Question

The equation of displacement due to a sound wave is $s=s_{0}si{n}^2(\omega t-kx)$. If the bulk modulus of the medium is B, then the equation of pressure variation due to that sound is :

• A. $Bk{s}_{0}sin(2\omega t-2kx)$
• B. $-Bk{s}_{0}sin(2\omega t-2kx)$
• C. $Bk{s}_{0}co{s}^2(\omega t-kx)$
• D. $-Bk{s}_{0}co{s}^2(\omega t-kx)$

Answer 1

Answer: (A)

$Bk{s}_{0}sin(2\omega t-2kx)$

The equation for pressure variation due to sound is
( Pressure in a sound wave is equal to product of elasticity of gas with the ratio of particle speed to wave speed)
$P=-B\frac{\partial S}{\partial x}$ where B is the Bulk modulus or elasticity of the gas.
Given, $S=S_0{\sin }^2\left(\omega t-kx\right)$
$P=-S_{0}B\sin \left(\omega t-kx\right)\cos \left(\omega t-kx\right)(-k))=(-S_{0}B)\times (-k)\times (2\times \sin \left(\omega t-kx\right)\cos \left(\omega t-kx\right))$
$=S_{0}Bksin\left(2\omega t-2kx\right)=Bk{S}_{0}sin\left(2\omega t-2kx\right)$