Sound and Acoustics: Sound Pressure Measurements
When sound pressure measurements are made, most commonly the actual pressure amplitudes are not measured directly. Instead, an effective pressure known as the root mean square pressure is used.
Root Mean Square Pressure Values
This is equivalent to the square root of the average of the pressure squared
which is not the same as the square root of the square of the average pressure
For a harmonic pressure wave given by
where is the associated period given by
These relationships are illustrated in Figure 12.2.
Note that if we have a sound composed of several harmonic waves
then it can be shown that the RMS pressure of the resultant wave is given by
Note that this only applies if the sound sources are uncorrelated.
Superposition of Sound Waves
When two sound waves are superimposed, the result is a linear combination of the two individual waves due to the linearity of the wave equation. Two interesting situations which may occur, depending on the frequencies and amplitude of the two waves are
- Standing waves.
Beating occurs when two sounds of nearly the same frequency are combined. Consider two pressure waves with identical amplitudes and slightly different frequencies
When these two pressure waves are added together the total pressure is
or, using trigonometric identities,
is the average of the two frequencies. If the difference between the two frequencies is small, will be small (much smaller than ) so that the response can be interpreted as a wave whose frequency is the average of the two close frequencies, but which has a slowly varying amplitude. The beating period is defined as
A typical beating response is shown in Figure 12.3. This is similar to the beating phenomenon discussed earlier.
Standing waves can occur when two waves travel in opposite directions. This can occur when a sound wave reflects off a surface perpendicular to the direction of the original wave propagation.
Consider two waves of equal frequency and amplitude traveling in opposite directions (These can both be shown to satisfy the wave equation 12.5.)
The superposition of these two waves results in
However, since and where is the wavelength, we see that
so that 12.20 can be written
Here we see that the amplitude of the pressure wave varies in space. In particular at locations given by
we see from 12.21 that so there will be no sound at these locations.
Consider the situation shown in Figure 12.4. As can be seen, over the distance of of the wavelength of the sound, the pressure amplitude can change from to 0. To put this in perspective, for sounds at 1000 Hz for example, \m so that \cm. This can introduce large errors in the measurement of sound pressures. These kinds of errors can occur in industrial settings where reflective surfaces are present.