Retardation (frequency domain)

It is often convenient to arrange the calculation of a wave to remove the effect of overall translation, thereby making the wave appear to ``stand still.'' It is easy enough to introduce the time shift t₀ of a vertically propagating wave in a hypothetical medium of velocity $\bar v(z)$, namely,  

$$t_0 \eq \int_0^z \ { dz \over \bar v (z) }$$ (67)

Instead of solving equations in coordinates (t,x,z) we could solve them in the so-called retarded coordinates (t',x',z') where t'=t-t₀(z), x'=x and z'=z. (For more details, see IEI sections 2.5-2.7.) A time delay t₀ in the time domain corresponds to multiplication by $\exp ( i \omega t_0 )$ in the $\omega$-domain. Thus, the wave pressure P is related to the time-shifted mathematical variable Q by  

$$P(z, \omega ) \eq Q(z, \omega ) \ \exp \left( \ i \omega \int_0^z \ {dz \over \bar v (z)} \ \right)$$ (68)

which is a generalization of equation (55) to depth-variable velocity. (Equations () and () apply in both x- and k_x-space). Differentiating with respect to z gives

$$\begin{eqnarray} {\partial P \over \partial z } &=& { \partial Q \over \partial ... ...over \partial z} \ +\ { i \omega \over \bar v (z) } \ \right) \ Q\end{eqnarray}$$ (69) (70)

Next, substitute () and () into Table .4 to obtain the retarded equations in Table .5.

 

$$5^\circ {\strut\partial Q\over \partial z} \eq +\ i\omega \left( \displaystyle {1\over v} - {\strut 1\over\overline{v}(z)} \right) Q$$

$$15^\circ {\strut\partial Q\over \partial z} \eq - \,i\, {\displaystyle {\strut v k_x^2\over 2\omega}} \ Q +\ i\omega \left( \displaystyle {1\over v} - {\strut 1\over\overline{v}(z)} \right) Q$$

$$45^\circ {\strut\partial Q\over \partial z} \eq - \,i\, {\displaystyle ... ...over\displaystyle {2\,{\omega\over v} - {\strut v k_x^2 \over 2\omega}}} \ Q +\ i\omega \left( \displaystyle {1\over v} - {\strut 1\over\overline{v}(z)} \right) Q$$

$${\strut\partial Q\over \partial z} \eq$$ general + thin lens