Boussinesq’s equations for (2+1)-dimensional surface gravity waves in an ideal fluid model

Abstract

We study the problem of gravity surface waves for the ideal fluid model in the (2+1)-dimensional case. We apply a systematic procedure for deriving the Boussinesq equations for a prescribed relationship between the orders of four expansion parameters, the amplitude parameter $\alpha$, the long-wavelength parameter $\beta$, the transverse wavelength parameter $\gamma$, and the bottom variation parameter $\delta$. We also take into account surface tension effects when relevant. For all considered cases, the (2+1)-dimensional Boussinesq equations cannot be reduced to a single nonlinear wave equation for surface elevation function. On the other hand, they can be reduced to a single, highly nonlinear partial differential equation for an auxiliary function f(x, y, t) which determines the velocity potential but is not directly observed quantity. The solution f of this equation, if known, determines the surface elevation function. We also show that limiting the obtained the Boussinesq equations to (1+1)-dimensions one recovers well-known cases of the KdV, extended KdV, fifth-order KdV, and Gardner equations.

Appendices

Surface tension terms

In the original, dimension variables, the term originating from additional pressure due to the surface tension has the following form [18]

$$\begin{aligned} \text {ST}=&-\frac{T}{\varrho }\;\nabla \cdot \left( \frac{\nabla \eta }{(1+\vert \nabla \eta \vert ^2)^{1/2}} \right) \nonumber \\ =&-\frac{T}{\varrho } \frac{(1+\eta _x^2)\eta _{yy}+(1+\eta _y^2)\eta _{xx}-2\eta _x \eta _y \eta _{xy}}{(1+\eta _x^2+\eta _y^2)^{3/2}} . \end{aligned}$$

(53)

In (53), $\nabla =(\partial _x,\partial _y)$ is two-dimensional gradient operator, and T is fluid’s surface tension coefficient. After transformation (6) to nondimensional scaled variables, we obtain (tildas are again dropped)

$$\begin{aligned} \text {ST}&\!=\! -\!\frac{T}{\varrho g H^{2}} \frac{ \beta \left( 1\!+\!\alpha ^{2}\beta \eta _{y}^{2}\right) \eta _{xx}\!+\!\gamma \left( 1\!+\!\alpha ^{2}\beta \eta _{x}^{2}\right) \eta _{yy}\!-\!2\alpha ^{2}\beta \gamma \eta _{x}\eta _{y}\eta _{xy} }{ \left( 1+\alpha ^{2}\beta \eta _{x}^{2}+\alpha ^{2}\gamma \eta _{y}^{2}\right) ^{3/2}} \nonumber \\&=\!-\!\tau \! \left[ \beta \left( 1\!+\!\alpha ^{2}\beta \eta _{y}^{2}\right) \eta _{xx}\!+\!\gamma \left( 1\!+\!\alpha ^{2}\beta \eta _{x}^{2}\right) \eta _{yy}\!\right. \nonumber \\&\quad \left. -\!2\alpha ^{2}\beta \gamma \eta _{x}\eta _{y}\eta _{xy} \right] \nonumber \\&\quad \times \left( 1 -\frac{3}{2}\alpha ^{2}\beta \eta _{x}^{2}- \frac{3}{2}\alpha ^{2}\gamma \eta _{y}^{2}+ O(\beta ^{6}) \right) \nonumber \\&=-\tau ( \beta \eta _{xx}+\gamma \eta _{yy} +O(\beta ^4)). \end{aligned}$$

(54)

In (54), we first expand the factor $1/\left( 1+\alpha ^{2}\beta \eta _{x}^{2}+\alpha ^{2}\gamma \eta _{y}^{2}\right) ^{3/2}$ assuming that terms $\alpha ^{2}\beta \eta _{x}^{2}$ and $\alpha ^{2}\gamma \eta _{y}^{2}$ are small.

Since we are interested in Boussinesq’s equations up to second order in small parameters, it is enough to use the surface term in the form

$$\begin{aligned} \text {ST} = -\tau \, ( \beta \eta _{xx}+\gamma \eta _{yy} ). \end{aligned}$$

(55)

First step towards solving Eq. (27)

We will try to find solution to Eq. (27) in two steps. First, assume that the function $f^{(0)}$ fulfils Eq. (27) reduced to zeroth order, that is, the following holds

$$\begin{aligned} f^{(0)}_{xx} + \frac{\gamma }{\beta } f^{(0)}_{yy} -f^{(0)}_{tt} =0. \end{aligned}$$

(56)

Next, postulate the solution to (27) in the form

$$\begin{aligned} f= & {} f^{(0)}+\alpha \, a(x,y,t)+\beta \, b(x,y,t)\nonumber \\&+\gamma \, g(x,y,t)+\delta \, d(x,y,t), \end{aligned}$$

(57)

where a, b, g, d are first-order correction functions. Inserting f in the form (57) into (27) and retaining terms up to first order in small parameters yield

$$\begin{aligned}&f^{(0)}_{xx} + \frac{\gamma }{\beta } f^{(0)}_{yy} -f^{(0)}_{tt} \nonumber \\&\quad +\alpha \left( a_{xx} +\frac{\gamma }{\beta } a_{yy} -a_{tt} -2 f^{(0)}_{x} f^{(0)}_{xt} - f^{(0)}_{t} f^{(0)}_{xx} \right. \nonumber \\&\quad \left. -\frac{\gamma }{\beta } \left( 2f^{(0)}_{y} f^{(0)}_{yt} +f^{(0)}_{t} f^{(0)}_{yy}\right) \right) \nonumber \\&\quad + \beta \left( b_{xx} + \frac{\gamma }{\beta } b_{yy} -b_{tt} +\frac{1}{2} f^{(0)}_{xxtt} -\frac{1}{6} f^{(0)}_{4x} \right) \nonumber \\&\quad + \delta \left( d_{xx} + \frac{\gamma }{\beta } d_{yy} - d_{tt} - \frac{\gamma }{\beta } \left( h\, f^{(0)}_{y}\right) _{y} -\left( h\, f^{(0)}_{x}\right) _{x} \right) \nonumber \\&\quad + \gamma \left( g_{xx} +\frac{\gamma }{\beta } g_{yy} - g_{tt} -\frac{\gamma }{\beta } f^{(0)}_{4y} \right. \nonumber \\&\quad \left. +\frac{1}{2} f^{(0)}_{yytt} -2 f^{(0)}_{xxyy} \right) =0. \end{aligned}$$

(58)

Since $f^{(0)}$ fulfils zeroth order Eq. (56) and small parameters can be arbitrary, condition (58) is equivalent to four inhomogeneous wave equations for the correction functions

$$\begin{aligned} a_{xx} +\frac{\gamma }{\beta } a_{yy} -a_{tt}&= 2 f^{(0)}_{x} f^{(0)}_{xt} + f^{(0)}_{t} f^{(0)}_{xx} \nonumber \\&\quad + \frac{\gamma }{\beta } \left( 2f^{(0)}_{y} f^{(0)}_{yt} +f^{(0)}_{t} f^{(0)}_{yy}\right) , \end{aligned}$$

(59)

$$\begin{aligned} b_{xx} + \frac{\gamma }{\beta } b_{yy} -b_{tt}&= -\frac{1}{2} f^{(0)}_{xxtt} +\frac{1}{6} f^{(0)}_{4x} , \end{aligned}$$

(60)

$$\begin{aligned} g_{xx} +\frac{\gamma }{\beta } g_{yy} - g_{tt}&= \frac{\gamma }{\beta } f^{(0)}_{4y} -\frac{1}{2} f^{(0)}_{yytt} +2 f^{(0)}_{xxyy} , \end{aligned}$$

(61)

$$\begin{aligned} d_{xx} + \frac{\gamma }{\beta } d_{yy} - d_{tt}&= \frac{\gamma }{\beta } \left( h\, f^{(0)}_{y}\right) _{y} +\left( h\, f^{(0)}_{x}\right) _{x} . \end{aligned}$$

(62)

For flat bottom case, $~\delta =0$, Eq. (62) does not appear.

If solutions to Eqs. (56), (59)–(62) are known, then the time-dependent surface profile is given by Eq. (28).