Sir Isaac Newton (1687) -- speed of sound in a gas:
R is the gas constant for the gas and T is the absolute temperature of the gas. For air (gas constant R = 287 J/kg-K) at 158C (288.2K or 598F), Newton’s equation would predict the speed of sound to be 288 m/s (944ft/sec), whereas the experimental value for the speed of sound at this temperatureis 340 m/s (1116 ft/sec). Newton’s expression was about 16% in error.
Pierre Simon Laplace 1816 -- Laplace proposed the correct expression for the speed of sound in a gas:
Where is the specific heat ratio for the gas. For air, = 1.40
B. BASIC ACOUSTICS
For an ideal gas, the speed of sound is a function of the absolute temperature of the gas:
where gc is the units conversion factor, gc = 1 kg-m/N-s2 = 32,174 lbm-ft/ lbf -sec2; is the specific heat ratio, = cp/cv; R is the specific gas constant for the gas, R = 287 J/kg-K = 53,35 ft-lbf /lbm-R for air; and T is the absolute temperature, K or R.
The speed of sound (or c2) in a fluid (liquid or gas), in general, is given by:
where B is the isothermal bulk modulus and is the fluid density.
For transverse (bulk) sound waves in a solid, the speed of sound is given by (Timoshenko, 1970):
where E is Young’s modulus and is Poisson’s ratio for the material.
For sound transmitted through a thin bar, the speed of sound expression reduces to:
The wavelength and speed of sound for a simple harmonic wave are related by:
Another parameter that is encountered in analysis of sound waves is the wave number (k), which is defined by:
ACOUSTIC PRESSURE AND PARTICLE VELOCITY
The acoustic pressure for a plane simple harmonic sound wave moving in the positive x-direction may be represented by the following.
The quantity pmax is the amplitude of the acoustic pressure wave.
Acoustic instruments, such as a sound level meter, generally do not measure the amplitude of the acoustic pressure wave; instead, these instruments measure the root-mean-square (rms) pressure, which is proportional to the amplitude.
The rms pressure is related to the pressure amplitude for a simple harmonic wave by:
The rms acoustic pressure and the rms acoustic particle velocity are related by the specific acoustic impedance (Zs):
Where u is instantaneous acoustic particle velocity and p is rms acoustic pressure.
The specific acoustic impedance for plane waves is called the characteristic impedance (Zo) and is given by:
ACOUSTIC INTENSITY AND ACOUSTIC ENERGY DENSITY
For plane sound waves, as shown in Fig. 2-3, the acoustic intensity is related to the acoustic power and the area (S) by:
Where I = acoustic intensity, W = acoustic power, S = area
For a spherical sound wave (a sound wave that moves out uniformly in all directions from the source), the area through which the acoustic energy is transmitted is 4_r2, where r is the distance from the sound source, so the intensity is given by:
For the general case in which the sound is not radiated uniformly from the source, but the acoustic intensity may vary with direction, the intensity is given by:
Where Q is called directivity factor (dimensionless quantity)
Realation of the acoustic intensity and the rms acoustic pressure.
where p = prms
For a plane sound wave, the acoustic energy density is:
where p = prms
(for a plane sound wave) the acoustic intensity and acoustic energy density are related:
For a spherical sound wave, the acoustic energy density is given by:
we find that the magnitude of the specific acoustic impedance for a spherical sound wave is given by:
Where k = (2/) = (2f/c) = wave number.
The phase angle () between the acoustic pressure and the acoustic particle velocity is found from:
The acoustic energy density for a spherical wave is given by:
DIRECTIVITY FACTOR AND DIRECTIVITY INDEX
The directivity factor (Q) is defined as the ratio of the intensity on a designated axis of a sound radiator at a specific distance from the source to the intensity that would be produced at the same location by a spherical source radiating the same total acoustic energy:
The directivity index (DI) is related to the directivity factor by:
For a spherical source, the directivity factor Q = 1 and the directivity index DI = 0.
If a spherical source of sound is placed near the floor or a wall, as shown in Fig. 2-5, sound is radiated through a hemispherical area, S = 2r2. In this case, the intensity is:
For this case, we see that the directivity factor is Q = 2, and the directivity index is:
Similarly, if the spherical source were placed on the floor near a wall, the energy is radiated through an area S = r2. For this case,
The directivity factor, in this case, is Q = 4 and the directivity index is 6.0.
By going through the same reasoning, we may show that if the spherical source were placed in a corner near the floor and two walls, Q = 8 and DI = 9,0.
LEVEL AND DECIBLE
The acoustic power level is defined by:
The factor 10 converts from bels to decibels. The reference acoustic power (Wref) is 10-12 watts or 1 pW.
The sound intensity level and sound energy density level are defined in a similar manner, since both of these quantities (I and D) are proportional to energy:
Where the reference quantities are:
The sound pressure level is defined by:
For sound transmitted from a directional source (or spherical source, with DI = 0 or Q = 1) outdoors in air at 258C, the sound pressure level and the sound power level are related by:
COMBINATION OF SOUND SOURCE
The general expression for determining the combination of any set of ‘‘level’’ quantities is:
The square of the pressure is proportional to energy (the intensity, for example), so the individual sound pressures must be combined in an ‘‘energy-like’’ manner.
For an octave:
f2 = 2f1 or f2/f1 = 2
WEIGHTED SOUND LEVEL
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C. ACOUSTIC MEASUREMENT
INTENSITY LEVEL METER
The error in the intensity measurement is proportional to the error in the derivative approximation:
OCTAVE BAND FILTER
An octave band is defined as a frequency range in which the ratio of the upper and lower frequency limits for the range is equal to 2.
Octave band (1/1) or 1/3 octave band filters are often used in basic acoustic engineering design and analysis work. By observing the frequency band in which the maximum sound pressure level occurs, the system characteristics that relate to the noise generation may be identified.
For general acoustic measurements, the measurement of the sound levels on the A- and C-scales, along with octave band or 1/3 octave band data, is usually sufficient. For detailed analysis and diagnosis, however, more information may be required. In addition, other acoustic parameters or data averaging may be needed. In these cases, acoustic analyzers are often used.
Most of the commercial sound level meters are digital, microprocessor-controlled units. Various levels of acoustic analysis may be provided as features of the sound level meter. In addition, many sound level meters have the capability of transferring the data to a personal computer (PC) for postprocessing using special-purpose software.
The dosimeter or noise-exposure meter is an instrument developed for measurement of the accumulated noise exposure of workers in an industrial environment.
D. BAB 5 NOISE SOURCE