Chapter 3.3, Problem 88E

To determine

To find: The loudness of music when two stereos are playing as compared to one stereo.

Answer to Problem 88E

  $3.01\text{ decibel}$ .

Explanation of Solution

Given information:

Two people are playing stereo with same intensity and at the same time.

Calculation:

The number of decibel and intensity of sound is given by −

  $\beta =10\log \left(\frac{I}{{10}^{-12}}\right)$ where $\beta$ is number of decibel and $I$ is intensity of sound in watt per square meter.

Let us suppose each individual stereo is playing at the intensity of $I$ .

Case $1$ : When two stereo are playing

Total intensity $=I+I=2I$ watt per meter square.

  $\begin{array}{l}\text{Loudness in decibel }{\beta }_1=10\log \left(\frac{2I}{{10}^{-12}}\right)\\ \Rightarrow {\beta }_1=10\left[\log 2I-\log {10}^{-12}\right]\\ \Rightarrow {\beta }_1=10\log 2+10\log I-10(-12)\log 10\\ \Rightarrow {\beta }_1=10\log 2+10\log I+120\end{array}$

Case $2$ : When one stereo is playing

  $\begin{array}{l}\text{Loudness in decibel }{\beta }_2=10\log \left(\frac{I}{{10}^{-12}}\right)\\ \Rightarrow {\beta }_2=10\left[\log I-\log {10}^{-12}\right]\\ \Rightarrow {\beta }_2=10\log I-10(-12)\log 10\\ \Rightarrow {\beta }_2=10\log I+120\end{array}$

Difference in loudness $=10\log 2=3.01\text{ decibel}$ .