When beats are produced by two progressive waves of nearly same frequency which one of the following

Question

When beats are produced by two progressive waves of nearly same frequency which one of the following is correct\

Professor PC Thomas Classes · Answer

Summary of beats\ 1) When two sound wave trains of nearly equal frequencies are sounded together, the resulting sound would wax (increase) and wane (decrease) in intensity at regular intervals of time. This phenomenon is called beats. One increase and one decrease is called a beat. Number of beats per second is beat frequency.\ Beat frequency = difference in frequency\ If $n_{1}$ and $n_{2}$ are the individual frequencies, then the number of beats $/$ second $=n_{1}-n_{2}$ or $n_{2}-n_{1}$.\ 2) Maximum number of beats which the human ear can distinguish per second is equal to 10 . ie the interval between two beats must be $1 / 10 \mathrm{~s}$ or more, because the relaxation time of the ear is $1 / 10$ seconds. (3) Two ultrasonic frequencies can be combined to produce beats. If the beat frequency is in the audible range we can hear the resulting sound.\ 4) Component waves $$ \begin{aligned} & \mathrm{y}_{1}=\mathrm{A} \sin 2 \pi \gamma_{1} \mathrm{t} \ & \mathrm{y}_{2}=\mathrm{A} \sin 2 \pi \gamma_{2} \mathrm{t} \end{aligned} $$ 5) Resultant wave or equation for beat is\ $\mathrm{Y}=\left[2 \mathrm{~A} \cos 2 \pi\left(\frac{\gamma_{1}-\gamma_{2}}{2}ight) \mathrm{t}ight] \sin 2 \pi\left(\frac{\gamma_{1}+\gamma 2}{2}ight) \mathrm{t}$\ 6) The first term is the amplitude term and the second term is the phase term.\ 7) Amplitude depends on (a) time ((b) the difference in frequencies. It is independent of the position where the observer is situated.\ 8) The amplitude varies simple harmonically at a frequency $\left(\gamma_{1}-\gamma_{2}ight)$.\ 9) Amplitude of component wave $=A$ Loudness of component wave $\propto A^{2}$\ Max. amplitude of beats $=2 \mathrm{~A}$\ Max. loudness of beats $\propto 4 \mathrm{~A}^{2}$\ $\frac{	ext { Max.loudness of beats }}{	ext { Loudness of component waves }}=\frac{4 \mathrm{~A}^{2}}{\mathrm{~A}^{2}}$\ 10) In the equation for beats both the terms contain t only and no x .