the depth slowly vary for water wave
set-1
show: \[ \begin{align} &\phi_{z z}+\delta^2\left(\phi_{x x}+\phi_{y y}\right)=0,\\ &\phi_z=\delta^2 \eta_t \text { and } \phi_t+\eta=0 \text { on } z=1 \text {, }\\ &\phi_z=\delta^2\left(\phi_x b_x+\phi_y b_y\right) \text { on } z=b(x, y) \end{align} \]
\(\widetilde{u} \rightarrow\) orginal \(\quad u \rightarrow\) current \[ \widetilde{u}=c u \quad \widetilde{x}=\lambda x \] we start from: \[ \tilde{u}=\frac{\partial \phi}{\partial \tilde{x} }, \quad u=\frac{\tilde{u} }{c}=\frac{\partial \phi}{c \partial x} \cdot \frac{\partial x}{\partial \tilde{x} }=\frac{1}{c \lambda} \phi_x \] Since: \[ \small\tilde{u}_{\tilde{x} }+\tilde{w}_{\tilde{z} }=0, \tilde{u}_{\tilde{x} }=\frac{\partial \tilde{u} }{\partial \tilde{x} }=\frac{\partial \tilde{u} }{\partial x} \frac{\partial x}{\partial \tilde{x} }=\frac{1}{\lambda} \frac{\partial \tilde{u} }{\partial x}=\frac{c}{\lambda} u_x \] then: \[ \widetilde{u}_{\tilde{x} }=\frac{c}{\lambda} \frac{\partial}{\partial x}\left(\frac{1}{c \lambda} \phi_x\right)=\frac{1}{\lambda^2} \phi_{x x} \] we have: \[ \widetilde{w}_{\tilde{z} }=\frac{\partial \tilde{w} }{\partial \tilde{z} }=\frac{\partial \tilde{w} }{\partial z} \cdot \frac{\partial z}{\partial \tilde{z} }=\frac{1}{h_0} \widetilde{w}_z,\left(\tilde{z}=h_0 z\right) \] from: \[ \tilde{w}=\frac{\partial \phi}{\partial \tilde z}=\frac{\partial \phi}{\partial z} \cdot \frac{\partial z}{\partial \tilde z}=\frac{1}{h_0} \phi_z \] therefore: \[ \tilde{w}_z=\frac{1}{h_0}\left(\frac{1}{h_0} \phi_z\right)=\frac{1}{h_0^2} \phi_{z z} \] for incompressible flow: \[ \widetilde{u}_{\tilde{x} }+\tilde{w}_{\tilde{z} }=0 \] consequently: \[ \frac{1}{h_0^2} \phi_{z z}+\frac{1}{\lambda^2} \phi_{x x}=0, \quad\left(\delta=\frac{h_0}{\lambda}\right) \] extend it to 3-D: \[ \phi_{z z}+\delta^2\left(\phi_{x x}+\phi_{y y}\right)=0. \quad \text{(equ 2.66)} \]
set-2
Show:
\[ \phi_z=\delta^2 \eta_t \]
we have: \[ w=\eta_t \quad\text{(equ.2.1)} \] note that: \(w\) here is dimensionless (not \(\tilde{w}\) ) we have: \(\tilde{w}=w h_0 \sqrt{gh_0}/\lambda\) and from previous derivation: \[ \widetilde{w}=\frac{1}{h_0} \phi_z, \quad \text{hence}. \quad \frac{h_0 \sqrt{g h_0} }{\lambda} w=\frac{1}{h_0} \phi_z \] \[ \small\Rightarrow \phi_z=\frac{h_0^2}{\lambda} \sqrt{g h_0} w=\delta h_0 \sqrt{g h_0} \eta_t \rightarrow=\delta^2 \eta_t\quad \text{in textbook} \] but probably wrong (only valid when \(\sqrt{g h_0}=1/\lambda\) )
set-3
Show: \[ \phi_t+\eta=0,\quad\text{on} \quad z=1 \]
we start from: \(p=\eta \quad\) ( \(W_e=0\) for gravity wave),for inviscid fluid: \(u_t=-p_x\) where \(u\) has been described: \[ u=\frac{1}{c \lambda} \phi_x, c=\sqrt{g h_0} \] then \[ u_t=\frac{1}{c \lambda} \phi_{t x}=-p_x \] \[ \Rightarrow \frac{1}{\sqrt{g h_0} \cdot \lambda} \phi_t+P=0 \] from previous equations only when \(\lambda=1/\sqrt{g h_0}\) \[ \phi_t+p=0 \]
set-4
Show: \[ \phi_z=\delta^2\left(\phi_x b_x+\phi_y b_y\right)\quad \text{on} \quad z=b(x, y) \]
we have: \[ w=u b_x+v b_y, \quad z=b \] Also: \[ \quad \delta^2 w=\phi_z \Rightarrow \phi_z=\delta^2\left(u b_x+v b_y\right) \] note that: \[ w=\frac{D z}{D t}=\frac{D}{D t}(1+\varepsilon\eta)=\varepsilon\left[\eta_t+\left(u_{\perp} \cdot \nabla\right) \eta\right] \] After scaling: \[ (u, v) \rightarrow \varepsilon(u, v) \quad w \rightarrow \varepsilon \ w \] then: \[ \varepsilon w=\varepsilon\left[\eta_t+\varepsilon\left(u_{\perp} \cdot \nabla\right)\eta\right] \] \[ w=\eta_t+\varepsilon\left(u_{\perp} \cdot \nabla\right) \eta \]