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Intensity Amplitude Relationship

The ratio of ...

Question

The ratio of amplitudes of two waves having same frequency is $3 : 1$ . Find the ratio of maximum and minimum intensities.

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Solution

Given : $\frac{A_{1}}{A_{2}} = \frac{3}{1}$
$⟹ A_{1} = 3 A_{2}$
Maximum intensity $I_{m a x} = (A_{1} + A_{2})^{2}$
Minimum intensity $I_{m i n} = (A_{1} - A_{2})^{2}$
Thus ratio of maximum to minimum intensity $\frac{I_{m a x}}{I_{m i n}} = (\frac{A_{1} + A_{2}}{A_{1} - A_{2}})^{2}$
Or $\frac{I_{m a x}}{I_{m i n}} = (\frac{3 A_{2} + A_{2}}{3 A_{2} - A_{2}})^{2}$
Or $\frac{I_{m a x}}{I_{m i n}} = (\frac{4}{2})^{2} = (2)^{2} = 4$

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