Exact and asymptotic analysis of waves generated by sea-floor disturbances on a sloping beach
- Arghya Bandyopadhyay^{1} and
- Maria Victoria Otero-Espinar^{2}Email author
https://doi.org/10.1186/s13661-015-0315-7
© Bandyopadhyay and Otero-Espinar; licensee Springer. 2015
Received: 29 November 2014
Accepted: 9 February 2015
Published: 26 March 2015
Abstract
A two-dimensional problem of small amplitude waves generated by some sea-bottom disturbance is studied on a sloping beach. The exact analytical solution for the wave height is provided, and with a periodic ground motion the asymptotic analysis of the waves is shown to exist for all time even in the vicinity of the shoreline. The novelty of this work lies in solving the corresponding Fredholm integral equation of the first kind and also to provide a uniform asymptotic estimate of the wave integral in the unsteady state involving both pole and saddle points for which Van der Waerden’s method is used. Within the framework of linear irrotational theory, explicit integral solutions of waves and their asymptotic and/or numerical computations presented here aim to provide an equivalent mathematical understanding to all such wave propagation which can be modelled in two dimensions at the open sea.
Keywords
small amplitude waves asymptotic analysis Van der Waerden’s method1 Introduction
The subject of waves on a fluid of variable depth has involved wide attention in a renewed way with the increased interest of researchers studying extreme waves, like tsunami, particularly to improve the related knowledge in an attempt to reduce the gap between theory and application [1–6]. In this article we deal with a two-dimensional problem of surface waves generated by a time dependent sea-bottom upheaval on a beach with uniform slope. Although we are not dealing with tsunami waves as such, at the outset it is sensible to note that tsunami waves in the mid ocean are, in fact, small amplitude waves moving with tremendous velocity. In the last ten to fifteen years, we have seen many works but probably not many theoretical works which analyse asymptotically waves on beaches with respect to space-time parameters; the most illuminating analytical works are those of Ehrenmark [7] and Constantin [8], the latter of which deals with non-linear governing equations in a Lagrangian approach. We are not here to discuss wave-current interaction on variable bed, but it is also another important aspect in which questions are many and answers are inadequate [9–11]. The scarcity of analytical solutions in this type of motion is perhaps due to the complex hydrodynamics involved, and even if we have the formal solution, the analysis of wave integrals is difficult, to say the least. Needless to say that numerical computation of wave integrals has its own demerits as a serious error may creep in due to the oscillatory nature of the integrals. In this paper we have resorted to asymptotic analysis of waves in the unsteady state generated by a ground motion perfectly arbitrary except for the periodicity. Suppose the ocean to be initially at rest and free from any underlying current. We assume that waves are excited only by the action of a time-dependent deformation of sea-bottom at time \(t = 0\), without any initial distribution of surface displacement and velocity. The governing equations and conditions used for the problem are usual linearised equations and conditions for small amplitude waves. Additional conditions such as boundedness conditions at infinity in the horizontal direction, usually required, are imposed. There is, of course, no a priori method of selecting mathematical solution best representing the physical phenomena but the approach to the solution here runs as follows: with a slight change of notation, we have used complex potential, and extensive application of integral transform is made. The main hurdle to a formal solution, as will be evident from the discussion in the subsequent sections, lies in the solution of a Fredholm integral equation of the first kind. For this we demonstrate that the solution is obtained for a 45^{∘} beach angle; for other beach angles, the present problem still remains unsolved.
Assuming an arbitrarily distributed ground motion oscillatory in time, the formal solution to the problem is subjected to an asymptotic analysis by Van der Waerden’s method [12] which provides uniform asymptotic estimate of the wave height in an unsteady motion even in the vicinity of the critical line. For an exponentially distributed ground motion, we show that it is possible to extract the exact course of variation of wave height at the origin (that is, at the shore line) with time t. In a concluding section some interesting features of the wave motion are discussed with interpretations and illustrations of the analytical and asymptotic results found.
The linear theory of generation of small amplitude waves in a potential problem is justified if the length scales in horizontal and vertical directions are comparable. Linear theory of wave generation and propagation is realistic even with tsunami waves as we find many tsunami source regions are elliptical, with a major axis beyond hundreds of kilometres corresponding to the active part of the fault. The fact that most of the tsunami energy is transmitted at right angles to the major axis allows researchers to regard in this context the propagation of the tsunami in the open sea as being two-dimensional [13]. By this we surmise that mathematical analysis of similar nature may be adopted while analysing wave integrals for the tsunami wave propagation at the open sea where the wave amplitudes are small. Last but not the least, for understanding and analysing ocean waves, some books and articles in dynamical oceanography remain authoritative even with the passing of time and should be mentioned [14–17].
2 The problem
3 The solution of the problem
- (i)
exist for \(\gamma>0\),
- (ii)
admit of a Fourier cosine transform with the parameter \(2m^{2}\ge0\), and
- (iii)
this transform, when used for \(B(m, t)\) in (3.4), makes the integral uniformly convergent in \(x\ge0\).
4 Detailed asymptotic analysis of η for a time-periodic ground motion
This is because the principal values (for \(k = 1\)) or ordinary values (for \(k =2\)) of the integrals on the right-hand side clearly exist. At \(\sigma=\omega\), the integrand in each term on the right-hand side is either continuous without a saddle point or has a pole (\(k = 1\)), while in the second term of \(I_{jk}\), there is a saddle point of combination \(mx - \sigma t\) only when \(|k-j|=1\). In the first three cases (that is, in \(I'_{12}\), \(I_{22}\), first part of \(I_{12}\)), the asymptotic contribution for large x and t may be determined by the method of integration by parts.
For the second part of the integral \(I_{21}\), the integrand has both a pole and a saddle point at \(s = gt / 2x\), which coincide with \(x = gt / 2\sigma\). We use a method essentially due to Van der Waerden for finding out the asymptotic contribution of this part which remains uniformly valid even in the neighbourhood of the strip \(x = gt / 2\sigma\).
5 Displacement at the origin
6 Physical interpretation of the results obtained in Sections 4 and 5 with numerical illustrations
The results obtained in (4.13)-(4.15) and (5.3)-(5.5) are particularly of interest. Asymptotic analysis done in (4.14) and (4.15) is obviously for different ranges of values for x, and we have considered the disturbance model as prescribed in (5.1).
The numerical computations done here are all with a hypothetical disturbance as mentioned in (5.1) at the sea-bed with values of the parameters being \(a = 0\), \(b = \sqrt {2}\), \(g=10\), \(\omega= 0.5\), \(k = 0.005\).
Mathematica software is used for the relevant programming to compute and illustrate the graphs.
7 Conclusion
The problem that we have examined here is a potential problem on a beach where waves are generated by a disturbance at the sea-bed with very little restrictions on the nature of the disturbance except for the oscillation. The solution and asymptotic analysis of waves are discussed at length. The beach angle here is 45^{∘}, which is a severe restriction in regard to a real life situation. The first author has found the solution for a similar type of problem for other beach angles too and also with a non-uniformly sloping beach of the form \(y = - qx^{r}\) but with the shallow water regime [6]. But with the potential problem the solution in the unsteady state for any arbitrary beach angle is not known to have been realised. Extracting wave form from a wave integral is always difficult in a mathematical context. The analysis presented here has addressed two basic issues. Firstly, it has shown the way to reach an exact solution when two decisive points, say one saddle point and a pole or two saddle points in a wave integral are in proximity. This situation is very common in the asymptotic analysis of wave integrals. Our results show quite categorically the nature of waves at large times and long distances in such a situation. From mathematical point of view, this is quite interesting. Secondly, in an oceanographic context these results may be used for all such similar cases of wave generation due to bottom disturbances where, under the influence of many parameters, an identical situation crops up and analytical solution fails to analyse the situation. For example, extreme waves like tsunami are essentially small amplitude waves at the deep sea so their analysis may be helpful in tsunami wave analysis more so as tsunamis are influenced by many oceanographic parameters. For the second point though we place one caveat that we have not dealt here with tsunami waves as such.
Declarations
Acknowledgements
The work is done in the period of an invited stay of the first author in the framework of Research Excellence Program USC - India (PEIN). Authors are deeply indebted to Professor AR Sen of the Department of Mathematics, Jadavpur University, Calcutta, for his help and suggestions. This research has been partially supported by Ministerio de Economía y Competitividad, project MTM 2010-15314, and Xunta de Galicia and FEDER.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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