Dispersion relations

Performing the substitutions of Table .1 into equation (10) gives dispersion relationships for comparison to the exact expression (8). These are shown in Table .2.

 

$$5^\circ k_z \eq \displaystyle {\strut\omega\over v}$$

$$15^\circ k_z \eq \displaystyle {\strut\omega\over v} - \displaystyle {\strut v k_x^2\over 2\omega}$$

$$45^\circ k_z \eq \displaystyle{\strut \omega\over v} -{\displaystyle {\strut k_x^2}\over \displaystyle {2\,{\omega\over v} - {\strut v k_x^2\over 2\omega}}}$$