Question

Plane harmonic wave of frequency $500Hz$ are produced in air with displacement amplitude of $10$ micron. Given that density of air is $1.29kg{m}^{-3}$ and speed of sound in air is $340m{s}^{-1}$ :

• A. The pressure amplitude is $13.76N{m}^{-2}$
• B. The energy density is $6.45\times 10^{-4}J{m}^{-3}$
• C. The energy flux is $0.22J{m}^{-2}{s}^{-1}$
• D. Only (A) and (C) are correct

Answer 1

Answer: (A), (B), (C)

Given that
$n=500Hz$
$A=10\times 10^{-6}m$
$\rho =1.29kg/m^3$
$v=340m/s$
We use $v=\sqrt{\frac{B}{\rho }}$
$\Rightarrow B=\rho v^2=1.29\times (340{)}^2=1.49\times 10^{5}N/m^2$
Wavelength $\lambda =\frac{v}{\lambda }=\frac{340}{500}=0.68m$
Pressure amplitude
$\Delta P=\frac{2\pi }{\lambda }AB$
$=\frac{2\pi }{0.68}\times 10\times 10^{-6}\times 1.49\times 10^5=13.76m$
Energy density $\frac{P}{Sv}=2{\pi }^2{n}^2{A}^2\rho$
$=2\times 10\times (500{)}^2\times {\left(10\times 10^{-6}\right)}^2\times 1.29$
$=6.45\times 10^{-4}J/m^3$
Energy flux $u_{\varphi }=\frac{P}{S}=2{\pi }^2{n}^2{A}^2\rho v$
$=0.22J/m^2-s$