Question

The human ear can detect a remarkable range of sound intensities. The quietest sound that we can hear has an intensity of 10−12W/m^2, and we begin to feel pain when the intensity reaches 1 W/m^2. Since the intensities that matter to people in everyday life cover a range of 12 orders of magnitude, intensities are usually converted to a logarithmic scale called the sound intensity level β, which is measured in decibels (dB).

Required:
Use this technique to find a formula for the intensity I of a sound, in terms of the sound level beta and the reference intensity Io.

Answer 1

I = Io 10^{β/10}

Step-by-step explanation:

To find a formula for the intensity of sound waves you use the fact that there is a great range of intensity that human can perceive.

The use of logarithms are useful for this kind of systems. For example, if you want a 10 scale for the measurement of the sound level you can write:

`\beta=10log((I)/(I_o))` (1)

I: intensity of sound

Io: hearing threshold

From the equation (1) you can find I in terms of Io and β. You use properties of the logarithms to obtain:

`\beta=log((I)/(I_o))^(10)\\\\10^(\beta)=10^{log((I)/(I_o))^(10)}\\\\10^(\beta)=((I)/(I_o))^(10)\\\\I=(I_o^(10)10^(\beta))^{(1)/(10)}=I_o10^{(\beta)/(10)}`