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Two waves of the same pitch have their amplitudes in the ratio 2:3.

(a) What will be the ratio of their loudness?

(b) What will be the ratio of their frequencies?

Sound

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Answer

(a) Given, amplitude ratio, a1 : a2 = 2 : 3

Loudness is directly proportional to square of amplitude

∴ L ∝ a2

⇒L1L2=a12a22⇒L1L2=2232⇒L1L2=49\phantom{\Rightarrow} \dfrac{\text{L}1}{\text{L}2} = \dfrac{\text{a}1^2}{\text{a}2^2} \\[1em] \Rightarrow \dfrac{\text{L}1}{\text{L}2} = \dfrac{2^2}{3^2} \\[1em] \Rightarrow \dfrac{\text{L}1}{\text{L}2} = \dfrac{4}{9}

Hence, the ratio of their loudness will be 4 : 9.

(b) Waves with same pitch have the same frequency.

Hence, the ratio of their frequencies will be 1 : 1.

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