty. We obtain a function localized in the neighborhood of the points S=0 which determine the front. The variable r and the angle y are similar to that ones appeared in the case of the basin with an uniform depth. The problem is to define the phase S(x,t) and the function g(r, y) in such a way to obtain the solution of (1, 2). The difficulty is that for t=0 the phase S corresponding to (1, 2) is not a smooth function, and the point x=0 is a (strong) focal point. Thus it is necessary to use the asymptotic representation different from WKB-solutions and we use the Maslov canonical operator [Maslov, 1965; Maslov and Fedoiuk, 1981; Dobrokhotov and Zhevandrov, 2003]. The realization of these ideas together with boundary layer expansions [Maslov, 1973; Vishik qnd Lusternik, 1962] near the wave fronts gives not only the phase S and function g(r, y)=(r)1/2 vtilde.gif(r,y) (vtilde.gif is the same as in (5)), but also global asymptotic solutions to problem (1, 2) which satisfies the initial conditions at t=0 and is correct in the cases with focal and self intersection points on the front etc (see [Dobrokhotov et al., 2006b]). Let us note that to construct the solutions with localized initial data one can try to use the asymptotics of the Green function (see e.g. [Brekhovskikh and Godin, 2006; Kiselev, 2006]), but during the realization of this approach one has to calculate quite complicated integrals (see [Dobrokhotov et al., 1991]). Our experience show that it is much easier to find explicit formulas applying the ideas mentioned above directly to the problem (1, 2). We present now the final formulas.

3.2. Rays and Wave Fronts

[23]  The Hamiltonian system in the case of a basin with non-uniform depth H(x) is:


where n(y) is the unit vector (18). Therefore, the family of trajectories X(t,y), P(t,y) of (20) starts from the point x=0 with unit momentum p= n(y) (with fixed y ). Let us indicate C(0)=C0. The Hamiltonian corresponding to (20) is H=C(x)|p|. From the conservation of the Hamiltonian on the trajectories we have the following equation


[24]  The projections x=X(y, t) of the trajectories on the plane mathbbrx.gif are called the "rays". Recall that the "front" in the plane mathbbrx.gif at the time t>0 is the curve gt ={xinmathbbr.gif2|x=X(t,y), yin [0,2p)}, (see e.g. [Arnold, 2001; Maslov, 1965]). The points on this curve are parameterized by the angle yin [0,2p). If partial X/partial yneq 0 in each point x of the wave front gt, then the wave front is a smooth curve. The points where partial X/partial y=0 are named "focal points". In these points the wave front looses its smoothness. In the situation in which the focal points appear, (they are very interesting from the point of view of tsunami), it is reasonable to introduce the concept of wave front in the phase space mathbbr.gif4p,x at the moment t>0, i.e. the curve Gt={p=P(t,y), x=X(t,y), yin [0,2p]}. We note that at least one of the component of the vector Py, Xy is different from zero and also the rays x=X(t,y) are orthogonal to the wave front gt: langleX,Xyrangle=0.

3.3. The Wave Profiles Before Critical Times.

[25]  There exist d>0 and t1>d such that a wave front exists but there are no focal points for t in [d, t1]. The first instant of time t1 at which focal points are formed is called "critical" and denoted tcr. First, we assume that dcr. In this case the asymptotic solution is derived in the following way. We define a neighborhood of the wave front for sufficiently small coordinate y, where |y| is the distance between the point x belonging to a neighborhood of the wave front and the wave front. For this aim we will take y > 0 for the external subset of the wave front and y < 0 and for the internal subset. Then a point x of the neighborhood of the wave front is characterized by two coordinates: y(t,x) and y(t,x), where y(t,x) is defined by the condition that the vector y=x-X(t,y) is orthogonal to the vector tangent to the wave front in the point X(t,y), so langle y,X<font face="Symbol">y(t,y)rangle=0.

[26]  The phase is defined by


where the second equality is a consequence of the equation (21).

[27]  Now we state the first main proposition of this paper.

Proposition 1. For tcr > t > d > 0, in some neighborhood of the wave front gt, not depending on m, h , the asymptotic elevation of the free surface, has the form:


Outside this region h=O(m3/2). The function F(z,y) is defined in (13).

[28]  In order to compute the elevation h(x,t) at the point x and time t, one has to find the trajectory of the Hamiltonian system starting from x=0 and arriving at time t in the point x. Then it is possible to compute the phase S(x,t) using the approximation written above. The trajectory is defined (see (20)) by the function H(x) and the angle y(x,t) which is the angle between the x1 axis and the ray arriving at the point x at the instant t from the origin x=0, where the ray was at the instant t=0. So y can be find by the solution of equation x=X(t,y). The solution exists and it is unique since the vector partial X/partial yneq 0 before the critical time.

[29]  Explicit formula (23) shows that the elevation of the free surface h(x,t) is defined by the form of the initial disturbance through the function F(z,y) and by the variation of the depth of the basin along the trajectories of the system.

[30]  It should be noted that despite of the simple and natural form of (23) its proof is not trivial at all. The main step is to prove the fact that the formula is the same as in the case of constant bottom, if the wave rays are found correctly.

[31]  Now we derive some consequences from formula (23). Since the phase S(x,t) is equal to zero on the wave front and |S(x,t)|/l increases rapidly going out from it, then maximum of |h| is attained in a neighborhood of the wave front. Moreover, h(x,t) can exhibit few oscillations depending on the properties of the function F(z,y) (which in turn, depend on the form of the initial disturbance, see Figures 1 and 2). The second factor in (23) can be interpreted as two dimensional analogue of the Green law, well known in the theory of tidal waves in the channels: amplitude of h increases as 1/(H(x))1/4 when the depth H(x) of the basin decreases; the factor 1/(|Xy|)1/2 is connected to the divergence of the rays, in other words if a smaller number of rays goes through a neighborhood of the point X(t,y), the smaller will be the amplitude of the wave field. The factor (H(0) / (H(X(t,y))) in the phase S(x,t) (see (22)) expresses the phenomena known as the "contraction'' of the wave profile and explains the fact that the wave length of a tsunami decreases when the wave approaches the coast.

[32]  We can imagine the following situation. Let two rays start from x= 0 with two very different angles y1 and y2, arrive to the wave front in two nearby points due to properties of the function H(x). Let also assume that the values of the function F(z,y) are very different for the angles y1, y2 and equal values of z (due to the form of the initial disturbance). Then the amplitudes of h(x,t) can be very different at these nearby points.

Figure 3
Figure 4

Figure 5
[33]  These effects are illustrated by Figures 3 and 4, where the wave field is pictured by the rays (red lines), wavefronts (blue lines) and wave profiles (green lines). The initial disturbance is shown as a black ellipse located near the origin. The black lines show the contours of H(x)= const, where dimensionless depth is H(x)= 1 +3 tanh (2x1 + x2 -11)2 / cosh 2 (4(x1 -7)2 + (x2 + 2)2/25)1/2 (3-D graph of H(x) see in Figure 5). In these figures the variables x1 and x2 are dimensionless and are equal to the same dimensional variables used in the text divided by l measured in km. The wave fronts are shown at the instants t=50 l, 100 l,...,400 l s, where l is measured in km. The wave profiles are shown at neighborhoods of the last wave fronts in the directions of the rays indicated by arrows. The numbers near the wave profiles show its maximum wave height (in m).

3.4. The Wave Profile After Critical Times.

[34]  At the instances t>tcr focal points appear on the wavefront. Now, the elevation h(x,t) of the wave in a point x belonging to a neighborhood of this point can be represented as a sum of the contributions coming from different yj(x,t), yj(x,t), and Sj(x,t) with index j, and with the so-called Maslov index mj=m(yj(x,t),t).

[35]  The Maslov index takes one of the following integer values: 0,1,2,3. It is defined in many ways and containing the topological information of the problem under considerations. We have shown in the paper [Dobrokhotov et al., 2006b] that for the problem (1, 2) one can simplify its calculation connecting mj=m(yj(x,t),t) with the Morse index which counts the number of focal point staying on a trajectory. So this is the Proposition generalizing the formula from Proposition 1:

Proposition 2. In a neighborhood of the front but outside of some neighborhood of the focal points the wave field is the sum of the fields


Outside this neighborhood of the front gt h(x,t)= O(m3/2). Again the function F(z,y) is defined in (13).

Figure 6
[36]  One can see that the indices mj change the behavior of the functions determining the wave profile (see Figure 6).

[37]  Let us emphasize that the number m has a pure topological and geometrical character and can be calculated without any relation with the asymptotic formulas for the wave field. From the Proposition 1, 2 it follows that, in order to construct the wave field at some time t in a point x, one has to know only the initial values h|t=0 and ht|t=0 and has not to know the wave field h for all previous time between 0 and t. The trajectories and the Maslov (Morse) index take into account all metamorphosis of the wave field during the evolution from zero time until time t. In the paper [Dobrokhotov et al., 2006b] some theorems have been shown for connecting those two indices and in the computer program which implements this algorithm there is a simple way for finding the focal points studying the change of the sign of the jacobian of the map. Note also that these formulas are easy to invert for finding the parameters of the shift V from the measures of the wave heights done at some stations.

3.5. Wave Field Asymptotic in a Neighborhood of Focal Point

3.5.1. Wave front singularities and focal points
[38]  To give the complete description of the asymptotic solution to problem (1, 2) one has to describe the asymptotic of the function h in the neighborhood of the focal points. These points are the singular ones on the fronts and one can see them on the Figure 5 on the upper part and on the Figure 4 near the right upper corner. They are located over underwater ridge from the Figure 5 and actually connected with the well known trapped waves. The wave field amplitude increases in the neighborhood of these points and depends on the degree of their degeneration. It seams that in real situation only the simplest situation can be realized, nevertheless we give the formulas in a general situation.

3.5.2. Focal points and coordinate system
[39]  So we consider the situation when for some t the point (PF,XF)=(P(t,yF(t)),X(t,yF(t))) corresponding to the angle yF(t) is a focal one. In this point Xy =0 and one has to use another asymptotic representation for the solution. Roughly speaking the neighborhood of the point X(t,yF(t)) on the plane mathbbrx.gif can include several arcs of gt with the angles y different from yF(t). This means that one has to take into account the contribution of all of these arcs in the final formulas for h in the neighborhood of the point x=X(t,yF(t)). The influence of nonsingular points are defined by formula (25) and the influence of the points from the neighborhood of the focal points are described by formulas (31) given below. Thus it is necessary to enumerate the focal points with nearby projections and write P(t,yFj(t)),X(t,yFj(t)). These points have the same position XF=X(t,yFj(t)), but different momentum PF=P(t,yFj(t)). To simplify the notation we discuss here the influence on h of only one focal point omitting the subindex j but keeping PF.

[40]  We present the corresponding formula under the assumption that some derivative


and the derivatives X(k)F<font face="Symbol">y=0 for 1 < k


and some characteristic quantities of the focal point (PF,XF):


[41]  Again the topological characteristic appears, i.e. the Maslov index of this focal point or its neighborhood (it is the same), but now it depends on the choice of the coordinates in the neighborhood of (PF,XF). It is natural to choose the new coordinates (x'1,x'2) associated with the nonzero vector XF= X(t,yF(t)); namely we assume that the direction of the new vertical axis x'2 coincides with the vector XF. We put k2=(k21,k22)T= XF/| XF| = XF/CF=PF CF/C0, k1=(k11,k12)T=(k22,-k21)T and introduce the new coordinates p',x' in the neighborhood of (PF,XF) in the phase space mathbbr.gif4p,x by the formulas:


[42]  It is easy to see that


3.5.3. The Maslov index of a focal point.
[43]  The determinant ytilde.gifFneq 0 in the focal point (PF,XF), hence the same inequality takes place in some of its neighborhood, thus ytilde.gif has a constant sign. On the contrary the Jacobian J changes sign in this neighborhood. We define the Maslov index m(PF,XF) of the non (completely) degenerate focal point (PF,XF)=(P,X)(t,yF(t)) as the index inline01.gif of a regular point inline01.gif in the neighborhood of (PF,XF) such that the signs of the determinants inline04.gif and inline05.gif coincide. For instance one can choose phitilde.gif=yF(t), ttilde.gif=tpmd, where d is small enough. This means that we compare the sign of J with the sign of ytilde.gif on the trajectory (P,X) crossing the curve Gt in the focal point (PF,XF) before and after this crossing.

3.5.4. The model functions and the wave profile in a neighborhood of the focal point.
[44]  Now we present the formulas for the wave field in the neighborhood of a focal point x=XF. Let us put inline06.gif and introduce the function (or more precisely the linear operator acting on the source function V(y1,y2) )


We put


Proposition 3. In a neighborhood of the front gt each focal point (PF,XF) gives the following contribution to the asymptotic values of the solution h


If several arcs of gt belong to the neighborhood of the point x, then one needs to sum over all the corresponding functions (32) and (25).


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