Question

The time period, loudness, and amplitude of a sound wave are 4 seconds, 5 dB, and 2 units respectively. Find the loudness and frequency of the sound wave when the amplitude is increased to 10 units and the time period is reduced to half the initial value.

• A. Loudness = 625 dB; Frequency = 0.5 Hz
• B. Loudness = 25 dB; Frequency = 0.25 Hz
• C. Loudness = 5 dB; Frequency = 0.25 Hz
• D. Loudness = 125 dB; Frequency = 0.5 Hz

Answer 1

The correct option is D Loudness = 125 dB; Frequency = 0.5 Hz

Initial loudness of the sound ($L_1$)= 5 dB
Initial amplitude ($A_1$) = 2 units
Final amplitude ($A_2$) = 10 units

As we know that,
Loudness ∝ ${\left(Amplitude\right)}^2$
Thus,
$\frac{L_1}{L_2}={\left(\frac{A_1}{A_2}\right)}^2$
=> $\frac{5}{L_2}={\left(\frac{2}{10}\right)}^2\Rightarrow L_2=25\times 5=125dB.$
Therefore, final loudness is 125 dB.

It is also given that,
Initial time period = 4 s
Final time period = $\frac{4}{2}s$ = 2 s

We know that, $Frequency=\frac{1}{Timeperiod}$
Final frequency = $\frac{1}{2}$ = 0.5 Hz
Therefore, final frequency is 0.5 Hz.