Two sources of sound are emitting progressive waves y 1 = 4 sin 710 π t and y 2 = 3 sin 702 π t . The sources are placed close to each other. The number of beats heard per second and intensity rat

Question

Two sources of sound are emitting progressive waves y 1 ​ = 4 sin 710 π t and y 2 ​ = 3 sin 702 π t . The sources are placed close to each other. The number of beats heard per second and intensity ratio between waxing and waning are respectively

TestHub · Accepted Answer

The given equations of the progressive waves are:

$y_{1} = 4 sin 710 π t$

$y_{2} = 3 sin 702 π t$

Comparing these with the standard equation $y = A sin 2 π f t$ , we get the frequencies and amplitudes as:

$2 π f_{1} = 710 π \Rightarrow f_{1} = 355 Hz$

$2 π f_{2} = 702 π \Rightarrow f_{2} = 351 Hz$

$A_{1} = 4$

$A_{2} = 3$

The number of beats heard per second (beat frequency) is:

$n = f_{1} - f_{2} = 355 - 351 = 4$

The intensity of sound is directly proportional to the square of its amplitude. The ratio of maximum intensity (waxing) to minimum intensity (waning) is:

$\frac{I _{ma x}}{I _{min}} = (\frac{A _{1} + A _{2}}{A _{1} - A _{2}})^{2}$

$\frac{I _{ma x}}{I _{min}} = (\frac{4 + 3}{4 - 3})^{2} = (\frac{7}{1})^{2} = \frac{49}{1}$

Thus, the number of beats is $4$ and the intensity ratio is $49 : 1$ .

Answer: $4, 49 : 1$

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