Wavelength of surface waves for a translating body
Contents
| Section | Topic |
|---|---|
| 1 | Wave component |
| 2 | Stationary wave pattern |
| 3 | Wavelength |
1. Wave component
- Steady problem
A generic wave component has a wavelength from a wavenumber \(\vec{k} = (k_x, k_y)\).
In polar coordinates:
\[ \left. \begin{aligned} k_x &= k\cos\theta \\ k_y &= k\sin\theta \end{aligned} \right\} \qquad k = |\vec{k}| = \sqrt{k_x^2 + k_y^2} \tag{1} \]
2. Stationary wave pattern
The wave pattern has to be stationary relative to the body:
\[ c = \frac{\omega}{k} \quad \rightarrow \quad \frac{\omega}{k_x} = u \qquad\qquad c = \text{phase speed} \tag{2} \]
\[ \frac{\omega}{k\cos\theta} = u \quad \rightarrow \quad \omega = uk\cos\theta \tag{3} \]
Dispersion for waves produced by a moving disturbance.
3. Wavelength
But:
\[ \omega^2 = gk \tag{4} \]
\[ (uk\cos\theta)^2 = gk \]
\[ u^2k^2\cos^2\theta = gk \]
\[ u^2k\cos^2\theta = g \]
\[ k = \frac{g}{u^2\cos^2\theta} \qquad\qquad k \rightarrow \frac{2\pi}{\lambda} \]
\[ \frac{2\pi}{\lambda} = \frac{g}{u^2\cos^2\theta} \]
\[ g\lambda = 2\pi u^2\cos^2\theta \]
\[ \boxed{\;\lambda = \frac{2\pi u^2\cos^2\theta}{g}\;} \tag{5} \]