13  Acoustics

13.1 Abbreviations and Terminology

Abbreviations

ANSI American National Standards Institute
\(\text{dB}\) decibels
\(f\) frequency, cycles/sec
\(p\) pressure
\(\text{pW}\) \(10^{-12}\) Watts
\(x\) \(\mathrm{RMS}\) value of quantity
\(x_0\) reference value of quantity
\(\mu Pa\) \(10^{-6}\) Pascals
\(\text{Hz}\) Hertz
\(nm\) \(10^{-9}\) meters
\(P\) sound power

Terminology

decade band with the upper frequency x10 that of the lower
decibels measure of a magnitude, \(\text{dB} = 10 \log_{10}\) (mag)
far field beyond the near field (region where sound level drops \(-6\,\text{dB}\) as distance from the source doubles).
Hertz frequency in cycles/second
narrow band band whose width is less than one-third octave but less than \(1\%\) of the center frequency near field range within a distance equal to the wavelength of the lowest frequency emitted or twice the greatest dimension of the subject
octave a band with the upper frequency exactly twice the lower frequency1
pink noise has equal energy in each octave from \(20\) to \(20\,000\) Hz, or with an energy content inversely proportional to frequency
random noise does not have a uniform frequency spectrum and has an amplitude, as a function of time, consistent with a Gaussian distribution curve
third-octave highest frequency \(= 1.26\) x lower frequency (ratio \(= 2^{\frac{1}{3}}\) )
white noise has a constant spectrum level over the entire band of audible frequencies (need not be random)

13.2 Acoustic Velocities, Spectrum, and Reference Levels

Acoustic Velocity (speed of sound)

Medium Approximate Velocity, \(\frac{\text{m}}{\text{s}}\)
Air (\(20° \text{C}\) ) \(343\)
Fresh water \(1\,480\)
Aluminum \(5\,150\)
Concrete \(3\,600\)
Glass \(5\,300\)
Steel \(6\,000\)

\[\begin{equation*} \text{Wavelength, } \lambda = \frac{\text{acoustic velocity}}{\text{frequency}} \end{equation*}\]

Human hearing range is approximately \(20\) to \(20\,000 \text{ Hz}\)

  • Ultrasound lies above \(20\,000 \text{ Hz}\)
  • Infrasound lies below \(20 \text{ Hz}\)

Acoustic Reference Levels

Quantity Formula Reference
Velocity \(\left( L_v \right)\) \(20 \log \left( \frac{v}{v_0} \right)\) \(v_0 = 10 \frac{\text{nm}}{\text{s}^2}\)
Intensity \(\left( L_I \right)\) \(10 \log \left( \frac{I}{I_0} \right)\) \(I_0 = 1 \frac{\text{pW}}{\text{m}^2}\)
Sound Power Level \(\left( L_W \right)\) \(10 \log \left( \frac{P}{P_0} \right)\) \(P_0 = 1 \text{ pW}\)
Sound Pressure Level, \(\mathrm{SPL}\) \(\left( L_p \right)\) \(20 \log \left( \frac{p}{p_0} \right)\) \(P_0 = 1\,\mu\text{Pa (air)}\)
Pressure Spectrum Level \(\left( \mathrm{PSL} \right)\) 2 \(\mathrm{SPL} - 10 \log \Delta f\) \(\text{dB}\)
Pressure Band Level \(\left( \mathrm{PBL} \right)\) \(\mathrm{PSL} + 10 \log \Delta f\) \(\text{dB}\)
Overall SPL \(\left( \mathrm{OASPL} \right)\) \(10 \log_{10} \sum 10^{\frac{\mathrm{SPL}}{10}}\) \(20\,\mu\text{Pa (air)}\)

13.3 Acoustic Pressure and Intensity

Sound Pressure from Sound Power

Transmission Environment \(L_p\)
Free Field \(L_W + \log Q - 20 \log r - 10.8 \text{ dB}\)
Reflecting Plane \(L_W + \log Q - 20 \log r - 7.8 \text{ dB}\)
Reverberant Room \(L_W + \log Q - 20 \log R - 6.2 \text{ dB}\)

where

\[\begin{align} r &= \text{distance from source}\\ Q &= \text{directivity index of source}\\ R &= \text{room constant}\\ \end{align}\]

Acoustic Intensity

\[\begin{equation*} I - \frac{\text{Imaginary} \left[ G_{yx} \left( f \right) \right]}{4 \pi \rho_0 \Delta r f} = \frac{\text{Im} \left[ G_{yx} \left( f \right) \right]}{16.25 \Delta r f} \left( \text{for air} \right) \end{equation*}\]

where

\[\begin{align} \rho_0 &= \text{fluid density} = 1.293 \frac{\text{kg}}{\text{m}^3} \text{ for air}\\ \Delta r &= \text{microphone spacing (meters)}\\ f &= \text{frequency} \end{align}\]

Intensity Spectrum Level (ISL)

Intensity level of a sound contained within a band 1 \(\text{Hz}\) wide

\[\begin{equation*} \mathrm{ISL} = 10 \log \frac{I}{I_0 \Delta f} = \mathrm{IL} - 10 \log \Delta f \left( \text{dB} \right) \end{equation*}\]

where

\[\begin{align} f &= \text{center frequency of band} \\ I &= \text{sound intensity} \left( \frac{\text{watts}}{\text{m}^2} \right) \\ I_0 &= 10^{-12} \frac{\text{watt}}{\text{m}^2} \text{reference intensity} \\ \Delta f &= \text{bandwidth} \left( \text{Hz} \right) \\ \end{align}\]

13.4 Acoustic Weighting Curves

(ANSI S1.4 1983)

Weighting for SPL

Nominal Freq Exact Freq A B C
\(\text{Hz}\) \(\text{Hz}\) \(\text{dB}\) \(\text{dB}\) \(\text{dB}\)
\(10.0\) \(10.00\) \(-70.4\) \(-38.2\) \(-14.3\)
\(12.5\) \(12.59\) \(-63.6\) \(-33.3\) \(-11.3\)
\(16.0\) \(15.85\) \(-56.4\) \(-28.3\) \(-8.4\)
\(20.0\) \(19.95\) \(-50.4\) \(-24.2\) \(-6.2\)
\(25.0\) \(25.12\) \(-44.8\) \(-20.5\) \(-4.4\)
\(31.5\) \(31.62\) \(-39.5\) \(-17.1\) \(-3.0\)
\(40.0\) \(39.81\) \(-34.5\) \(-14.1\) \(-2.0\)
\(50.0\) \(50.12\) \(-30.3\) \(-11.6\) \(-1.3\)
\(63.0\) \(63.10\) \(-26.2\) \(-9.4\) \(-0.8\)
\(80.0\) \(79.43\) \(-22.4\) \(-7.3\) \(-0.5\)
\(100.0\) \(100.00\) \(-19.1\) \(-5.6\) \(-0.3\)
\(125.0\) \(126.90\) \(-16.2\) \(-4.2\) \(-0.2\)
\(160.0\) \(158.50\) \(-13.2\) \(-2.9\) \(-0.1\)
\(200.0\) \(199.50\) \(-10.8\) \(-2.0\) \(0.0\)
\(250.0\) \(251.20\) \(-8.7\) \(-1.4\) \(0.0\)
\(315.0\) \(316.20\) \(-6.6\) \(-0.9\) \(0.0\)
\(400.0\) \(398.10\) \(-4.8\) \(-0.5\) \(0.0\)
\(500.0\) \(501.20\) \(-3.2\) \(-0.3\) \(0.0\)
\(630.0\) \(631.00\) \(-1.9\) \(-0.1\) \(0.0\)
\(800.0\) \(794.30\) \(-0.8\) \(0.0\) \(0.0\)
\(1\,000.0\) \(1\,000.00\) \(0.0\) \(0.0\) \(0.0\)
\(1\,250.0\) \(1\,259.00\) \(0.6\) \(0.0\) \(0.0\)
\(1\,600.0\) \(1\,585.00\) \(1.0\) \(0.0\) \(-0.1\)
\(2\,000.0\) \(1\,995.00\) \(1.2\) \(-0.1\) \(-0.2\)
\(2\,500.0\) \(2\,512.00\) \(1.3\) \(-0.2\) \(-0.3\)
\(3\,150.0\) \(3\,162.00\) \(1.2\) \(-0.4\) \(-0.5\)
\(4\,000.0\) \(3\,981.00\) \(1.0\) \(-0.7\) \(-0.8\)
\(5\,000.0\) \(5\,012.00\) \(0.6\) \(-1.2\) \(-1.3\)
\(6\,300.0\) \(6\,310.00\) \(-0.1\) \(-1.9\) \(-2.0\)
\(8\,000.0\) \(7\,943.00\) \(-1.1\) \(-2.9\) \(-3.0\)
\(10\,000.0\) \(10\,000.00\) \(-2.5\) \(-4.3\) \(-4.4\)
\(12\,500.0\) \(12\,589.00\) \(-4.3\) \(-6.1\) \(-6.2\)
\(16\,000.0\) \(15\,849.00\) \(-6.7\) \(-8.5\) \(-8.6\)
\(20\,000.0\) \(19\,953.00\) \(-9.3\) \(-11.2\) \(-11.3\)

 

13.5 1/3 Octave Center Frequencies

(ANSI S1.6 1984)

 Band No. Nominal Center Exact Center Octave Center
# \(\text{Hz}\) \(\text{Hz}\) \(\text{Hz}\)
\(1\) \(1.25\) \(1.26\)
\(2\) \(1.60\) \(1.58\)
\(3\) \(2.00\) \(2.00\) \(2\)
\(4\) \(2.50\) \(2.51\)
\(5\) \(3.15\) \(3.16\)
\(6\) \(4.00\) \(3.98\) \(4\)
\(7\) \(5.00\) \(5.01\)
\(8\) \(6.30\) \(6.31\)
\(9\) \(8.00\) \(7.94\) \(8\)
\(10\) \(10.00\) \(10.00\)
\(11\) \(12.50\) \(12.59\)
\(12\) \(16.00\) \(15.58\) \(16\)
\(13\) \(20.00\) \(19.95\)
\(14\) \(25.00\) \(25.12\)
\(15\) \(31.50\) \(31.62\) \(32\)
\(16\) \(40.00\) \(39.81\)
\(17\) \(50.00\) \(50.12\)
\(18\) \(63.00\) \(63.10\) \(63\)
\(19\) \(80.00\) \(79.43\)
\(20\) \(100.00\) \(100.00\)
\(21\) \(125.00\) \(125.89\) \(125\)
\(22\) \(160.00\) \(158.49\)
\(23\) \(200.00\) \(199.53\)
\(24\) \(250.00\) \(251.19\) \(250\)
\(25\) \(315.00\) \(316.23\)
\(26\) \(400.00\) \(398.11\)
\(27\) \(500.00\) \(501.19\) \(500\)
\(28\) \(630.00\) \(630.96\)
\(29\) \(800.00\) \(794.33\)
\(30\) \(1\,000.00\) \(1\,000.00\) \(1\,000\)
\(31\) \(1\,250.00\) \(1\,258.90\)
\(32\) \(1\,600.00\) \(1\,584.90\)
\(33\) \(2\,000.00\) \(1\,995.30\) \(2\,000\)
\(34\) \(2\,500.00\) \(2\,511.90\)
\(35\) \(3\,150.00\) \(3\,162.30\)
\(36\) \(4\,000.00\) \(3\,981.10\) \(4\,000\)
\(37\) \(5\,000.00\) \(5\,011.90\)
\(38\) \(6\,300.00\) \(6\,309.60\)
\(39\) \(8\,000.00\) \(7\,943.30\) \(8\,000\)
\(40\) \(10\,000.00\) \(10\,000.00\)
\(41\) \(12\,500.00\) \(12\,589.30\)
\(42\) \(16\,000.00\) \(15\,848.90\) \(16\,000\)
\(43\) \(20\,000.00\) \(19\,952.60\)

 

13.6 References

13.1 Beranek, Leo L., “Acoustic Measurements,” John Wiley & Sons, New York, New York, 1956.
13.2 Peterson, Arnold P.G. and Gross, Ervin E., Jr., “Handbook of Noise Measurement,” GenRag Incorporated, Concord, Massachusetts, 1978.
13.3 “Measuring Sound,” (Pamphlet), Bruel & Kjaer, Naerum, Denmark, September 1984.
13.4 “Pocket Handbook, Noise, Vibration, Light, Thermal Comfort,” Bruel & Kjaer, Naerum, Denmark, 1986.

Additional Reading

Hunter, Joseph L., “Acoustics,” Prentice-Hall Incorporated, Englewood Cliffs, New Jersey, 1957.

  1. Common octaves include, in \(\text{kHz}\) \[\begin{align} 0.0375&-0.0750\\0.075&-0.15\\0.15&-0.30\\0.30&-0.60\\0.6&-1.2\\1.2&-2.4\\2.4&-4.8\\4.8&-9.6\end{align}\]↩︎

  2. The \(\mathrm{SPL}\) contained within a band \(1 \text{ Hz}\) wide↩︎