Question

Two sinusoidal sound waves with frequencies $108Hz$ and $112Hz$ at your ear simultaneously. Each wave has an amplitude of $1.5\times {10}^{-8}m$ when it reaches your ear.
What is the maximum amplitude of the total sound wave? What is the minimum amplitude?

Answer 1

When two sound waves of slightly different frequencies and equal amplitudes superimpose on each other, then their resultant amplitude is given by

$A_{net}=2a\cos 2\pi \frac{f_1-f_2}{2}t$

here $f_1=112Hz$ and $f_2=108Hz$ and $a=1.5\times {10}^{-8}m$

$A_{net}=2a\cos (2\pi \frac{112-108}{2}t)$

$A_{net}=2a\cos \left(2\pi \times 2t\right)$

$A_{net}=2a\cos \left(4\pi t\right)$

Amplitude is maximum when $\cos 4\pi t$ is maximum i.e 1

$A_{ne{t}_{max}}=2a$

$A_{ne{t}_{max}}=2\times 1.5\times {10}^{-8}m$

$A_{ne{t}_{max}}=3\times {10}^{-8}m$

Amplitude is minimum when $\cos 4\pi t$ is minimum i.e 0

$A_{ne{t}_{min}}=2a\times 0$

$A_{ne{t}_{min}}=0$

$A_{ne{t}_{min}}=0$

Maximum Amplitude of total sound wave is $3\times {10}^{-8}m$ and minimum amplitude is $0$.