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A longitudinal standing wave $y = A \sin k x \sin ω t$ is maintained in a homogeneous medium of density $ρ$ . The total energy per unit volume at $x$ and $t$ is

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$\frac{1}{2} ρ A^{2} ω^{2}$

$\frac{1}{2} ρ A^{2} ω^{2} (\sin^{2} k x \cos^{2} ω t + \cos^{2} k x \sin^{2} ω t)$

$\frac{1}{2} ρ A^{2} ω^{2} (\sin^{2} k x + \sin^{2} ω t)$

$\frac{1}{2} ρ A^{2} ω^{2} (\cos^{2} k x + \cos^{2} ω t)$

answer is B.

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Detailed Solution

$k = \frac{1}{2} ρ {(\frac{d y}{d t})}^{2} u = \frac{1}{2} ρ v^{2} {(\frac{d y}{d x})}^{2}$

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