Two waves of the same pitch have amplitudes in the ratio 1 : 3. What will be the ratio of their (i) loudness and (ii) frequencies?

Given,

a₁ : a₂ = 1 : 3

(i) As we know, that loudness is proportional to the square of the amplitude

Loudness ∝ (amplitude) ²

Therefore,

$\dfrac{L$1}{L2} = \Big(\dfrac{a1}{a2}\Big)^2 \\[0.5em]

Substituting the values, we get

$\dfrac{L$1}{L2} = \Big(\dfrac{1}{3}\Big)^2 \\[0.5em] \dfrac{L1}{L2} = \dfrac{1}{9}

Hence, ratio of loudness between the two waves = 1 : 9

(ii) Pitch of a wave depends on its frequency and if two waves have same frequencies then their pitch will be same.

Hence, ratio of pitch between the two waves = 1 : 1.