Question

Two sound sources produce progressive waves given by ${y_1} = 12\cos (100\pi t)$ and${y_2} = 4\cos (102\pi t)$ near the ear of an observer. When sounded together, the observer will hear:
A. 2 beat per second with an intensity ratio of maximum to minimum equal to 4
B. 1 beat per second with an intensity ratio of maximum to minimum equal to 3
C. 2 beat per second with an intensity ratio of maximum to minimum equal to 9
D. 1 beat per second with an intensity ratio of maximum to minimum equal to 4

Answer 1

Hint: A progressive wave is described as the onward propagation of a body's vibratory motion from one particle to the next particle in an elastic medium. In a medium through which a wave moves, an equation can be developed to generally describe the displacement of a vibrating particle. Thus, each particle of a progressive wave performs simple harmonic motion of the same time and amplitude that varies from each other in phase.

Formula used:
$\dfrac{{{I_{\max }}}}{{{I_{\min }}}} = \dfrac{{{{({a_1} + {a_2})}^2}}}{{{{({a_1} - {a_2})}^2}}}$

Complete answer:
Given:
${y_1} = 12\cos (100\pi t)$ and ${y_2} = 4\cos (102\pi t)$

From the general equation of progressive wave, we can conclude that
${\omega _1} = 100\pi $and${\omega _2} = 102\pi $
Also, we know that $\omega = 2\pi f$
$ \Rightarrow {f_1} = 50Hz$
Similarly, ${f_2} = 51Hz$

Now for beat it is calculated by subtracting the frequencies of two waves.
Beat$ = {f_1} - {f_2} = 50 - 51 = 1$
Also, the ratio $\dfrac{{{I_{\max }}}}{{{I_{\min }}}} = \dfrac{{{{({a_1} + {a_2})}^2}}}{{{{({a_1} - {a_2})}^2}}}$
$ \Rightarrow \dfrac{{{I_{\max }}}}{{{I_{\min }}}} = \dfrac{{{{(12 + 4)}^2}}}{{{{(12 - 4)}^2}}} = 4$

So, 1 beat per second with an intensity ratio of maximum to minimum equal to 4. So, option D is correct.

Note: Progressive Wave properties
(a) Each medium particle performs a vibration of its mean position. From one atom to another, the disruption continues onwards.
(b) Medium particles vibrate at the same amplitude in comparison to their mean location
(c) Each successive particle of the medium travels along the propagation of the wave in a motion identical to that of its ancestor, but later in time.
(d) Every particle's phase shifts from 0 to 2π.