Question

(b) If the pipe is filled with air at pressure $1.00\times {10}^5$ Pa and density $1.20kg/m^3$, what will be the amplitude A and wavelength $\lambda$ of a longitudinal wave with intensity $3.00\times {10}^{-6}W/m^2$ and frequency 3400 Hz?

Answer 1

Given- $P=1.00\times {10}^{5}Pa,I=3.00\times {10}^{-6}W/m^2,f=3400Hz,\rho =1.20kg/m^3,$ ,

speed of sound in a medium is given by ,

$v=\sqrt{\left(\gamma P/\rho \right)}$ , ($\gamma =1.4$ , as in air most of the gases are diatomic )

$v=\sqrt{(1.4\times 1.00\times {10}^5/1.20)}=340.5m/s$ ,

we have , $v=f\lambda$ ,

or $\lambda =v/f=340.5/3400=0.1m$ ,

now intensity of longitudinal wave is given by ,

$I=2{\pi }^2{f}^2{A}^{2}v\rho$ ,

or $A^2=I/\left(2{\pi }^2{f}^{2}v\rho \right)=\frac{3.00\times {10}^{-6}}{2\times {3.14}^2\times {3400}^2\times 340.5\times 1.20}=3.22\times {10}^{-17}m$ ,

or $A=5.67\times {10}^{-9}m$