Question

A sound wave is propagating through a medium and the displacement equation for the particles of the medium is given by $y=-A_0\sin (kx-\omega t)$ then which of the following options represents a possible pressure wave through this medium?

• A. $\Delta P_o\sin (kx-\omega t)$
• B. $\Delta P_o\cos (kx-\omega t)$
• C. $-\Delta P_o\cos (kx-\omega t)$
• D. $-\Delta P_o\cos (kx+\omega t)$

Answer 1

The correct option is B $\Delta P_o\cos (kx-\omega t)$

We know that relation between pressure wave and displacement wave is given by,
$P=-B\frac{\delta y}{\delta x}$ where $P$ is the pressure, $B$ is the bulk modulus of the medium and $y$ is displacement of the particle.
Given,
$y=-A_0\sin (kx-\omega t)$
Now, $\frac{\delta y}{\delta x}=-A_{0}k\cos (kx-\omega t)$
So, $P=-B\frac{\delta y}{\delta x}=B{A}_{0}k\cos (kx-\omega t)$
$\Rightarrow P=\Delta P_o\cos (kx-\omega t)$