Question

You stand 5 meters away from two machines. One machine operates at a sound intensity level that is 50 dB at your location. The other machine operates at a sound intensity level of 70 dB at your location. What sound intensity level do you hear (measure in dB) when both machines are turned on?

Answer 1

suppose intensity of individual wave is I1 and I2 respectively for 50 dB and 70dB.

As the frequency is same, path difference is zero, the net intensity is $I = (\sqrt{I_{1}}+\sqrt{I_{2}})^{2}$

Now. the formula of sound level is $\beta = 10log_{10}\frac{I}{I_{o}}$ [where Io is reference intensity which is a constant = $10^{-12}W/m^{2}$ .

For, I1 sound level is 50 dB :

$50=10log_{10}\frac{I_{1}}{I_{o}}$

$\Rightarrow I_{1}=10^{5}*I_{o}=10^{5}*10^{-12}=10^{7}W/m^{2}$

for I2 sound level is 70 dB :

$70=10log_{10}\frac{I_{2}}{I_{o}}$

$\Rightarrow I_{2}=10^{7}*I_{o}=10^{7}*10^{-12}=10^{5}W/m^{2}$

when both is turned on :

sound level is = $\beta=10log_{10}\frac{(\sqrt{I_{1}}+\sqrt{I_{2}})^{2}}{I_{o}} = 10log_{10}\frac{(10^{\frac{7}{2}}+10^{\frac{5}{2}})^{2}}{10^{-12}}=190.83 dB$ [answer]