Head-on Collision Between Two Hydroelastic Solitary Waves in Shallow Water

Abstract

A comprehensive theoretical study on the head-on collision between two solitary waves in a thin elastic plate floating on an inviscid fluid of finite depth is investigated analytically by means of a singular perturbation method. The effects of plate compression are also taken into account. The Poincaré–Lighthill–Kuo method has been used to derive the solution up to the fourth order of the resulting nonlinear differential equation, which in principle gives the asymptotic series solution. It is found that after collision, both the hydroelastic solitary waves preserves their original shape and positions. However a collision does have imprints on colliding waves with non-uniform phase shift up to the third order which creates tilting in the wave profile. Maximum run-up amplitude, wave speed, phase shift and distortion profile have also been calculated and plotted for two colliding solitary waves.

Appendices

Appendix A: Equation of $O(\epsilon ^3 )$

$$\begin{aligned}&4 \bar{k} \frac{\partial \alpha _2}{\partial \eta } + k\frac{\partial \alpha _1}{\partial \xi } \left( 4 \bar{k} \frac{\partial \theta _0}{\partial \eta } - 2aR_1 + 3\alpha _0 -\beta _0 \right) + \bar{k}\frac{\partial \alpha _1}{\partial \eta } \big (2bR_1+3\alpha _0-\beta _0\big ) \nonumber \\&\quad -\, (\alpha _0 + \beta _0 ) \left( k \frac{\partial \beta _1}{\partial \xi } + \bar{k}\frac{\partial \beta _1}{\partial \eta }\right) + \frac{H_0^2}{3} \left( k^3 \frac{\partial ^3}{\partial \xi ^3} - 2\bar{k}^3\frac{\partial ^3}{\partial \eta ^3} \right) (\alpha _1-\beta _1)\nonumber \\&\quad -\, \bar{k} \frac{\partial \beta _0}{\partial \eta } \left[ -2\bar{k}^2 H_0^2 \frac{\partial ^2 \beta _0}{\partial \eta ^2} + \frac{1}{3}k^3H_0^2\frac{\partial ^3 \varphi _0}{\partial \xi ^3} + k\frac{\partial \varphi _0}{\partial \xi } (\alpha _0 +\beta _0) + \alpha _1+ \beta _1 \right] \nonumber \\&\quad +\, k \frac{\partial \alpha _0}{\partial \xi } \left[ k^2 H_0^2 \frac{\partial ^2 \alpha _0}{\partial \xi ^2} + \bar{k}^2 H_0^2 \frac{\partial ^2 \beta _0}{\partial \eta ^2} - \frac{2}{3}\bar{k}^3H_0^2\frac{\partial ^3 \theta _0}{\partial \eta ^3} \right. \nonumber \\&\quad \left. +\, 4 \bar{k} \frac{\partial \theta _1}{\partial \eta } + 2 \bar{k} R_1 \frac{\partial \theta _0}{\partial \eta } -2a^2 R_2 + \bar{k} \frac{\partial \theta _0}{\partial \eta } (3\alpha _0 - \beta _0 ) + 3 \alpha _1 - \beta _1\right] \nonumber \\&\quad +\, k^3 H_0^2 \frac{\partial ^3 \alpha _0}{\partial \xi ^3 } \left( \frac{1}{2}aR_1 + \beta _0 \right) + k^2 \bar{k}^2H_0^2 \frac{\partial ^2 \alpha _0}{\partial \xi ^2} \frac{\partial ^2 \theta _0}{\partial \eta ^2} + \bar{k}^3 H_0^2\nonumber \\&\quad \times \left( \frac{1}{2} bR_1 + \beta _0 + 2\alpha _0 + k \frac{\partial \varphi _0}{\partial \xi } \right) \nonumber \\&\quad - \,\frac{H_0^4}{30} \left( k^5 \frac{\partial ^5 \alpha _0}{\partial \xi ^5} + \frac{3}{2}\bar{k}^5 \frac{\partial ^5 \beta }{\partial \eta ^5} \right) + \Gamma H_0^4 \left( k^5 \frac{\partial ^5 \alpha _0}{\partial \xi ^5} + \bar{k}^5 \frac{\partial ^5 \beta _0}{\partial \eta ^5} \right) \nonumber \\&\quad +\, \sigma H_0^2 \left\{ \left( k \frac{\partial }{\partial \xi } + \bar{k} \frac{\partial }{\partial \eta } \right) \left[ \left( k^2\frac{\partial ^2 }{\partial \xi ^2} + \bar{k}^2 \frac{\partial ^2}{\partial \eta ^2}\right) (\alpha _1+\beta _1) \right. \right. \nonumber \\&\quad \left. \left. +\, 2\bar{k} k \left( k \frac{\partial \theta _0}{\partial \eta }\frac{\partial ^2 \alpha _0}{\partial \xi ^2} +\bar{k} \frac{\partial \varphi _0}{\partial \xi } \frac{\partial ^2 \beta _0}{\partial \eta ^2} \right) \right] + \bar{k} k \left( k^3 \frac{\partial \theta _0}{\partial \eta } \frac{\partial ^3 \alpha _0}{\partial \xi ^3} + \bar{k}^3 \frac{\partial \varphi _0}{\partial \xi } \frac{\partial ^3 \beta _0}{\partial \eta ^3} \right) \right\} \nonumber \\&\quad +\, \Lambda H_0^2 \left\{ \left( k \frac{\partial }{\partial \xi } + \bar{k} \frac{\partial }{\partial \eta }\right) \left[ \left( k^2 \frac{\partial ^2 }{\partial \xi ^2 } + \bar{k}^2 \frac{\partial ^2}{\partial \eta ^2} \right. \right. +\, 2\bar{k} k \frac{\partial ^2}{\partial \xi \partial \eta }\right) (\alpha _1+\beta _1) \nonumber \\&\quad \left. \left. +\, 2\bar{k} k \left( k \frac{\partial \theta _0}{\partial \eta }\frac{\partial ^2 \alpha _0}{\partial \xi ^2} +\bar{k} \frac{\partial \varphi _0}{\partial \xi } \frac{\partial ^2 \beta _0}{\partial \eta ^2} \right) \right] - \bar{k} k \left( k^3 \frac{\partial \theta _0}{\partial \eta } \frac{\partial ^3 \alpha _0}{\partial \xi ^3} + \bar{k}^3 \frac{\partial \varphi _0}{\partial \xi } \frac{\partial ^3 \beta _0}{\partial \eta ^3} \right) \right\} =0. \end{aligned}$$

(A1)

Appendix B: Expressions of the Coefficients

$$\begin{aligned} c_0&= \frac{3}{4\gamma } \left( \frac{2}{3}+\sigma +\Lambda \right) . \end{aligned}$$

(B1)

$$\begin{aligned} c_1&= \frac{3}{10}-\frac{9\Gamma }{\gamma ^2}+\frac{1}{4\gamma }[4c_0 (3\sigma +3\Lambda -1)-3-2\sigma ], \end{aligned}$$

(B2)

$$\begin{aligned} c_2&= \frac{1}{8\gamma ^2}\left[ 540\Gamma -\gamma \lbrace 17+6\sigma +12\Lambda +4c_0 (45\sigma -15+45\Lambda +2\gamma ) \rbrace \right] , \end{aligned}$$

(B3)

$$\begin{aligned} c_3&= \frac{1}{8\gamma } \left[ -540\Gamma +\gamma \lbrace 25+\gamma +15\gamma (\sigma +\Lambda )+12c_0 (15\sigma -5+15\Lambda +\gamma ) \rbrace \right] , \end{aligned}$$

(B4)

$$\begin{aligned} c_4&=\frac{ 1 }{4}\left( \frac{1}{12}+\frac{11}{2\gamma }+c_0-\frac{c_3}{6}-\frac{5c_3}{4\gamma } \right) , \end{aligned}$$

(B5)

$$\begin{aligned} c_5&=\frac{2}{3}c_2+c_3, \end{aligned}$$

(B6)

$$\begin{aligned} c_6&=\frac{27 }{32 \gamma ^2}, \end{aligned}$$

(B7)

$$\begin{aligned} c_7&= \frac{1}{4} \left( -1+\frac{27}{32\gamma }(\sigma +\Lambda )-\frac{21}{4\gamma } -\frac{9}{2}c_0 -\frac{c_3}{2} \right) , \end{aligned}$$

(B8)

$$\begin{aligned} c_8&= \frac{1}{4} \left( \frac{11}{8}-\frac{9\left( 1-\Lambda +\sigma \right) }{4\gamma }+\frac{3}{2}c_0\right) , \end{aligned}$$

(B9)

$$\begin{aligned} c_9&= \frac{1}{4}\left( \frac{3}{4}+3c_0-\frac{29}{4\gamma }+\frac{9}{4\gamma } (-\Lambda -\sigma )\right) , \end{aligned}$$

(B10)

$$\begin{aligned} c_{10}&= \frac{1}{4}\left( \frac{3}{8\gamma }c_2 +1-c_0+\frac{3}{2\gamma }(1+c_3)+\frac{3}{2\gamma }(\sigma -\Lambda )\right) , \end{aligned}$$

(B11)

$$\begin{aligned} c_{11}&= \frac{1}{4}\left( - \frac{1}{8}+\frac{11}{8\gamma }-\frac{7c_0}{4} +\frac{2c_2}{\gamma }+\frac{8c_3}{\gamma } + \frac{c_4}{2}\right) , \end{aligned}$$

(B12)

$$\begin{aligned} c_{12}&=\frac{1}{32}, \end{aligned}$$

(B13)

$$\begin{aligned} c_{13}&= \frac{1}{4} \left( -\frac{3}{4\gamma }+c_0-\frac{4c_2}{3\gamma }-\frac{2c_3}{\gamma }\right) , \end{aligned}$$

(B14)

$$\begin{aligned} c_{14}&=\frac{9}{80\gamma ^2}. \end{aligned}$$

(B15)