Linear wave reflection by trench with various shapes

Abstract

Two types of analytical solutions for waves propagating over an asymmetric trench are derived.One is a long wave solution and the other is a mild-slope solution which is applicable to deeper water. The water depth inside the trench varies in proportion to a power of distance from the center of the trench (where the center means the deepest water depth point and the origin of -coordinate in this study). The mild-slope equation is transformed into a second order ordinary differential equation with variable coefficients based on the longwave assumption or Hunt's (1979) Reply to reviewers' comments on revision of "Linear Wave Reflection by Trench with Various Shapes" by Tae-Hwa Jung, Kyung-Duck Suh, Seung Oh Lee, and Yong-Sik Cho First, we appreciate the reviewers' interest and comments on our manuscript. In the following, we provide responses to the reviewers' comments in a question-answer format. The answers are typed in italic. The page numbers and figure numbers referred to in this reply are those in the revised manuscript in which all numbers of equations were rearranged according to the revision. * Detailed Response to ReviewersAnswer to comments of reviewer 1 I suggest that this paper can be accepted for publication if the following minor modification is made. Minor modification:Please delete all the contents after Eq.(B-8) in Appendix B. The reason is 1) It is unnecessary to expand A(x), B(x), C(x). 2) Even if they need to be expanded, those expansions can be obtained very easily by using the Maple or some other softwares. A) We removed all the contents after Eq. (B-8) as the reviewer suggested.Answer to comments of reviewer 2 1. General comments. The authors have taken care of almost all of my remarks. However, they still want to stick to their extended solutions. They only changed their model into solution. I strongly object to the use of the notation of extended solution to deeper water in this paper. I said it before, the solutions of the mild-slope equation are valid for both deep and shallow water.In the paper two different models are considered:• The long-wave model as such, and Under the condition that the notation and wording of extended is removed from the paper and heed is taken on my previous remarks on this matter, the paper may be published, but only under this condition.We removed the word 'extended' in the paper and replaced following expressions as reviewer suggested. The extended solution  the mild-slope solution The other is a solution extended to deeper water  the other is the mild-slope solution

Transformation of Long Surface and Tsunami-Like Waves in the Ocean with a Variable Bathymetry

“…We have studied also wave scattering on underwater barriers and trenches whose slopes can be described by the same functions (linear, piecewise-quadratic, and tanh-functions). The results obtained are in a good agreement with the results previously derived by different authors for the similar models (Kajiura Massel 1989;Mei 1990;Dingemans 1997;Rey 1992;Rey et al 1992;Ka ˆnog ˇlu and Synolakis 1998;Jung et al 2008;Xie et al 2011). In the limiting case when the wave frequency goes to zero we obtained the same transformation coefficients which are predicted by Lamb's theory (1932) for step-wise bottom.…”

“…Besides these limiting cases the problem of long wave propagation can be solved analytically for some particular bottom profiles (Dean 1964;Mei 1990; Ka ˆnog ˇlu and Synolakis 1998; Jung et al 2008;Xie et al 2011) (see also Dingemans 1997), moreover for the linear bottom profile there is an exact solution of linearised hydrodynamic equations for waves of arbitrary wavelength (Stoker 1957;Sretenskii 1977). It has also been shown that in some particular cases exact solutions can be obtained for the reflectionless propagation of long linear and nonlinear waves in a fluid with a special bottom profiles (see Pelinovsky 2013, 2016 andreferences therein).…”

Asymptotic Formulas for the Reflection/Transmission of Long Water Waves Propagating in a Tapered and Slender Harbor

“…On the other hand, the linear long water wave approximation has been used to estimate the reflection and transmission coefficients for different submerged trapezoidal breakwaters, Chang and Liou [14]. In this direction, Jung et al [15] obtained an approximated analytical solution for the mild-slope equation, applied to the analysis of wave reflection for an asymmetric trench. The aforementioned work was improved by Xie and Liu [16] by obtaining an exact analytical solution to the modified mild-slope equation.…”

On the weak viscous effect of the reflection and transmission over an arbitrary topography

“…Jung and Suh 21 further took the high-order terms of the bottom effect into account and obtained an analytical solution for long waves propagating over an axisymmetric pit. Moreover, analytical solutions of waves propagating over trenches of various shapes were also investigated by Jung et al 22 In addition, Bender and Dean 23 used several different methods to study the reflection and transmission of normally incident waves travelling over two-dimensional trenches and shoals of finite width with sloped transitions between two different depths. Madsen et al 24 used a spatial expansion to derive a new equation based on a Boussinesq-type method for investigating reflection and transmission for different geometric bottom topographies.…”