I am trying to understand some notes my professor has typed up. He writes that we can write a sound wave in terms of the variation of pressure in a medium over time using the function $$ p(x, t) = P_0 \cos(kx - \omega t) \text{ where } P_0 = kp_0v_x^2A.$$
I am trying to tie this in with my book. The book intially writes the sound wave as a displacement $$ y(x, t) = A \sin(kx - \omega t). $$ Then shows that $$ p(x, t) = -B \frac{\partial y(x, t)}{\partial x} \text{ where $B$ is the bulk modulus}.$$ This gives $$ p(x, t) = -BAk\cos(kx - \omega t). $$ Equating this with what is in my professor's notes, I get $-B = p_0v_x^2$.
I am not sure though why this equality would hold, or if it even does, I could have made a mistake.
Can anyone give me a point in a direction? Thanks!
