paper core
Mechanistic explanation
when a small obstacle, such as a fishing line, is moved forward slowly through still water, or is held stationary in moving water, the surface is covered with a beautiful wave pattern, fixed relatively to the obstacle. On the up-stream side the wave-length is short, and, the force governing the vibrations is principally cohesion. On the down-stream side the waves are longer, and are governed principally by gravity. Both sets of waves move with the same velocity relatively to the water, that says, that required in order that they may maintain a fixed position relatively to the obstacle.
Core
notation
the angle between this direction of motion(ship or water flow direction) and the normal to the wave front be called \(\theta\)
\(c_p=U \cos\theta\), where \(c_p\) is phase velocity, \(U\) represents the velocity of water relatively to the fixed obstacle
some useful formula
steady state water wave \[ \frac{\partial}{\partial t}=0 \tag{1} \] assume disturbance moving along the \(x\) axis, it follows that \[ \phi=U x+\varphi \tag{2} \] assume \(\varphi\) is small
consider the boundary condition on surface and assume the amplitude is small as well \[ \begin{aligned} &U \eta_x-\varphi_z=0\\ &U \varphi_x+g \eta-\frac{T}{\rho} \nabla_{\perp}^2 \eta=P \frac{\delta(r-a)}{2 \pi a} \end{aligned} \] here we assume
\[ \begin{aligned} & \omega_0^2=g+\frac{T}{\rho} k^2 \\ & \omega=k_x U+\omega_0 \end{aligned} \]
where \(k=\abs{\vec{k}}\), and \(\vec{k}=(k_x,k_y)\)
group velocity && phase velocity && wavevector(wavenumber) && rays
Below is a concise overview of how the wavevector \(\mathbf{k}\), wavefronts, rays, group velocity, and phase velocity all relate to each other, especially in a dispersive medium (such as water waves with nonconstant \(\omega / k\) ). 1. Wavevector \(\mathbf{k}\) 1. A wave in two or three dimensions often has a phase given by
\[ \theta(\mathbf{x}, t)=\mathbf{k} \cdot \mathbf{x}-\omega t \]
where \(\mathbf{k}\) is the wavevector and \(\omega\) is the angular frequency. 2. Direction: \(\mathbf{k}\) points normal (perpendicular) to the wavefronts (lines or surfaces of constant phase \(\theta=\) const).
- Wavevector \(\mathbf{k}\) :
- Normal to wavefronts.
- Has magnitude \(k\).
- Phase Velocity \(\mathbf{v}_p\) :
- Moves crests (constant phase) along \(\mathbf{k}\).
- \(\mathbf{v}_p=\frac{\omega}{k} \hat{\mathbf{k}}\).
- Group Velocity \(\mathbf{v}_g\) :
- Velocity of energy / wave packets.
- \(\mathbf{v}_g=\nabla_{\mathbf{k}} \omega(\mathbf{k})\).
- In isotropic media with no flow, \(\mathbf{v}_g\) is collinear with \(\mathbf{k}\). In dispersive media, \(\left\|\mathbf{v}_g\right\| \neq\left\|\mathbf{v}_p\right\|\).
- Wavefronts:
- Surfaces where \(\mathbf{k} \cdot \mathbf{x}-\omega t=\) const.
- Orthogonal to \(\mathbf{k}\).
- Rays:
- Paths of wave energy propagation.
the direction of group velocity and wavevector
- Basic Reminder: Group Velocity is \(\nabla_{\mathbf{k}} \omega\)
Given
\[ \omega(\mathbf{k})=\underbrace{\mathbf{U} \cdot \mathbf{k}}_{\text {flow Doppler shift }}+\underbrace{\omega_0(|\mathbf{k}|)}_{\text {intrinsic frequency }} \]
the group velocity is
\[ \mathbf{v}_g=\nabla_{\mathbf{k}} \omega(\mathbf{k})=\nabla_{\mathbf{k}}[\mathbf{U} \cdot \mathbf{k}]+\nabla_{\mathbf{k}}\left[\omega_0(k)\right] \]
- The gradient of \(\mathbf{U} \cdot \mathbf{k}\) is simply \(\mathbf{U}\).
- If \(\omega_0\) depends only on \(k=\sqrt{k_x^2+k_y^2}\), then
\[ \nabla_{\mathbf{k}} \omega_0(k)=\omega_0^{\prime}(k) \frac{\mathbf{k}}{k} \]
which points along \(\mathbf{k}\).
Hence,
\[ \mathbf{v}_g(\mathbf{k})=\mathbf{U}+\omega_0^{\prime}(k) \hat{\mathbf{k}}, \quad \hat{\mathbf{k}}=\frac{\mathbf{k}}{k} \] Hence, the presence of a nonzero \(\mathbf{U}\) breaks the simple collinearity between the group velocity and \(\mathbf{k}\). That sum \(\mathbf{v}_g\) is generally not collinear with \(\mathbf{k}\).
- Phase Lines and the Wavevector
A wave's phase is given by
\[ \theta(\mathbf{x}, t)=\mathbf{k} \cdot \mathbf{x}-\omega t \]
- Phase lines (or wave crests/troughs in 2D) are the sets of points where \(\theta(\mathbf{x}, t)\) is constant.
- Geometrically, these lines are always perpendicular to \(\mathbf{k}\) because \(\mathbf{k}\) is the gradient of \(\theta\) w.r.t. space.
Therefore, no matter whether there is flow or not, \(\mathbf{k}\) remains the normal to the wave's phase lines.
Fourier-transform approach and Galilean-shift
Fourier transform
water flow with velocity \(\mathbf{U}\) and ship is not moving
we have intrinsic frequency if water were at rest for Capillary-Gravity wave which is \[ \omega_0(\mathbf{k})=\sqrt{g|\mathbf{k}|+\frac{T}{\rho}|\mathbf{k}|^3 } \] When we write linearized free surface problem with a uniform flow \(\mathbf{U}\), the time derivative appear as \[ \frac{\partial}{\partial t}+\mathbf{U} \cdot \nabla \] After using Fourier transform we get factors like \[ -i(\omega-\mathbf{U} \cdot \mathbf{k}) \] Solving the free-surface boundary conditions then leads to \[ \omega(\mathbf{k})=\mathbf{U} \cdot \mathbf{k}+\omega_0(\mathbf{k}) \] the ship's velocity is oriented in the negative \(x-\)direction
the steady wave is defined as the forms in the following
If you define "steady wave" as "the wave crests do not move relative to the obstacle," you get a phase-velocity condition, typically \(c_p=U \cos \theta\).
usage
3D or Oblique Waves:
The wave crests may form at an angle \(\theta\) to the flow.
A common "steady pattern" argument is that each crest line is stationary relative to the obstacle, which leads to a phase-velocity component condition \(c_p=U\cos \theta\).
Meanwhile, the group velocity \(\mathbf{c}_g\) is a vector that can differ in direction from \(\mathbf{k}\). As a result, \(\left\|\mathbf{c}_g\right\|\) or any single component is not typically forced to equal \(U\).
If you define "steady wave" as "the wave energy packet is pinned to the obstacle," you get a group-velocity condition, \(c_g=U\).
usage
2D Forcing (Purely along \(x\), no variation across-stream):
The wave is essentially a 1D wave train behind or around the obstacle.
To keep this entire wave packet attached in the obstacle's frame, its energy (which moves at group velocity \(c_g\) ) cannot drift off.
Hence the condition \(c_g=U\) (in that one-dimensional sense) is often cited.
In three dimensions: \[ \mathbf{c}_g(\mathbf{k})=\nabla_{\mathbf{k}} \omega(\mathbf{k}) \]
For water waves on a uniform flow, e.g. \(\omega(\mathbf{k})=\mathbf{U} \cdot \mathbf{k}+\omega_0(k)\), the group velocity becomes
\[ \mathbf{c}_g=\mathbf{U}+\omega_0^{\prime}(k) \hat{\mathbf{k}} \quad \text { (in deep water, say). } \]
Galilean Shift
In lab frame the water is at rest, we have \[ \omega(\mathbf{k})=\omega_0(\mathbf{k}) \] where ship velocity is \(-\mathbf{U}\)
the equations 2.23
the velocity form \[ \phi=U x+\varphi(x, y, z ; t)-\frac{1}{2} U^2 t \] \(\varphi\) is small disturbance, on \(z=0\) we have \[ \begin{aligned} & \frac{p}{\rho}+g \eta+\varphi_t+U \varphi_x-\frac{T}{\rho} \nabla_{\perp}^2 \eta=0 \\ & \eta_t+U \eta_x-\varphi_z=0 \end{aligned} \] and
\[ \left(2 p_0 / \sqrt{2 \pi}\right)(\sin s a) / s \]