The problem of generation and propagation of tsunami waves is mainly focused on plane beach, there are very few analytical works where wave generation is considered on non-uniformly sloping beach and as a result those works might have failed to capture important facts which are influenced by bottom-slope of the beach. Some researchers provided solution to the forced long linear waves but on a beach with uniform slope while the importance of including variable bottom topography is mentioned by few researchers but they also stayed away from considering continuous variability of the ocean bed as they were studying runup problem. This paper analyzes tsunami waves which are generated by instantaneous bottom dislocation on a ocean floor with variable slope of the form y=-qxr, q > 0, r > 0. Attempts are made to find analytical solution of the problem and along the way tsunami forerunners are identified while investigating the short time wave behavior, not found even with constant slope beaches. In our study a rather significant phenomenon with regard to energy transmission to the waves at steady-state are observed with some notable features.
The evaluation of the terminal effect of natural hazards remains one of the holy grails of geophysical research [
We take the vertical upward direction as the y-axis, and the undisturbed horizontal surface of the sea as the xz -plane of which the axis Oz is along the shoreline. The sea is supposed to be bounded by a beach of variable slope given by the equation y = h0 (x) at equilibrium (
We assume a two-dimensional motion in which long waves are exited by a sudden bottom upheaval of height h0 (x, t) accompanied by an initial surface displacement h1(x) together with an initial vertical surface velocity h2 (x). If u(x, t) is the horizontal velocity, h(x, t) is the surface displacement and
is the depth at the point x, at time t > 0, the non-linear shallow water equations are
At t = 0-, we have
If and are small compared to and is small compared with the local wave speed, Equations (2) and (3), after using (1), may be linearised to
Eliminating u(x, t) from (5) and (6), and using suffix notation for partial differentiation, we obtain the partial differential equation satisfied by:
when and are given, it is required to determine as the solution of (7) subject to the initial condition (4). The horizontal velocity u is then found from (5); for this purpose, we may impose a physically reasonable boundary condition at x = 0, namely
when, q > 0, r > 0, Equation (7) suggests that we consider the solution of the ordinary differential equation
for the determination of.
For, the general solution of this equation is [
where denote respectively Bessel functions of first and second kind of order, and
For, that is r = 2 the general solution of (9) is
Equations (10) and (12) show that cannot be both finite at (in other words, and u cannot be both finite at x = 0 unless
, and. (13)
To solve the Equation (7) subject to the given initial and boundary conditions, we assume that
Using this in (7), we obtain, by means of (9) and (10), with, the integral equation of first kind
where
Then solution of h is obtained with the help of Hankel inversion theorem [
where
where H is the Heaviside unit function.
We note that for r =1 this expression reduces to that of h found for constant slope beach [
Taking a time-dependent bottom dislocation like the following
we have been able to evaluate the above integral for all time t with the help of a very nice result [11, p. 58]
.
In the above the t-integral reduces to
where
Following this evaluation of the t-integral of (17) we spilt the x-integral of (17) in two parts one from x = 0 to
[the first part] which can be evaluated with the help of another result that combines product of two Bessel functions as a terminating hypergeometric series [11, p. 11, 7.2.7 (47)]
This spilt corresponds to h11, say, of h which after some tricky manipulation takes a nice form and it corresponds to the free vibration that can be treated as the forerunners. These waves in this spectrum dominate for first few minutes, to be precise for the half period of the quake forcing.
where
and
On the other hand, the second part of x-integral from x = x0 to ¥ contribute h12, say, of h representing the forced wave part is also analytically calculated with its expression being a little complicated is not shown here. We hope to analyze their character in a subsequent paper. At this stage, we remark qualitatively that h12 catch up the free waves beyond half period t and dominate the wave spectrum gradually for t > t.
Before we proceed further and discuss the steady-state nature of the waves and the energy transmission let us provide some illustrative figures showing the nature of h11 and h12 in an attempt to distinguish them for small time. We will take for a particular type of bottom dislocation of the following form
with, and
The following two figures (Figures 2 and 3) depicting h11 and h12 for small time when r = 0.7 and r = 0.8, that is for two different sloppiness of the ocean bed and in both the cases we find the prominence of h11 over h12 as it was shown analytically and we call this h11 as the forerunners.
The next two graphs (Figures 4 and 5) illustrate nature of h for two different values of r, namely r = 0.7 and r = 0.8,
here we find h increasing indefinitely with increase of time. Our analytical result also has shown this. It indicates that there might be some sort singularity at t = t, the reason of which may be the sudden disappearance of the bottom vibration at t = t. For any such definite conclusion, though, we require further analytical investigation of the motion for t > t.
The spilt of h namely h12 which comes from the second part of x-integral in (17) while integrating it from x = x0 to ¥ consists of three parts: one of which has a wave form and the other two are standing disturbances, analytical expressions of which is valid for 2/3 < r < 4/3. We restrain ourselves of writing those complicated expressions rather give some illustration of h12 below for two different arbitrary sloppiness of the ocean floor.
The Figures 6 and 7 indicate the nature of h12 in the wave spectrum it actually corresponds to the forced wave part of h which will dominate the spectrum over those which are small and corresponds to the natural frequencies of wave motion.
We assume
,;
, t > 0 and show that a steady state exists and also determine the corresponding values of and u.
Steady-State Value of (When the integration with respect to s in (17) is completed, we get
where
We spilt the -range in (21) into the sub-intervals and. By the help of known results on Fourier integrals, the part of the integral in (21) over the interval is asymptotically equal to
The remaining part of the integral in (21) is written as
Combining (24) and (25), we get for the integral in (21) the expression
Here the symbol indicates the Cauchy Principal value of the integral in question. Following Bochner [
The results in (27) hold provided 1) is differentiable with respect to in;
2) exists;
3) and are each absolutely integrable in.
Equation (21) then gives
Writing, we have
By Oberhettinger, F. [13, Section 4.32] the second term is
where denotes Bessel function of the second kind, and is Lommel’s function. We note that since 0 < r < 2 in our discussion, imposes the further restriction.
Using the results (21b) and (21c) in (21a), we finally obtain the steady-state value of:
For the same periodic ground motion and following the same procedure as described above to obtain, we find the steady-state value of u as
Here denotes Bessel function of the second kind, and is Lommel’s function. The first term of both and, as given below, represent progressive waves:
We also note that h* is an integral of the hyperbolic equation
The rest part of as well as represent clearly standing waves. Since g > 0 in our case, we may use the asymptotic expansion of for z ≥ 1 to obtain h* for large x:
The wave described by (28) propagates towards x ® +¥ according to the equation
Thus this wave moves with a variable acceleration unless r = 1 when the acceleration is constant [cp. [
(which is equivalent to). Since the depth increases as x, this corresponds to Green’s law of shallow water waves.
A notable feature of the steady-state solution is that no energy is transmitted through the liquid for frequencies
which make, and hence h* = 0, u*
= 0, the part and being a standing wave. That these critical frequencies may form a countably infinite set as shown by the following example:
with
Then
The zeros of, for and x real, are known to be countably infinite. If be the n-th position zero of, the critical frequencies are given by
After the evaluation of s-integral in (17) the splitting of the x-integral in two parts is not arbitrary but instead a necessary one, so the existence of the forerunners. Tsunami forerunners arrive before the arrival of the main tsunami waves with typically smaller amplitude, existence of such waves and a correct analytical expression of which is tried to put up here over a variable sloping beach. Another interesting observation is that the choice of the depth profile, at distance x from the shoreline, makes the dispersive relation [viz. Equation (16)] dispersive while in the classical shallow water theory it is non-dispersive. Leaving aside the actual physical dislocation of the sea floor the solution provided here is correct for all t. In fact if we apply the sea bed deformation due to earthquake as given by Okada’s solution [
The author is deeply indebted to Prof. (Retd.) Asim Ranjan Sen, Department of Mathematics, Jadavpur University, Calcutta, India for his help and suggestions during the preparation of this paper. The author is also grateful for the anonymous reviewer for suggesting some improvements which have since been incorporated in the paper.
Some portion of this work was presented by the author at the ISOPE 2012 conference [