## Sound and Acoustics: Frequency Analysis of Noise

To understand and quantify noise, we need to know about both the pressure levels involved as well as the frequency characteristics. The frequency is important for a number of reasons:

- Human hearing is frequency dependent. A noise with a certain sound pressure level at one frequency does not necessarily sound as loud as another noise with the same sound pressure level at a different frequency.
*Weighting scales*have been developed to try to take these differences into account. - Largely as a result of the above, many codes and regulations are expressed in terms of specific frequencies.

The human hearing range is between approximately 20~HZ and 20 000~HZ. As opposed to the decibel (logarithmic) scale used when dealing with sound pressures, a geometric scale is typically used when dealing with frequencies. In this approach, the audio spectrum is broken down into a number of convenient *bands* and the acoustic energy in each band is used. There are many types of frequency bands, but they share some common features.

### Frequency Bands

A frequency band is defined by a lower frequency () and an upper frequency (). The *bandwidth* associated with a particular frequency band is simply the difference between the upper and lower frequencies

(12.43)

Each band is typically named after its \emph{center} frequency which is defined as

(12.44)

Note that this isn’t the average of the upper and lower frequencies but is the \emph{geometric mean} with the property that

In absolute terms, \ensuremath{f_{\text{C}}} is closer to \ensuremath{f_{\text{L}}} than to \ensuremath{f_{\text{U}}}, however, the term center frequency is commonly used.

The relationship between the upper and lower frequencies for each band follows a common rule, typically

(12.45)

where the value of determines the type of band used.

More generally, for the \nTH band \mbox{(\ensuremath{f_{\text{L}}} = , \ensuremath{f_{\text{U}}} = )},

(12.46)

This is illustrated for four bands schematically in \fref{fig:Band_Relationship}.

As a result, the center frequency for each band becomes

(12.47)

(12.48)

(12.49)

As can be seen, for a given , the bandwidth for any given band is a fixed percentage of the center frequency for that band. As a result, these types of bands are often referred to as *percentage bands*.

There are also systems which break up the frequency spectrum into bands with *constant* bandwidth. These are known as *narrow bands*. FFT analyzers fall into this category. These approaches are however not yet as standardized as the percentage band approach.

### Octave Bands

If in 12.45, then

so that the upper frequency is twice the lower one. These bands are known as \emph{octave bands}. We also find from 12.49 that

so that the bandwidth is approximately 71% of the center frequency.

Most work with octave bands uses a standardized set of bands which are shown in Table 12.5. Note once again that each band is typically named after its center frequency, so we could talk about the 1000~HZ band, for example, which would include all frequencies between 707 and 1414~HZ. Characterizing noise in terms of octave bands is the crudest form of analysis. More refinement can be obtained by using fractional octave bands.

### One-Third Octave Bands

For one-third octave bands, we set in 12.45 so that

so that the upper frequency is only 26\% larger than the lower. In terms of the center frequency we find that

so that the bandwidth is

which is approximately 23% of the center frequency. The standard one-third octave bands commonly used are shown in Table 12.6.

#### Example

The octave band sound pressure levels for a centrifugal fan measured at 3~m are shown below.

- Determine the overall noise level (SPL) at 3~m.
- Estimate the acoustic power of the fan. Assume that

a) The fan is acting as a point source and acoustic energy is being radiated in all directions equally.

b) The fan is located in the middle of a smooth floor which does not absorb any of the acoustic energy.