Question

The intensity level of the sound wave is said to be 4 decibels. If the intensity of the wave is doubled, then the intensity level of sound is expressed in decibel would be
A) 8.
B) 16.
C) 7.
D) 14.

Answer 1

Intensity is defined as the power carried per unit area by a wave. Power can be said to be the rate at which a wave is transferring energy. We can write this in an equation as, intensity I is $I = \dfrac{P}{A}$, where P is the power through an area A. To calculate the increase in intensity, first we have to understand how intensity is calculated in case of sound.

Formula used: Sound Intensity, $\beta = 10{\log _{10}}(\dfrac{I}{{{I_0}}})$, where ${I_0}$ is the initial intensity and $I$is the final intensity.

Complete step by step answer:
Intensity is defined as the power carried per unit area by a wave. Power can be said to be the rate at which a wave is transferring energy. We can write this in an equation as, intensity I is $I = \dfrac{P}{A}$, where P is the power through an area A. The intensity of a sound wave can be related to its amplitude squared shown in the following equation:

$I = \dfrac{{{{(\Delta p)}^2}}}{{2\rho {v_w}}}$
Here $\Delta p$ is the pressure variation or pressure , $\rho$ is the density of the material in which the sound wave travels, and ${v_w}$ is the speed of sound in the medium. This relationship remains valid with the fact that the sound wave is produced by some vibration the more the air is compressed in the sound it creates.

The sound intensity level $\beta$ in decibels of a sound having an intensity is defined to be
$\beta = 10{\log _{10}}(\dfrac{I}{{{I_0}}})$, where ${I_0}$ is the initial intensity and $I$is the final intensity.
Given in the question $I = 2{I_0}$ .
Thus, we have $\beta = 10{\log _{10}}(\dfrac{{2{I_0}}}{{{I_0}}}) = 10{\log _{10}}(2) = 10 \times 0.301 = 3$.

So, the final output will be 4+3=7dB. Thus, the correct option is (C).

Note: For this question to be solved we have to know the values of some common logs like ${\log _{10}}2$, ${\log _{10}}3$ and so on. Without knowing the value of this we will not be able to calculate the answer on any of these types of questions.