P & SV plane waves about the Z-axis

Next: P plane waves about Up: ELLIPTIC APPROXIMANTS FOR TI Previous: SH plane waves about

Begin with the exact dispersion relations for P and SV waves.

	$\begin{eqnarray} 2W(\theta) = (W_{33} + W_{44}) C^2 + (W_{11} + W_{44}) S^2 \non... ...- (W_{11} - W_{44}) S^2)^{2} + 4(W_{13} + W_{44})^{2} C^{2}S^{2})\end{eqnarray}$
		(2)

Now convert the cosines under the surd to sines.

	$\begin{eqnarraystar} 2W(\theta) = (W_{33} + W_{44}) C^2 + (W_{11} + W_{44}) S^2 ... ...- 2W_{44}) S^2)^{2} + 4(W_{13} + W_{44})^{2} (1 -S^{2})S^{2})\end{eqnarraystar}$

Neglecting terms in S⁴.

	$\begin{eqnarraystar} 2W(\theta) = (W_{33} + W_{44}) C^2 + (W_{11} + W_{44}) S^2 ... ... (W_{11} + W_{33} - 2W_{44}) - 2(W_{13} + W_{44})^{2})S^{2})\end{eqnarraystar}$

Take constant out of bracket.

	$\begin{eqnarraystar} 2W(\theta) = (W_{33} + W_{44}) C^2 + (W_{11} + W_{44}) S^2 ... ..._{44}) - 2(W_{13} + W_{44})^{2}}{(W_{33} - W_{44})^{2}}S^{2})\end{eqnarraystar}$

Expand the surd in powers of S².

	$\begin{eqnarraystar} 2W(\theta) = (W_{33} + W_{44}) C^2 + (W_{11} + W_{44}) S^2 ... ..._{44}) - 2(W_{13} + W_{44})^{2}}{(W_{33} - W_{44})^{2}}S^{2})\end{eqnarraystar}$

Return factor to original location

	$\begin{eqnarraystar} 2W(\theta) = (W_{33} + W_{44}) C^2 + (W_{11} + W_{44}) S^2 ... ...} - 2W_{44}) - 2(W_{13} + W_{44})^{2}}{W_{33} - W_{44}}S^{2})\end{eqnarraystar}$

Simplify.

	$\begin{eqnarraystar} 2W(\theta) = (W_{33} + W_{44}) C^2 + (W_{11} + W_{44}) S^2 ... ...S^{2} + \frac {2(W_{13} + W_{44})^{2}}{W_{33} - W_{44}}S^{2})\end{eqnarraystar}$

Reintroduce cosines.

	$\begin{eqnarraystar} 2W(\theta) = (W_{33} + W_{44}) C^2 + (W_{11} + W_{44}) S^2 ... ...S^{2} + \frac {2(W_{13} + W_{44})^{2}}{W_{33} - W_{44}}S^{2})\end{eqnarraystar}$

Next: P plane waves about Up: ELLIPTIC APPROXIMANTS FOR TI Previous: SH plane waves about

Stanford Exploration Project
12/18/1997