the depth slowly vary for water wave
set-1
show (p94-1):
\[ \phi_z=a \delta^2\left(\phi_x B_{\bar{x} }+\phi_y B_{\bar{y} }\right) \text { on } z=B(\bar{x}, \bar{y}) \]
Start from: \[ \begin{aligned} & \left\{\begin{array}{l} \phi_z=\delta^2\left(\phi_x b_x+\phi_y b_y\right) \\ b(x, y)=B(a x, a y), \quad \bar{x}=a x, \bar{y}=a y \end{array}\right. \\ \end{aligned} \] Also,we have \[ b_x=\frac{\partial B}{\partial x}=\frac{\partial B}{\partial \bar{x} } \cdot \frac{\partial \bar{x} }{\partial x}=a B_{\bar{x} } \\ \] Similarly: $ b y=a B {y}$ hence: \[ \phi_z=a \delta^2\left(\phi_x B_{\bar{x} }+\phi_y B_{\bar{y} }\right) \]
set-2
show(p95-1): \[ \small \begin{aligned} \phi_{z z}+\delta^2\left[\left(k^2+l^2\right) \phi_{\theta \theta}\right. & + 2a\left(k \phi_{\theta \bar{x} }+l \phi_{\theta \bar{y} }\right) \\ & \left.+a(k \bar{x}+l \bar{y})\phi_\theta+\small a^2\left(\phi_{\bar{x} \bar{x} }+\phi_{\bar{y} \bar{y} }\right)\right]=0 \end{aligned} \] consider that: \[ \phi_{z z}+\delta^2\left(\phi_{x x}+\phi_{y y}\right)=0 \] \[ \phi_{x x}=\frac{\partial}{\partial x}\left(\frac{\partial}{\partial x} \phi\right)=\frac{\partial}{\partial x}\left(\frac{\partial \bar{x} }{\partial x} \phi_{\bar{x} }+\frac{\partial \theta}{\partial x} \phi_\theta\right) \] Since : \[ \partial \bar{x} / \partial x=a, \partial \theta / \partial x=k(\bar{x}, \bar{y}, \bar{t}) \] \[ \Rightarrow \phi_{x x}=\frac{\partial}{\partial x}\left(a \phi_{\bar{x} }+k \phi_\theta\right) \]
\[ \begin{split} \phi_{x x}=\frac{\partial \bar{x} }{\partial x} \frac{\partial}{\partial \bar{x} }\left(a \phi_{\bar{x} }\right) &+ \frac{\partial \theta}{\partial \bar{x} } \frac{\partial}{\partial \theta}\left(a \phi_{\bar{x} }\right) \\ & +\frac{\partial \bar{x} }{\partial x} \frac{\partial}{\partial \bar{x} }\left(k \phi_\theta\right)+\frac{\partial \theta}{\partial \bar{x} } \frac{\partial}{\partial \theta}\left(k \phi_\theta\right) \end{split} \]
\[ \begin{split} \ &=a^2 \phi_{\bar{x} \bar{x} }+k a \phi_{\bar{x} \theta}+a\left(k_{\bar{x} } \phi_\theta+k \phi_{\theta \bar{x} }\right)+k^2 \phi_{\theta \theta}\\ & =a^2 \phi_{\bar{x} \bar{x} }+k^2 \phi_{\theta \theta}+2 a k \phi_{\theta \bar{x} }+a k_{\bar{x} } \phi_\theta \end{split} \] Similarly: \[ \phi_{y y}=a^2 \phi_{\bar{y} \bar{y} }+l^2 \phi_{\theta \theta}+2 a l \phi_{\theta \bar{y} }+a l_{\bar{y} } \phi_\theta \] Q.E.D
set-3
show（p95-2 ）: \[ \beta_{0 z z}-\delta^2\left(k^2+l^2\right) \beta_0=0 \] We have: \[ \small \begin{aligned} \beta_{z z}+\delta^2[-\left(k^2+l^2\right) \beta & + 2 i a\left(k \beta_{\bar{x} }+l\beta_{\bar{y} }\right)\\ & + i a\left(k_{\bar{x} }+l_y\right) \beta+a^2\left(\beta_{\bar{x} \bar{x} }+\beta_{\bar{y} \bar{y} }\right)]=0 \\ \end{aligned} \] we write: \[ \beta \backsim \sum_{n=0}^{\infty} a^n \beta_n(\bar{x}, \bar{y}, \bar{t}, z) \text { as } a \rightarrow 0 \] then the term of \(\large a^0\) \[ \beta_{z z}=a^0 \beta_{0 z z} \]
\[ \delta^2\left[-\left(k^2+l^2\right)\right] \beta=-\delta^2\left(k^2+l^2\right) a^0 \beta_0 \] others are vanish for \(a^0\) consequently: \[ \beta_{0 z z}-\delta^2\left(k^2+l^2\right) \beta_0=0 \quad \text { (equ. 2.74) } \] Q.E.D