In dispersive water waves (e.g., surface gravity waves on deep or shallow water), several concepts describe how waves travel, how energy moves, and how the wave patterns evolve. Below is an overview of wave fronts, wave crests, the wave number vector \(\mathbf{k}\), phase velocity \(\mathbf{c}_p\), group velocity \(\mathbf{c}_g\), and rays, along with how their directions compare.
1. Wave Front
- A wave front is a surface (in 3D) or a curve (in 2D) of constant phase.
- In simpler terms, if you take a specific phase value (for instance, a zero crossing or a maximum point of oscillation), the collection of all points in space that share that phase value at a given time forms the wave front.
- Direction: The normal to the wave front points along the wave number vector \(\mathbf{k}\). The condition is universally valid!!!!! no matter \(\mathbf{\omega}= \mathbf{\omega}_0+\mathbf{k}\cdot\mathbf{U}\) or other case.
2. Wave Crest
- A wave crest is a line (in 2D) or surface (in 3D) where the wave displacement is at its maximum (peak).
- Wave crests are special examples of wave fronts, since a crest is also a constant-phase surface (often the phase is \(\pi/2\) or \(3\pi/2\), depending on the definition).
- Direction: The normal to the crest is likewise parallel to \(\mathbf{k}\), because the crest is a particular phase location (where the wave is at a maximum).
3. Wave Number Vector \(\mathbf{k}\)
Definition
The wave number vector \(\mathbf{k}\) points in the direction of phase propagation. Its magnitude is
\[ k = \|\mathbf{k}\| = \frac{2\pi}{\lambda}, \]
where \(\lambda\) is the wavelength.
Direction
- \(\mathbf{k}\) is perpendicular (normal) to the wave fronts (and thus to the wave crests).
- In a uniform and isotropic medium (e.g., deep water of constant depth), \(\mathbf{k}\) is simply the direction the wave pattern (phase) moves.
4. Phase Velocity \(\mathbf{c}_p\)
Definition
The phase velocity \(c_p\) describes the speed at which individual wave crests (or a single phase point) move. For surface gravity waves in deep water, the dispersion relation is
\[ \omega^2 = g\,k, \]
where \(\omega\) is the angular frequency and \(g\) is gravitational acceleration. From this,
\[ c_p = \frac{\omega}{k} = \sqrt{\frac{g}{k}}. \]
Direction
- The phase velocity vector \(\mathbf{c}_p\) has the same direction as \(\mathbf{k}\) (normal to wave crests).
- If you watch one crest in time, it moves with speed \(c_p\) along \(\mathbf{k}\).
more explanations
The usual way to define the phase velocity for a plane wave with wave vector \(\mathbf{k}\) is to track a surface of constant phase: \[ \mathbf{k} \cdot \mathbf{x}-\omega t=\text { constant } \]
Taking the time derivative of both sides (and calling \(\mathbf{c}_p\) the velocity of that constant-phase point):
\[ \mathbf{k} \cdot \frac{d \mathbf{x}}{d t}-\omega=0 \quad \Longrightarrow \quad \mathbf{k} \cdot \mathbf{c}_p=\omega \]
Since \(\mathbf{c}_p\) is parallel to \(\mathbf{k}\) by definition, we can write
\[ \mathbf{c}_p=\frac{\omega}{|\mathbf{k}|^2} \mathbf{k} \]
or equivalently in scalar form,
\[ \left|\mathbf{c}_p\right|=\frac{\omega}{|\mathbf{k}|} \]
Thus, even if the dispersion relation is
\[ \omega(\mathbf{k})=\mathbf{U} \cdot \mathbf{k}+\omega_0(\mathbf{k}) \]
the phase velocity remains
\[ \mathbf{c}_p(\mathbf{k})=\frac{\omega(\mathbf{k})}{|\mathbf{k}|^2} \mathbf{k}=\frac{\mathrm{U} \cdot \mathbf{k}+\omega_0(\mathbf{k})}{|\mathbf{k}|^2} \mathbf{k} \] Hence: 1. Direction: still parallel to \(\mathbf{k}\). 2. Magnitude: \(\left|\mathbf{c}_p\right|=\omega(\mathbf{k}) /|\mathbf{k}|\).
5. Group Velocity \(\mathbf{c}_g\)
Definition
The group velocity \(c_g\) describes how the envelope of a wave packet (and thus the energy) travels. It is defined by
\[ c_g = \frac{\partial \omega}{\partial k}. \]
In deep-water surface gravity waves,
\[ c_g = \frac{1}{2} c_p. \]
Direction
In the more general situation where a wave propagates within (or atop) a uniform flow \(\mathbf{U}\), the dispersion relation for a wave with wave vector \(\mathbf{k}\) often takes the form
\[ \omega(\mathbf{k})=\mathbf{U} \cdot \mathbf{k}+\omega_0(\mathbf{k}) \]
where \(\omega_0(\mathbf{k})\) is the intrinsic (or "rest-frame") dispersion relation that you would have in the absence of the background flow. In that case, the group velocity is given by
\[ \mathbf{c}_g=\nabla_{\mathbf{k}} \omega=\nabla_{\mathbf{k}}(\mathbf{U} \cdot \mathbf{k})+\nabla_{\mathbf{k}} \omega_0(\mathbf{k}) \]
Since \(\mathbf{U} \cdot \mathbf{k}\) is a simple dot product, we have
\[ \nabla_{\mathbf{k}}(\mathbf{U} \cdot \mathbf{k})=\mathbf{U} \]
Hence,
\[ \mathbf{c}_g=\mathbf{U}+\nabla_{\mathbf{k}} \omega_0(\mathbf{k}) \] therefore the direction of \(\mathbf{c}_g\) in this more general scenario is not necessarily parallel to \(\mathbf{k}\). Even if \(\nabla_{\mathbf{k}} \omega_0(\mathbf{k})\) itself were parallel to \(\mathbf{k}\) (which happens, for instance, in an isotropic, no-flow situation), the extra term \(\mathbf{U}\) can tilt or shift the resultant vector away from \(\mathbf{k}\). Consequently:
- If \(\mathbf{U} \neq \mathbf{0}\) and is not collinear with \(\mathbf{k}\), the sum \(\mathbf{U}+\nabla_{\mathbf{k}} \omega_0(\mathbf{k})\) points in some new direction, not simply along \(\mathbf{k}\).
- Only in special cases-such as \(\mathbf{U}=\mathbf{0}\) and an isotropic dispersion \(\omega_0(\mathbf{k})\) (meaning \(\omega_0\) depends solely on \(|\mathbf{k}|\), so \(\nabla_{\mathbf{k}} \omega_0\) is strictly parallel to \(\left.\mathbf{k}\right)\)-do we get \(\mathbf{c}_g\) parallel to \(\mathbf{k}\).
Thus, group velocity is parallel to the wave vector \(\mathbf{k}\) is strictly true only under more limited conditions (e.g., no background flow and isotropic dispersion). When a uniform current \(\mathbf{U}\) is present, the group velocity almost always departs from the direction of \(\mathbf{k}\).
6. Rays (Wave Rays)
Definition
A ray is the path along which wave energy (the group) propagates. In a nonuniform medium (e.g., where water depth changes), these rays can bend (refraction).
Direction
- In a uniform medium, rays coincide with the direction of group velocity, which is also the direction of \(\mathbf{k}\) when there is no refraction.
- In varying depth, rays bend according to Snell’s law–type refraction relationships, but they still align locally with the group velocity vector (the energy flow).
7. Summary of Direction Relationships
- Wave Number Vector \(\mathbf{k}\):
- Points in the direction of phase propagation.
- Normal to wave fronts (and crests).
- Points in the direction of phase propagation.
- Phase Velocity \(\mathbf{c}_p\):
- Same direction as \(\mathbf{k}\).
- Represents how fast crests (or a particular phase) move.
- Same direction as \(\mathbf{k}\).
- Group Velocity \(\mathbf{c}_g\):
- the direction part sees in the preceding's content
- Represents how fast the wave energy (wave envelope) travels.
- Rays:
- Paths of energy flow, which follow the group velocity direction in a uniform medium.
- Can bend if the medium changes (e.g., varying depth).
- Paths of energy flow, which follow the group velocity direction in a uniform medium.
- Wave Front vs. Wave Crest:
- Both are surfaces (or lines in 2D) of constant phase.
- A crest is a specific maximum displacement, so it’s one particular wave front among many possible ones.
- Normals to these surfaces point along \(\mathbf{k}\).
- Both are surfaces (or lines in 2D) of constant phase.
In deep and uniform water: - \(\mathbf{k}\), \(\mathbf{c}_p\), and \(\mathbf{c}_g\) are all aligned (collinear). - \(c_g = \tfrac{1}{2}c_p\). - Rays are straight and coincide with \(\mathbf{c}_g\).
In nonuniform media (e.g., nearshore waves in varying depth): - The magnitudes and directions can change.
- Rays bend (refraction), but locally still follow \(\mathbf{c}_g\).