Beats — Question 2

Question 2

Incomplete beats reveal the amplitudes of the two tones

Two nearby frequencies produce pressure-envelope maximum $1.00\,\mathrm{Pa}$ and minimum $0.20\,\mathrm{Pa}$. Both amplitudes are constant and positive, with $P_1\ge P_2$. The medium impedance is $Z=400\,\mathrm{Pa\,s/m}$.

1. Infer the component peak pressure amplitudes.

2. Find the maximum and minimum carrier-averaged intensities and the individual intensity ratio.

Question 2 - Solution

Incomplete beats reveal the amplitudes of the two tones

(a) Method and calculation

Extrema occur when phasors align or oppose:$$P_1+P_2=1.00,\quad P_1-P_2=0.20.$$$$P_1=0.60\,\mathrm{Pa},\qquad P_2=0.40\,\mathrm{Pa}.$$The nonzero minimum signals unequal amplitudes rather than failure of superposition.

(b) Interpretation and further calculation

Using peak amplitudes for locally copropagating waves,$$I_{\max}=\frac{1.00^2}{2Z}=1.25\times10^{-3}\,\mathrm{W/m^2},$$$$I_{\min}=\frac{0.20^2}{2Z}=5.0\times10^{-5}\,\mathrm{W/m^2}.$$$$I_2/I_1=(0.40/0.60)^2=4/9.$$The envelope contrast is $I_{\min}/I_{\max}=0.04$, which differs from the component intensity ratio.

Physical check

Equal amplitudes would drive the envelope minimum to zero; different amplitudes retain a finite minimum at opposition.