I am looking at total internal reflection for an acoustic wave, defined in terms of its pressure such that
$$p = p_1 \,exp\left[-i\{\omega t-\vec{k} \cdot\vec x\}\right]$$
Using the definition of mean-square pressure, given by
$$ (\Delta p)^2 = \frac{1}{T}\int_0^T [Re(p)]^2 dt$$
I am required to thus show that
$$(\Delta p)^2 = \frac{1}{2}|p_1|^2$$
Given that $p_1$ is complex, I have observed that if
$$p_1 = Re(p_1) + i\,Im(p_1)$$
and further if
$$\omega t-\vec{k} \cdot\vec x = \varphi$$
then
$$Re(p) = Re(p_1)cos(\varphi) + Im(p_1)sin(\varphi)$$
So then I see that
$$(\Delta p)^2 = \frac{1}{T}\int_0^T \left[Re(p_1)^2cos^2(\varphi) + Im(p_1)^2 sin^2(\varphi) + 2Re(p_1)Im(p_1)cos(\varphi)(\sin(\varphi)\right]dt$$
I am slightly lost at this point. Do I need to equate $Re(p_1)$ with $Im(p_1)$? But how would such a move be legal?