Question

A sound wave travelling through a medium of bulk modulus $B$ is represented as $y(x, t)=A \sin (k x-\omega t)$ 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. $B(A k)^{1 / 2}$

Answer 1

Answer: (A)

Here $y(x, t)=A \sin (k x-\omega t) $
$\therefore \frac{\partial y}{\partial x}=k A \cos (k x-\omega t)$
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
$\Delta P=-B A k \cos (k x-\omega t)$
Pressure amplitude, $P_{0}=B A k$
Also, $\Delta P=-B A k \sin \left[(k x-\omega t)-\frac{\pi}{2}\right]$
also indicates a phase lag of $\frac{\pi}{2}$ with respect to displacement.