Question

The loudest painless sound produces a pressure amplitude of $28$ $N{m}^{-2}$. Calculate the intensity of this sound wave at S.T.P. Density of air at S.T.P. $=1.3$ kg $m^{-3}$, speed of sound at S.T.P. $=332$ m/s.

Answer 1

$y=a\sin (\omega t-kx)$ is the equation of progressive wave and $1/2$ $p{a}^2{\omega }^{2}v$ is its intensity.
$P=\frac{Edy}{dx}$, $v=\sqrt{\frac{E}{\rho }}$
$=v^2\rho (-ak)(\cos \omega t-kx)=v^2\rho ak\sin (\omega t-kx-\pi /2)$
$\therefore P_{max}=v^2\rho ak$
$\therefore I=\frac{1}{2}\rho \frac{P^2}{v^4{\rho }^2{k}^2}{v}^2{k}^v$ $\{\frac{\omega }{k}=v\}$
$=\frac{1}{2}\frac{P_{max}^2}{\rho v}=\frac{1}{2}\times \frac{{28}^2}{1.3\times 332}=0.91$ W-$m^{-2}$.