Question

Two soundwaves of same frequency have amplitudes 3cm and 4 cm. Find the ratio of maximum and minimum intensity when they interfare​

Answer 1

Given :

Two sound waves of same frequency have amplitude 3cm and 4cm.

To Find :

The ratio of maximum and minimum intensity.

Solution :

Amplitude of wave A = 3 cm

Amplitude of wave B = 5 cm

By taking ratio of amplitudes;

➠ A₁ / A₂ = 3 / 5

➠ A₁ = 3A₂ / 5

We know that intensity is directly proportional to the square of its amplitude.

★ Maximum intensity is given by;

$:\implies\:\underline{\boxed{\bf{\purple{I_{max}=(A_1+A_2)^2}}}}$

★ Minimum intensity is given by;

$:\implies\:\underline{\boxed{\bf{\orange{I_{min}=(A_1-A_2)^2}}}}$

Ratio of maximum to the minimum intensity,

$:\implies\sf\:\dfrac{I_{max}}{I_{min}}=\dfrac{(A_1+A_2)^2}{(A_1-A_2)^2}$

$:\implies\sf\:\dfrac{I_{max}}{I_{min}}=\dfrac{(\frac{3A_2}{5}+A_2)^2}{(\frac{3A_2}{5}-A_2)^2}$

$:\implies\sf\:\dfrac{I_{max}}{I_{min}}=\dfrac{(3A_2+5A_2)^2}{(3A_2-5A_2)^2}$

$:\implies\sf\:\dfrac{I_{max}}{I_{min}}=\dfrac{(8A_2)^2}{(-2A_2)^2}$

$:\implies\sf\:\dfrac{I_{max}}{I_{min}}=\dfrac{64}{4}$

$:\implies\:\underline{\boxed{\bf{\red{I_{max}:I_{min}=16:1}}}}$