PX285 - L2 - potential dipole

let $ϕ = a \cos θ / r$ in cylindrical coordinates
the components of u:

\begin{matrix} u_{r} = \frac{\partial ϕ}{\partial r} = - \frac{a \cos θ}{r^{2}} \\ u_{θ} = \frac{\partial ϕ}{\partial θ} = - \frac{a \sin θ}{r} \\ u_{z} = \frac{\partial ϕ}{\partial z} = 0 \end{matrix}

\vec{\nabla} \cdot \vec{u} = \frac{a \cos θ}{r^{2}} - \frac{a \cos θ}{r^{3}} = 0

this means it is incompressible

for time dependent irrotational flows: $\vec{\nabla} \times \vec{u} = 0$
the vector identity:

(\vec{u} \cdot \vec{\nabla}) \vec{u} = \frac{1}{2} \vec{\nabla} u^{2} - \vec{u} (\vec{\nabla} \times \vec{u}) = \frac{1}{2} \vec{\nabla} u^{2}

the euler equation becomes:

\frac{\partial \vec{u}}{\partial t} + \frac{1}{2} \vec{\nabla} u^{2} = \vec{\nabla} (\dot{ϕ} + \frac{u^{2}}{2}) = - g \hat{z} - \frac{1}{ρ} \vec{\nabla} p

\vec{\nabla} (\dot{ϕ} + \frac{u^{2}}{2} + g z + \frac{p}{ρ}) = 0 ⟹ \dot{ϕ} + \frac{u^{2}}{2} + g z + \frac{p}{ρ} = constant

this is generalized bernoulli's principle
here, $\vec{u}$ is incompressible, irrotational and inviscid
eg: gravity waves (different from gravitational waves)

considering flows that satisfy laplace's equation: $\nabla^{2} ϕ = 0$
also, the boundary conditions: $u_{z} (z = 0) = 0$
collision on free surface ( $z ≃ d$ )
let $ϕ (x, z) = f (x) g (z)$

\nabla^{2} ϕ = \frac{f_{x x}^{″}}{f} + \frac{g_{z z}^{″}}{g} = 0 ⟹ \frac{f_{x x}^{″}}{f} = \frac{- g_{x x}^{″}}{g} = - k^{2}

\begin{matrix} f_{x x}^{″} + k^{2} f = 0 ⟹ f = A_{1} \cos k x + B_{1} \sin k x \\ g_{z z}^{″} - k^{2} g = 0 ⟹ g = A_{2} e^{- k z} + B_{2} e^{k z} \end{matrix}

considering a propagating wave:

ϕ = ϕ_{0} \cos (k x - ω t) (e^{k z} + B e^{- k z})

\begin{matrix} u_{x} = \frac{\partial ϕ}{\partial x} = - k ϕ_{0} \sin (k x - ω t) (e^{k z} + B e^{- k z}) \\ u_{z} = \frac{\partial ϕ}{\partial z} = k ϕ_{0} \cos (k x - ω t) (e^{k z} - B e^{- k z}) \\ u_{z} (z = 0) = 0 ⟹ 1 - B = 0 ⟹ B = 1 \end{matrix}

considering waves of small amplitude (amplitude of displacement $≪ 2 π / k$ , the wavelength)
the displacement of the fluid element by wave

\begin{matrix} X (t) = \int u_{x} d t = - \frac{k ϕ_{0}}{ω} \cos (k x - ω t) (e^{k z} + e^{- k z}) + X_{0} \\ Z (t) = \int u_{z} d t = - \frac{k ϕ_{0}}{ω} \sin (k x - ω t) (e^{k z} - e^{- k z}) + Z_{0} \\ ⟹ \frac{(X (t) - X_{0})^{2}}{a (z)} + \frac{(Z (t) - Z_{0})^{2}}{b (z)} = 1 \end{matrix}

$(X (t), Z (t))$ gives the coordinates of fluid elements which was $(X_{0}, Z_{0})$

\begin{matrix} a (z) = {[- \frac{k ϕ}{ω} (e^{k z} + e^{- k z})]}^{2} \\ b (z) = {[- \frac{k ϕ}{ω} (e^{k z} - e^{- k z})]}^{2} \end{matrix}

the path of a fluid element in the wave is an ellipse
the 'aspect ratios' of the ellipse depends upon $a$ via $a (z)$ and $b (z)$
considering the trajectory of fluid element near the surface, $z ≃ d$
let $k d ≫ 1 ⟹ d ≫ λ$ , ie. 'deep water approximation'

\begin{matrix} e^{k d} + e^{- k d} ≃ e^{k d} \\ e^{k d} - e^{- k d} ≃ e^{k d} \\ ⟹ a (z) ≃ b (z) \end{matrix}

so, the trajectory is a circle
at $z = 0 :$

\begin{matrix} e^{k 0} + e^{- k 0} = 2 \\ e^{k 0} - e^{- k 0} = 0 \\ ⟹ a (0) ≫ b (0) \end{matrix}

this is a piece of straight line

to determine the boundary condition at the free surface, $z ≃ d$ , using the generalized bernoulli's principle:

\dot{ϕ} + \frac{P_{a t m}}{ρ} + g z + \frac{u^{2}}{z} = constant

considering waves of very low amplitude the governing equations can be linearized
$u^{2} / 2$ can be neglected with respect to $\dot{ϕ}$ , as $u^{2} / 2 \sim ϕ_{0}^{2}$ , and $\dot{ϕ} \sim ϕ_{0}$

\dot{ϕ} (t) + g z (t) + \frac{P_{a t m}}{ρ} = constant ⟹ \dot{ϕ} (t) + g z (t) = constant

let the constant be zero

\begin{matrix} ω ϕ_{0} \sin (k x - ω t) (e^{k d} + e^{- k d}) + g [- \frac{k ϕ_{0}}{ω} \sin (k x - ω t) (e^{k d} - e^{- k d})] = 0 \\ ω^{2} = g k \frac{e^{k d} - e^{- k d}}{e^{k d} + e^{- k d}} = g k \tanh (k d) \end{matrix}

this is the dispersion relation for surface waves
$ω (k)$ is independent of the amplitude, $ϕ_{0}$ , because of linear approximation
considering two limiting cases:
firstly, let $k d ≪ 1 ⟹ d ≪ λ$ , ie. 'shallow water limit', eg: tsunami waves

\begin{matrix} lim_{k d \to 0} \tanh (k d) \to k d ⟹ ω \to \sqrt{g h} k \\ v_{p} = \frac{ω}{k} = \sqrt{g h} \\ v_{g} = \frac{d ω}{d k} = \sqrt{g k} \end{matrix}

this is called a 'dispersion-less wave'
secondly, $k d ≫ 1 ⟹ d ≫ λ$ , ie. 'deep water limit'

\begin{matrix} lim_{k d} \tanh (k d) = 1 \\ ω = \sqrt{g k} \\ v_{p} = \frac{ω}{k} = \sqrt{\frac{g}{k}} \sim λ^{1 / 2} \\ v_{g} = \frac{d ω}{d k} = \frac{1}{2} \sqrt{\frac{g}{k}} \end{matrix}

this is a dispersive wave