# Sound waves. Relating the displacement to the pressure function

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!

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