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

Answer:

I = Io 10^{β/10}

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:

$\beta=10log(\frac{I}{I_o})$    (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:

$\beta=log(\frac{I}{I_o})^{10}\\\\10^{\beta}=10^{log(\frac{I}{I_o})^{10}}\\\\10^{\beta}=(\frac{I}{I_o})^{10}\\\\I=(I_o^{10}10^{\beta})^{\frac{1}{10}}=I_o10^{\frac{\beta}{10}}$