Question

(a) A longitudinal wave propagating in a water-filled pipe has intensity $3.00\times {10}^{-6}W/m^2$ and frequency 3400 Hz. Find the amplitude A and wavelength $\lambda$ of the wave. Water has density $1000kg/m^3$ and bulk modulus $2.18\times {10}^9$ Pa.

Answer 1

Given- $K=2.18\times {10}^{9}Pa,I=3.00\times {10}^{-6}W/m^2,f=3400Hz,\rho =1000kg/m^3,$ ,

speed of sound in a medium is given by ,

$v=\sqrt{\left(K/\rho \right)}$ ,

$v=\sqrt{(2.18\times {10}^9/1000)}=1.48\times {10}^{3}m/s$ ,

we have , $v=f\lambda$ ,

or $\lambda =v/f=1.48\times {10}^3/3400=0.43m$ ,

now intensity of longitudinal wave is given by ,

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

or $A^2=I/\left(2{\pi }^2{f}^{2}v\rho \right)=\frac{3.00\times {10}^{-6}}{2\times {3.14}^2\times {3400}^2\times 1.48\times {10}^3\times 1000}=8.89\times {10}^{-21}m$ ,

or $A=9.43\times {10}^{-11}m$