Steady waves in water of constant finite depth

Water dynamics in finite depth are more complex and interesting than in infinite depth, even under simple conditions with zero surface pressure and steady motion.

Hypotheses

1. Surface pressure is zero: \(p=0\), motion is steady: \(\frac{\partial}{\partial t}=0\). note: steady flow implies the physical quantities of the fluid do not vary with time, at fixed point in space.
   notes

   If motion is uniform: \(\nabla \phi=0\) which indicates: The properties are the same at every point in space. therefore the convective derivative vanish as well.

Boundary conditions

1. incompressible flow: \(\nabla^2 \varphi=0\)
2. dynamic and kinematic condition on free surface \[ \left.\begin{array}{rl} \frac{p}{p}+g \eta+\varphi_t+U \varphi_x+\frac{1}{2} U^2 & =0 \\ \eta_t+U \eta_x-\varphi_y & =0 \end{array}\right\} \quad \text { at } y=0 \]
3. bottom condition

\[ \varphi_y=0 \quad \text { ant } y=-h \]

Solution

under these assumptions. the expressions becomes. \[ \begin{aligned} & \nabla^2 \varphi=0 \quad(1) \\ & \frac{p}{\rho}+g \eta+U \varphi_x+\frac{1}{2} U^2=\left.0 \quad\right|_{y=0} \\ & \quad U \eta_x-\varphi_y=\left.0 \quad\right|_{y=0} \\ & \varphi_y=\left.0 \quad\right|_{y=-h} \end{aligned} \] thus it follows that

\[ \begin{aligned} & \varphi_y+\frac{U^2}{g} \varphi_{x x}=0 \quad \left. \quad\right|_{y=0}\quad (2) \\ & \varphi_y=0 \quad \left. \quad\right|_{y=-h}\quad(3) \end{aligned} \]

A solution that conforms to all conditions is \[ \varphi(x, y)=A \cosh m(y+h) \cos (m x+a) \]

where \(A\) and \(a\) are constants, moreover, \(m\) is a root of the equation

\[ \frac{U^2}{g h}=\frac{\tanh m h}{m h} \]

Eqn (2) is extended to \(y<0\) by analytic continuation

\[ \varphi_y+\frac{U^2}{g} \varphi_{x x}=0 \quad y \leqslant 0 \]

also

\[ \begin{aligned} & \varphi_{x x}+\varphi_{y y}=0 \\ \Rightarrow \quad & \varphi_y-\frac{u^2}{g} \varphi_{y y}=0 \quad(y \leq 0) \end{aligned} \] The selection of an appropriate solution is outlined as follows.

\[ \varphi=c(x) \cosh \frac{g}{U^2} y \quad \text{or}\quad c(x) e^{\frac{g}{U^2} y} \quad\text{or others?} \] - case1: \(\varphi=c(x) e^{\frac{g}{U^2}y}\) and given condition (3)

\[ \begin{aligned} & \varphi_y=\frac{g}{U^2} c(x) e^{\frac{g}{U^2} y} \\ & \varphi_{y y}=\left(\frac{g}{U^2}\right)^2 c(x) e^{\frac{g}{U^2} y} \\ & \varphi_{x x}=c^{\prime \prime}(x) e^{\frac{g}{U^2} g} \\ & \frac{g}{U^2} c(x) e^{-\frac{g}{U^2} h}=0 \end{aligned} \]

\(\Rightarrow c(x)=0 \quad\) clearly incorrect.

• case 2: \(\varphi=c(x) \cosh \frac{9}{U^2} y\)

  \[ \begin{aligned} & \varphi_y=\frac{g}{U^2} c(x) \sinh \frac{g}{U^2} y \\ & \varphi_{y y}=\left(\frac{g}{U^2}\right)^2 c(x) \cosh \frac{g}{U^2} y \end{aligned} \]

  and consider equation (2)

  \[ \begin{aligned} & \varphi_y-\frac{U^2}{g} \varphi_{y y}=0 \\ \Rightarrow & \frac{g}{U^2} c(x) \sinh 0-\frac{U^2}{g}\left(\frac{g}{U^2}\right)^2 c(x) \cosh 0=0 \\ & \frac{g}{u^2}=0 ? \quad \text { as } y=0 \end{aligned} \]

  Clearly incorrect!

• case 3: The solution may require the addition of arbitrary constants to ensure equality, thus we give

  \[ \varphi=c(x) \cosh m(y+n) \]

  Here, constants are added to the terms containing \(y\). Similarly. substitute it into equation (3) \[ \begin{aligned} & \varphi_y=m c(x) \sinh m(y+n) \\ \varphi_{y y} & =m^2 c(x) \cosh m(y+n) \\ \Rightarrow & y=-h \quad m c(x) \sinh m(n-h)=0 \\ \Rightarrow & h=n \end{aligned} \]

  also apply \(\varphi\) into equation (2)

  \[ \begin{aligned} & \varphi_y-\frac{U^2}{g} \varphi_{y y}=0 \quad \text { at } \quad y=0 \\ \Rightarrow & mc(x) \sinh m h-\frac{U^2}{g} m^2 c(x) \cosh m h=0 \\ \Rightarrow & \tanh m h=\frac{U^2}{g} m^2 \end{aligned} \]

Next, substitute \(\varphi\) into equation (D)

\[ \begin{aligned} & \varphi_{x x}=c^{\prime \prime}(x) \cosh m(y+h) \\ \Rightarrow & c^{\prime \prime}(x) \cosh m(y+h)+m^2 c(x) \cosh m(y+h)=0 \\ \Rightarrow & c^{\prime \prime}(x)+m^2 c(x)=0 \end{aligned} \]

two roots: \(\sin m(x+\beta), \cos m(x+\beta)\). Given that we are interested in a specific solution any solution will be adequate. As such \(\cos m(x+\beta)\) is chosen

\[ \varphi=A \cosh m(y+h) \cos m(x+\beta) \]

Investigation \((m, h)\)

Let \(\zeta=\tanh \xi ; \xi=m h\) and Eqn (4) implies that \[ \zeta=\frac{U^2}{g h} \zeta \]

As for the two equation related to \(\zeta\) and \(\xi\).

\[ \left\{\begin{array}{l} \zeta=\tanh \xi \\ \zeta=\frac{U^2}{g h} \xi \end{array}\right. \]

• Case 1. \(\frac{U^2}{g h}<1 \Rightarrow 2\) real roots and \(\xi=0\) is always a root

• Case 2. \(\frac{U^2}{g h} \geqslant 1\) no real root besides 0

• Case 3. \(\tan i \xi=i \tanh \xi \Rightarrow\) many pure imaginary roots

if \(\frac{U^2}{g h}<1\) there is other type motion other than steady flow. The discussed wave seems to hold no condition at $\infty$ other than boundedness conditions. note that only include boundedness conditions

Investigation of boundary conditions

For steady waves due to disturbances, the boundary conditions may contain 2:

1. disturbance die out upstream
2. boundedness condition
   notes

   boundedness conditions \(\left\{\begin{array}{l}\varphi \text { and } \varphi_y \text { bounded at } \infty \\ \varphi \text { bounded and } \varphi_y=O\left(p^{-1+\varepsilon}\right), \varepsilon>0 \text { at } x= \pm a\end{array}\right.\) \(x \pm a \rightarrow\) disturbance source

behavior \(t \rightarrow \infty\)

➡️ \(\frac{U^2}{g h}>1\), waves die out at infinity both upstream and downstream

➡️ \(\frac{U^2}{g h}<1\), waves die out upstream but not downstream at infinity

➡️ \[ \frac{U^2}{g h}=1 \rightarrow\left\{\begin{array}{l} \text {1) no steady state as } t \rightarrow \infty \\ \text {2) disturbance potential }(\varphi) \rightarrow \infty \rightarrow\left\{O\left(t^{\frac{2}{3}}\right)\right\}\\ \text { at all points }(t \rightarrow \infty) \\ \text{3) }\vec{u} \quad(t \rightarrow \infty) \end{array}\right. \]