Question

A sound wave travelling through a medium of bulk modulus $B$ is represented as $y(x,t)=A\sin (kx-\omega t)$ where symbols have their usual meanings. Then, the corresponding pressure amplitude is

• A. B A k
• B. $B(A/k{)}^{1/2}$
• C. B
• D. $B(Ak{)}^{1/2}$

Answer 1

Answer: (A)

Here $y(x,t)=A\sin (kx-\omega t)$
$\therefore \frac{\partial y}{\partial x}=kA\cos (kx-\omega t)$
Now use the expression $\Delta P=-B\frac{\partial y}{\partial x}$ where $B$ is the bulk modulus. This gives an expression for the pressure change
$\Delta P=-BAk\cos (kx-\omega t)$
Pressure amplitude, $P_0=BAk$
Also, $\Delta P=-BAk\sin \left[(kx-\omega t)-\frac{\pi }{2}\right]$
also indicates a phase lag of $\frac{\pi }{2}$ with respect to displacement.