Frequency dependence of the ray path equation

The frequency-dependent eikonal equation is,  

$$K^2 = k^2+ \frac{\nabla^2 A}{A} = \nabla \phi \cdot \nabla \phi,$$ (10)

where A and $\phi$ are the amplitude and phase functions, respectively. Defining parameter $\gamma = \frac{\nabla^2 A}{A}$ and using $k=\omega s$, where $\omega$ is angular frequency and s is slowness, one may use the definition of K in equation (10) to write the following as the ray path equation (see equation (4) )  

$$\nabla K = \frac{\omega s \nabla s + \nabla \frac{\gamma}{2}... ...}}{{\rm d} s}\left( K \frac{{\rm d}}{{\rm d} s}{\bf r} \right).$$ (11)

To examine how equation (11) varies as a function of frequency, a frequency derivative is applied to yield,

$$\frac{\partial}{\partial \omega}\left( \nabla K \right)= \fr... ...a s\right)}{\left(\omega^2 s^2 + \gamma\right)}^{\frac{3}{2}}.$$ (12)

 

• High frequency approximation