Let $c_{L},c_{T},\omega$ be positive constants with $c_{L}>c_{T}$. Define
$p=\sqrt{\frac{\omega^{2}}{c_{L}^{2}}-\xi^{2}}\qquad q=\sqrt{\frac{\omega^{2}}{c_{T}^{2}}-\xi^{2}}$
Consider the function $D_{S}\left(\xi\right)$ defined as follows:
$D_{S}=4\xi^{2}pq\sin p\cos q+\left(\xi^{2}-q^{2}\right)^{2}\cos p\sin q$
How can I prove that all the real zeros of $D_{S}\left(\xi\right)$ are first-order? Thanks.