Wave scattering by uneven porous bottom in a three layered channel

Abstract

The scattering of plane surface waves by bottom undulations in a three-layer channel of different constant densities is investigated here by using a simplified perturbation analysis. In such a three layer fluid there exists waves of two different modes propagate along each of the interfaces. In the process of obtaining solution for the problem a Fourier transform technique is employed to derive the first-order corrections of the reflection and transmission coefficients in terms of integrals involving the shape function representing the bottom elevation. For sinusoidal undulations, these coefficients are plotted graphically to illustrate the energy transfer between the waves of different modes induced by the bottom undulations. It is shown that the scattering phenomena effected by the change in different parameters, viz, the porous effect parameter, the number of ripple present at the bottom and the depth of the lower layer.

Appendix

1.1 Roots of the dispersion equation

We find the number of real and purely imaginary roots of the dispersion equation in a three layered finite depth water wave scattering by undulating porous bed given in Eq. (6). The dimensionless dispersion equation can be written as

$$\begin{aligned} F_1(x)=F_2(x) \end{aligned}$$

(55)

where

$$\begin{aligned} F_1(x)= & {} (x-G_1\coth {xh_1})[(1-s_1)(1-s_2)x^2\nonumber \\&-K_1 x\{(1-s_1s_2)\coth {xh'_1}+(1-s_1)s_2\coth {x(H_1-h'_1)}\}\nonumber \\&+s_1 K_1^2\{1+s_2\coth {xh'_1}\coth {x(H_1-h'_1)}\}] \end{aligned}$$

(56)

$$\begin{aligned} F_2(x)= & {} K_1(G_1-x\coth {xh_1})\nonumber \\&\left[ K_1\{\coth {xh'_1}+s_2\coth {x(H_1-h'_1)}\}-(1-s_2)x\right] \end{aligned}$$

(57)

where $G_1=Ga$, $K_1=Ka$, $h_1=\frac{h}{a}$, $h'_1=\frac{h'}{a}$, $H_1=\frac{H}{a}$, a being the amplitude of the ripple.

The plot of the functions $F_1(x)$ and $F_2(x)$ are depicted in Fig. 23 for $G_1=0.05$, $K_1=0.3$, $h_1=5$, $h'_1=2$, $H_1=5$, $s_1=0.5$ and $s_2=0.7$. Figure 23 showing the fact that, the equation have only two real roots given by ±0.870283 and ±1.70393.

Fig. 23

Real roots of the dispersion equation

Fig. 24

Purely imaginary roots of the dispersion equation

Similarly, the pure imaginary roots of the dispersion equation (6) are shown in Fig. 24. In this case we take $x=i\widehat{x}$ and plotting two curves as a function of $\widehat{x}$ for the same values of the dimensionless parameters as in Fig. 23, where the functions $\Gamma _1$ and $\Gamma _2$ are given by

$$\begin{aligned} \Gamma _1(\widehat{x})= & {} (\widehat{x}+G_1\cot {\widehat{x}h_1})\nonumber \\&[(1-s_1)(1-s_2)\widehat{x}^2+K_1\widehat{x}\{(1-s_1s_2)\cot {\widehat{x}h'_1}\nonumber \\&+(1-s_1)s_2\cot {\widehat{x}(H_1-h'_1)}\}\nonumber \\&-s_1 K_1^2\{1-s_2\cot {\widehat{x}h'_1}\cot {\widehat{x}(H_1-h'_1)}\}] \end{aligned}$$

(58)

and

$$\begin{aligned} \Gamma _2(\widehat{x})= & {} K_1(G_1-\widehat{x}\cot {\widehat{x}h_1})\nonumber \\&\left[ K_1\{\cot {\widehat{x}h'_1}+s_2\cot {\widehat{x}(H_1-h'_1)}\}+(1-s_2)\widehat{x}\right] \end{aligned}$$

(59)

From Fig. 24 it is clear that these two curves intersect at infinitely many number of points and consequently, the dispersion equation has an infinite number of purely imaginary roots.

It is possible to show that no other root exists for dispersion equation (6) by using Rauche’s theorem (cf. Das and Mandal [6]).