Chapter 12, Problem 5P

What is the intensity of a sound at the pain level of 120 dB? Compare it to that of a whisper at 20 dB

To determine

The intensity of the sound at the pain level of

$120dB$

Answer to Problem 5P

Solution:

The intensity of sound at the pain level is ${10}^{10}$ times that of a whisper at $20$ dB

Explanation of Solution

Given:

$I=120$ dB

$I_0=20$ dB

Find the intensity of the $120$ dB

$\begin{array}{l}B=10\log \left(l/l_0\right)\end{array}$

$\begin{array}{l}120dB=10\log \left(\frac{l}{{10}^{-12}}\right)\end{array}$

$\begin{array}{l}12dB=\log \left(\frac{l}{{10}^{-12}}\right)\end{array}$

$\begin{array}{l}{10}^{12}=\frac{l}{{10}^{-12}}\\ {10}^0=1\end{array}$

$\begin{array}{l}1W/m^2=1\end{array}$

Now 20dB

$\begin{array}{l}B=10\log \left(\frac{l}{l_0}\right)\end{array}$

$\begin{array}{l}120dB=10\log \left(\frac{l}{{10}^{-12}}\right)\\ 2=\log \left(\frac{l}{{10}^{-12}}\right)\end{array}$

$\begin{array}{l}{10}^2=l/{10}^{-12}\end{array}$

$\begin{array}{l}{10}^{-10w}/m^2=1\end{array}$