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Sound

1. Intensity is defined as power per unit area.
2. Using the reference intensity I₀ ≡ 10^-12 W/m², we define the sound level in decibels as

   db ≡ 10 log(I / I₀).

3. For a sound source approaching an observer at speed u_S, the observed wavelength is shortened by the source translation distance in one period:

   λ_O = λ_S - u_S T_S

   or, with T_S = λ_S / c:

   λ_O = λ_S (c - u_S) / c

4. If the observer is approaching the source at speed u_O, the observed wavelength is shortened by the observer translation distance in one period as observed by the observer:

   λ_O = λ_S - u_O T_O

   or, with T_O = λ_O / c:

   λ_O = λ_S c / (c + u_O)

5. Putting these together, we have the general equation for the one-dimensional Doppler Effect: the shift in observed wavelength as a function of the velocities of the source and observer:

   λ_O = λ_S (c - u_S) / (c + u_O)

   or, in terms of frequency:

   ν_O = ν_S (c + u_O) / (c - u_S)

   Note that if the source is moving away from the observer, u_S < 0. Likewise, if the observer is moving away from the source, u_O < 0.
6. For sound waves in air:

   c = √ (γ R T / M),

   where (as we will later find), γ is 1.4 for air (whose molecules are assumed to be diatomic), R is the molar gas constant, T is the temperature in Kelvin and M is the molar mass (0.02895 kg/mole for air). This equation approximates to

   c = 20.05 m/s * √ T

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©2010, Kenneth R. Koehler. All Rights Reserved. This document may be freely reproduced provided that this copyright notice is included.

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