Integrals with a large parameter: water waves on finite depth due to an impulse

Clarisse, Jean-Marie; Newman, J. N.; Ursell, F.

doi:10.1098/rspa.1995.0072

Cited by 13 publications

(21 citation statements)

References 5 publications

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“…(8) are available (see Mei, 1989;Clarisse et al, 1995). However, it is more instructive to study sources which are more realistic representations of submarine earthquakes.…”

Section: Portugal (1969)mentioning

confidence: 99%

Dispersion of tsunamis: does it really matter?

Glimsdal

Pedersen

Harbitz

et al. 2013

Nat. Hazards Earth Syst. Sci.

192

131

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Abstract. This article focuses on the effect of dispersion in the field of tsunami modeling. Frequency dispersion in the linear long-wave limit is first briefly discussed from a theoretical point of view. A single parameter, denoted as "dispersion time", for the integrated effect of frequency dispersion is identified. This parameter depends on the wavelength, the water depth during propagation, and the propagation distance or time. Also the role of long-time asymptotes is discussed in this context. The wave generation by the two main tsunami sources, namely earthquakes and landslides, are briefly discussed with formulas for the surface response to the bottom sources. Dispersive effects are then exemplified through a semi-idealized study of a moderate-strength inverse thrust fault. Emphasis is put on the directivity, the role of the "dispersion time", the significance of the Boussinesq model employed (dispersive effect), and the effects of the transfer from bottom sources to initial surface elevation. Finally, the experience from a series of case studies, including earthquake- and landslide-generated tsunamis, is presented. The examples are taken from both historical (e.g. the 2011 Japan tsunami and the 2004 Indian Ocean tsunami) and potential tsunamis (e.g. the tsunami after the potential La Palma volcanic flank collapse). Attention is mainly given to the role of dispersion during propagation in the deep ocean and the way the accumulation of this effect relates to the "dispersion time". It turns out that this parameter is useful as a first indication as to when frequency dispersion is important, even though ambiguity with respect to the definition of the wavelength may be a problem for complex cases. Tsunamis from most landslides and moderate earthquakes tend to display dispersive behavior, at least in some directions. On the other hand, for the mega events of the last decade dispersion during deep water propagation is mostly noticeable for transoceanic propagation.

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“…(8) are available (see Mei, 1989;Clarisse et al, 1995). However, it is more instructive to study sources which are more realistic representations of submarine earthquakes.…”

Section: Portugal (1969)mentioning

confidence: 99%

Dispersion of tsunamis: does it really matter?

Glimsdal

Pedersen

Harbitz

et al. 2013

Nat. Hazards Earth Syst. Sci.

192

131

View full text Add to dashboard Cite

show abstract

“…After substituting (15) and (14) in (6-10), we equate coefficients at each power of . Moreover, coefficients depending on τ and t are equated separately.…”

Section: Formal Asymptotic Expansionmentioning

confidence: 99%

“…We developed the following algorithm for finding terms in the asymptotic expansion (15). Solutions to the recurrent sequence of timeindependent boundary-value problems (26), (27) must be found first, and then (25) determines ϕ m , m = 2, 3, .…”

Section: Formal Asymptotic Expansionmentioning

confidence: 99%

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Asymptotic analysis of rapid forward accelerations of a free-surface pressure

Kuznetsov

2006

J Eng Math

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Linear unsteady water waves due to a pressure distribution in the forward motion over the free surface of an inviscid, incompressible, heavy fluid are considered. The motion of pressure system is assumed to be rectilinear and non-uniform starting from rest. The effect of a rapid acceleration is analysed asymptotically. A two-scale expansion is developed for the velocity potential, and estimates for the remainder are established. Hydrodynamic corollaries are derived from the asymptotics obtained. In particular, it is shown how the resistance, which is the horizontal component of the fluid's reaction to the system's motion, depends on the bottom topography varying in the direction of motion.

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Uniform asymptotic approximations for transient waves due to an initial disturbance

et al. 2016

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In this work, we first present a semianalytical method for the evolution of linear fully dispersive transient waves generated by an initial surface displacement and propagating over a constant depth. The procedure starts from Fourier and Hankel transforms and involves a combination of the method of stationary phase, the method of uniform asymptotic approximations and various Airy integral formulations. Second, we develop efficient convolution techniques expressed as single and double summations over the source area. These formulations are flexible, extremely fast, and highly accurate even for the dispersive tail of the transient waves. To verify the accuracy of the embedded dispersion properties, we consider test cases with sharp‐edged disturbances in 1‐D and 2‐D. Furthermore, we consider the case of a relatively blunt Gaussian disturbance in 2‐D. In all cases, the agreement between the convolution results and simulations with a high‐order Boussinesq model is outstanding. Finally, we make an attempt to extend the convolution methods to geophysical tsunami problems taking into account, e.g., uneven bottom effects. Unfortunately, refraction/diffraction effects cannot easily be incorporated, so instead we focus on the incorporation of linear shoaling and its effect on travel time and temporal evolution of the surface elevation. The procedure is tested on data from the 2011 Japan tsunami. Convolution results are likewise compared to model simulations based on the nonlinear shallow water equations and both are compared with field observations from 10 deep water DART buoys. The near‐field results are generally satisfactory, while the far‐field results leave much to be desired.

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Integrals with a large parameter: water waves on finite depth due to an impulse

Cited by 13 publications

References 5 publications

Dispersion of tsunamis: does it really matter?

Dispersion of tsunamis: does it really matter?

Asymptotic analysis of rapid forward accelerations of a free-surface pressure

Uniform asymptotic approximations for transient waves due to an initial disturbance

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