In this paper Käse and Bleckmann describe how to determine the distance from the wave source by the wave frequency. I'm trying to understand how they developed their final equation.
They give (eq. 5): $$D = \frac{t_2 - t_1}{\frac{1}{c_{g2}}-\frac{1}{c_{g1}}}$$ and end up with (eq. 7): $$D = - \frac{9}{2} c_{ph} \frac{\omega}{d \omega/d t}.$$
I can follow their first step which is a first order Taylor-Series expansion ($c_{g2}(\omega) = c_{g1}(\omega + \Delta \omega)$ and $c_{g1}(\omega + \Delta \omega) \approx c_{g1}(\omega) +(d c_{g1}/d \omega)\Delta \omega$) and gives (eq. 6): $$D = c_{g1} \frac{\frac{-(t_2 - t_1)}{\Delta \omega}}{\frac{1}{c_{g1}} \frac{d c_{g1}}{d \omega}}.$$
They further say
- $t_2 - t_2 = \Delta t \rightarrow 0$ (I assume this is the reason why this term is neglected in the Taylor-Series expansion)
- $\Delta t /\Delta \omega \rightarrow \frac{1}{d \omega/d t}$
and that they used $$c_g = \frac{d \omega}{d K} = c_{ph} - \lambda \frac{d c_{ph}}{d \lambda}.$$
But I did not find a way to get from there to the final equation. Does anybody have an idea?
Other (possibly) necessary equations:
- $c_{ph}^2 = \frac{\omega^2}{K^2} = \left(g + \frac{T K^2}{\rho} \right) K^{-1}$
- $K = \frac{2 \pi}{\lambda}$
quantitities:
- $c_{ph}$ - phase velocity
- $c_g$ - wave number
- $\omega$ - angular frequency
- $\lambda$ - wave length
- $K$ - wavenumber
- $\rho$ - density
- $g$ - gravity
- $T$ - surface tension