Question

Two source of sound placed closed to each other, are emitting progressive wave given by \[{y_1} = 4\sin {\rm{ 600}}\pi {\rm{t}}\] and \[{y_2} = 5\sin {\rm{ 608}}\pi {\rm{t}}\]. An observed located near these two sources will hear:
A. 8 beats per second with intensity ratio 81:1 between waxing and waning
B. 4 beats per second with intensity ratio 81:1 between waxing and waning
C. 4 beats per second with intensity ratio 25:16 between waxing and waning
D. 8 beats per second with intensity ratio 25:16 between waxing and waning

Answer 1

Hint:The number of beats produced per second by the waves or beat frequency is defined as the difference in frequency of two waves. The ratio of maximum to minimum intensity due to superposition is given by the formula with the amplitudes of the two waves.

Formula used:
The general equation for progressive wave is given as,
\[y = a\sin {\rm{ 2}}\pi f{\rm{t}}\]
Where a is the amplitude, f is the frequency and t is the time.
The intensity ratio is given as,
\[\dfrac{{{I_{\max }}}}{{{I_{\min }}}} = {\left( {\dfrac{{{a_1} + {a_2}}}{{{a_1} - {a_2}}}} \right)^2}\]
Where \[{a_1}\] and \[{a_2}\] are the amplitudes of the two waves.

Complete step by step solution:
Given progressive wave of two source of sound is,
\[{y_1} = 4\sin {\rm{ 600}}\pi {\rm{t}} \\ \]
\[\Rightarrow {y_2} = 5\sin {\rm{ 608}}\pi {\rm{t}} \\ \]
As we know the general equation for progressive wave is,
\[y = a\sin {\rm{ 2}}\pi f{\rm{t}} \\ \]
After comparing the given equation of progressive wave of two source with the general equation for progressive wave, we get
\[{a_1} = 4\], \[{a_2} = 5\]
\[\Rightarrow {f_1} = 300\], \[{f_2} = 304 \\ \]
Number of beats = 304 – 300 = 4 beats/sec
Now intensity ratio is given as,
\[\dfrac{{{I_{\max }}}}{{{I_{\min }}}} = {\left( {\dfrac{{{a_1} + {a_2}}}{{{a_1} - {a_2}}}} \right)^2} \\ \]
\[\Rightarrow \dfrac{{{I_{\max }}}}{{{I_{\min }}}} = {\left( {\dfrac{{4 + 5}}{{4 - 5}}} \right)^2} \\ \]
\[\Rightarrow \dfrac{{{I_{\max }}}}{{{I_{\min }}}} = \dfrac{{81}}{1} \\ \]
\[\therefore {I_{\max }}:{I_{\min }} = 81:1\]
Therefore, an observer located near these two sources will hear 4 beats per second with intensity ratio 81:1 between waxing and waning.

Hence option B is the correct answer

Note: The principle of superposition can be applied to waves when two or more waves travelling through the same medium at the same time. The intensity of the wave is directly proportional to the amplitude of the given wave. Intensity is proportional to the square of the amplitude. When amplitude increases then intensity will also increase.