Sound Wave Intensity and Amplitude – Rankers Physics

Sound Wave and its Characteristics: Practice Problem & Solution

Assertion (A): Two sound waves of same intensity in a particular medium will have displacement amplitude in ratio of 2:1 if they have frequency in the ratio 1:2. Reason (R): Two wave of same velocity and amplitude in a particular medium have equal intensity.  
Both (A) & (R) are true and the (R) is the correct explanation of the (A)
Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
(A) is true but (R) is false
Both (A) and (R) are false

Solution Explained:

To solve this problem, we apply the core principles of Sound Wave and its Characteristics. Understanding the underlying formula is key to arriving at the correct answer below:

Intensity of a sound wave is \( I = \frac{1}{2} \rho v \omega^2 A^2 \), where \( \omega = 2\pi f \). If \( I_1 = I_2 \) and the medium is the same (so \( \rho \) and \( v \) are constant), then \( \omega_1^2 A_1^2 = \omega_2^2 A_2^2 \). Given \( f_1 : f_2 = 1 : 2 \), so \( \omega_1 : \omega_2 = 1 : 2 \). This yields \( (1)^2 A_1^2 = (2)^2 A_2^2 \), so \( A_1^2 = 4 A_2^2 \), meaning \( A_1 : A_2 = 2 : 1 \). Thus, A is true. Reason R is false because intensity also depends on frequency (\( \omega \)), so waves with the same velocity and amplitude but different frequencies will have different intensities.

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