Fundamentals and Basic Terminology
1.11 COMBINING SOUND PRESSURES .1 Coherent and Incoherent Sounds.1 Coherent and Incoherent Sounds
∆f ' fC 21/N&1
21/ 2N ' 0.2316fC for 1/3 octave bands
' 0.7071fC for octave bands
(1.91) The information provided thus far allows calculation of the bandwidth, ∆f of every band, using the following equation:
It will be found that the above equations give calculated numbers that are always close to those given in the table.
When logarithmic scales are used in plots, as will frequently be done in this book, it will be well to remember the one-third octave band centre frequencies. For example, the centre frequencies of the 1/3 octave bands between 12.5 Hz and 80 Hz inclusive, will lie respectively at 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8 and 0.9 of the distance on the scale between 10 and 100. The latter two numbers in turn will lie at 1.0 and 2.0, respectively, on the same logarithmic scale.
Instruments are available that provide other forms of band analysis (see Section 3.12). However, they do not enjoy the advantage of standardisation so that the comparison of readings taken on such instruments may be difficult. One way to ameliorate the problem is to present such readings as mean levels per unit frequency.
Data presented in this way are referred to as spectral density levels as opposed to band levels. In this case the measured level is reduced by ten times the logarithm to the base ten of the bandwidth. For example, referring to Table 1.2, if the 500 Hz octave band which has a bandwidth of 354 Hz were presented in this way, the measured octave band level would be reduced by 10 log10 (354) = 25.5 dB to give an estimate of the spectral density level at 500 Hz.
The problem is not entirely alleviated, as the effective bandwidth will depend upon the sharpness of the filter cut-off, which is also not standardised. Generally, the bandwidth is taken as lying between the frequencies, on either side of the pass band, at which the signal is down 3 dB from the signal at the centre of the band.
The spectral density level represents the energy level in a band one cycle wide whereas by definition a tone has a bandwidth of zero.
There are two ways of transforming a signal from the time domain to the frequency domain. The first requires the use of band limited digital or analog filters.
The second requires the use of Fourier analysis where the time domain signal is transformed using a Fourier series. This is implemented in practice digitally (referred to as the DFT – discrete Fourier transform) using a very efficient algorithm known as the FFT (fast Fourier transform). Digital filtering is discussed in Appendix D.
1.11 COMBINING SOUND PRESSURES
p2 ' p102 cos2(ωt%β1)%p202 cos2(ωt%β2)
%2p10p20cos(ωt%β1) cos(ωt%β2) (1.92)
p2 ' 1
2p102 1%cos2(ωt%β1) % 1
2p202 1%cos2(ωt%β2)
% p10p20cos(2ωt%β1%β2)%cos(β1&β2) (1.93) relationships with each other and as observed in Section 1.4.4, their summation is strongly dependent upon their phase relationship. Such sounds are known as coherent sounds.
Coherent sounds are sounds of fixed relative phase and are quite rare, although sound radiated from different parts of a large tonal source such as an electrical transformer in a substation are an example of coherent sound. Coherent sounds can also be easily generated electronically. When coherent sounds combine they sum vectorially and their relative phase will determine the sum (see Section 1.4.4).
It is more common to encounter sounds that are characterised by varying relative phases. For example, in an orchestra the musical instruments of a section may all play in pitch, but in general their relative phases will be random. The violin section may play beautifully but the phases of the sounds of the individual violins will vary randomly, one from another. Thus the sounds of the violins will be incoherent with one another, and their contributions at an observer will sum incoherently. Incoherent sounds are sounds of random relative phase and they sum as scalar quantities on an energy basis. The mathematical expressions describing the combining of incoherent sounds may be considered as special limiting cases of those describing the combining of coherent sound.
Sound reflected at grazing incidence in a ground plane at large distance from a source will be partially coherent with the direct sound of the source. For a source mounted on a hard ground, the phase of the reflected sound will be opposite to that of the source so that the source and its image will radiate to large distances near the ground plane as a vertically oriented dipole (see Section 5.3). The null plane of a vertically oriented dipole will be coincident with the ground plane and this will limit the usefulness of any signalling device placed near the ground plane. See Section 5.10.2 for discussion of reflection in the ground plane.
1.11.2 Addition of Coherent Sound Pressures
When coherent sounds (which must be tonal and of the same frequency) are to be combined, the phase between the sounds must be included in the calculation.
Let p ' p1%p2 and pi ' pi0cos(ωt%βi), i'1, 2; then:
where the subscript, 0, denotes an amplitude.
Use of well known trigonometric identities (Abramowitz and Stegun, 1965) allows Equation (1.92) to be rewritten as follows:
¢p2¦ ' ¢p12¦ % ¢p22¦ % 2¢p1p2¦cos(β1&β2) (1.94)
p1%p2 ' A1cosωt%A2cos(ω%∆ω)t (1.95)
p1%p2 ' Acos(∆ω/2)tcos(ω%∆ω/2)t
%Bcos(∆ω/2&π/2)t cos(ω%∆ω/2&π/2)t (1.96) Substitution of Equation (1.93) into Equation (1.58) and carrying out the indicated operations gives the time average total pressure, +p2,. Thus for two sounds of the same frequency, characterised by mean square sound pressures +p12, and +p22, and phase difference β1 - β2, the total mean square sound pressure is given by the following equation:
1.11.3 Beating
When two tones of very small frequency difference are presented to the ear, one tone, which varies in amplitude with a frequency modulation equal to the difference in frequency of the two tones, will be heard. When the two tones have exactly the same frequency, the frequency modulation will cease. When the tones are separated by a frequency difference greater than what is called the “critical bandwidth”, two tones are heard. When the tones are separated by less than the critical bandwidth, one tone of modulated amplitude is heard where the frequency of modulation is equal to the difference in frequency of the two tones. The latter phenomenon is known as beating.
For more on the beating phenomenon, see Section 2.2.6.
Let two tonal sounds of amplitudes A1 and A2 and of slightly different frequencies, ω and ω + ∆ω be added together. It will be shown that a third amplitude modulated sound is obtained. The total pressure due to the two tones may be written as:
where one tone is described by the first term and the other tone is described by the second term in Equation (1.95).
Assuming that A1 $ A2 , defining A = A1 + A2 and B = A1 - A2 , and using well known trigonometric identities, Equation (1.95) may be rewritten as follows:
When A1 = A2 , B = 0 and the second term in Equation (1.96) is zero. The first term is a cosine wave of frequency (ω + ∆ω) modulated by a frequency ∆ω/2. At certain values of time, t, the amplitude of the wave is zero; thus, the wave is described as fully modulated. If B is non-zero as a result of the two waves having different amplitudes, a modulated wave is still obtained, but the depth of the modulation decreases with increasing B and the wave is described as partially modulated. If ∆ω is small, the familiar beating phenomenon is obtained (see Figure 1.9). The figure shows a beating phenomenon where the two waves are slightly different in amplitude resulting in partial modulation and incomplete cancellation at the null points.
¢pt2¦ ' ¢p12¦ % ¢p22¦ (1.97)
Lpt ' 10 log1010L1/ 10%10L2/ 10 ... %10LN/ 10 (1.98)
x
t
Figure 1.9 Illustration of beating.
It is interesting to note that if the signal in Figure 1.9 were analysed on a very fine resolution spectrum analyser, only two peaks would be seen; one at each of the two interacting frequencies. There would be no peak seen at the beat frequency as there is no energy at that frequency even though we apparently "hear" that frequency.
1.11.4 Addition of Incoherent Sounds (L ogarithmic Addition)
When bands of noise are added and the phases are random, the limiting form of Equation (1.96) reduces to the case of addition of two incoherent sounds:
which may be written in a general form for the addition of N incoherent sounds as:
Incoherent sounds add together on a linear energy (pressure squared) basis. The simple procedure embodied in Equation (1.98) may easily be performed on a standard calculator. The procedure accounts for the addition of sounds on a linear energy basis and their representation on a logarithmic basis. Note that the division by 10, rather than 20 in the exponent is because the process involves the addition of squared pressures.
Ex ample 1.1
Assume that three sounds of different frequencies (or three incoherent noise sources) are to be combined to obtain a total sound pressure level. Let the three sound pressure levels be (a) 90 dB, (b) 88 dB and (c) 85 dB. The solution is obtained by use of Equation (1.81b).
¢p12¦ ' pref2 × 1090/10 ' pref2 × 10 × 108
¢p22¦ ' pref2 × 6.31 × 108
¢p32¦ ' pref2 × 3.16 × 108
¢pt2¦ ' ¢p12¦ % ¢p22¦ % ¢p32¦ ' pref2 × 19.47 × 108
Lpt ' 10 log10[¢pt2¦/pref2] ' 10 log10[19.47 × 108] ' 92.9 dB
Lpt ' 10 log101090/10 % 1088/10 % 1085/10 ' 92.9 dB
Lpt ' 10 log10 10L1/ 10%10L2/ 10
Lpt ' L1%10 log10 1%10(L2&L1) / 10
10(L2&L1) / 10#1
Solution For source (a):
For source (b):
For source (c):
The total mean square sound pressure is:
The total sound pressure level is:
Alternatively, in short form:
Some useful properties of the addition of sound levels will be illustrated with two further examples. The following example will show that the addition of two sounds can never result in a sound pressure level more than 3 dB greater than the level of the louder sound.
Ex ample 1.2
Consider the addition of two sounds of sound pressure levels L1 and L2 where L1 $ L2. Compute the total sound pressure level on the assumption that the sounds are incoherent; for example, that they add on a squared pressure basis:
then,
Since,
Lpt # L1%3 dB
Lpt ' 10 log10 j
N i'1
10(Li±∆) / 10
Lpt ' 10 log10 j
N i'1
10Li/ 10 ±∆
Lpm ' 10 log101092/10 & 1088/10 ' 89.8 dB(A) then,
Ex ample 1.3
Consider the addition of N sound levels each with an uncertainty of measured level ±∆.
Show that the level of the sum is characterised by the same uncertainty:
Evidently the uncertainty in the total is no greater than the uncertainty in the measurement of any of the contributing sounds.
1.11.5 Subtraction of Sound Pressure Levels
Sometimes it is necessary to subtract one noise from another; for example, when background noise must be subtracted from total noise to obtain the sound produced by a machine alone. The method used is similar to that described in the addition of levels and is illustrated here with an example.
Ex ample 1.4
The sound pressure level measured at a particular location in a factory with a noisy machine operating nearby is 92.0 dB(A). When the machine is turned off, the sound pressure level measured at the same location is 88.0 dB(A). What is the sound pressure level due to the machine alone?
Solution
For noise-testing purposes, this procedure should be used only when the total sound pressure level exceeds the background noise by 3 dB or more. If the difference is less
Lpi ' LpR & NRi (1.99)
Lp ' LpR % 10 log10j
n i'1
10&(NRi/10) (1.100)
NR ' 10 log10j
nA i'1
10&(NRA i/10) & 10 log10j
nB i'1
10&(NRBi/10) (1.101)
than 3 dB a valid sound test probably cannot be made. Note that here subtraction is between squared pressures.
1.11.6 Combining Level Reductions
Sometimes it is necessary to determine the effect of the placement or removal of constructions such as barriers and reflectors on the sound pressure level at an observation point. The difference between levels before and after an alteration (placement or removal of a construction) is called the noise reduction, NR. If the level decreases after the alteration, the NR is positive; if the level increases, the NR is negative. The problem of assessing the effect of an alteration is complex because the number of possible paths along which sound may travel from the source to the observer may increase or decrease.
In assessing the overall effect of any alteration, the combined effect of all possible propagation paths must be considered. Initially, it is supposed that a reference level LpR may be defined at the point of observation as a level which would or does exist due only to straight-line propagation from source to receiver. Noise reduction due to propagation over any other path is then assessed in terms of this reference level.
Calculated noise reductions would include spreading due to travel over a longer path, losses due to barriers, reflection losses at reflectors and losses due to source directivity effects (see Section 5.11.3).
For octave band analysis, it will be assumed that the noise arriving at the point of observation by different paths combines incoherently. Thus, the total observed sound level may be determined by adding together logarithmically the contributing levels due to each propagation path.
The problem that will now be addressed is how to combine noise reductions to obtain an overall noise reduction due to an alteration. Either before alteration or after alteration, the sound pressure level at the point of observation due to the ith path may be written in terms of the ith path noise reduction, NRi, as:
In either case, the observed overall noise level due to contributions over n paths, including the direct path, is:
The effect of an alteration will now be considered, where note is taken that, after alteration, the propagation paths, associated noise reductions and number of paths may differ from those before alteration. Introducing subscripts to indicate cases A (before alteration) and B (after alteration) the overall noise reduction (NR = LpA - LpB) due to the alteration is:
NR ' 10 log1010&0/10 % 10&5/10
& 10 log1010&4/10 % 10&6/10 % 10&7/10 % 10&10/10 ' 1.2 % 0.2 ' 1.4 dB
Ex ample 1.5
Initially, the sound pressure level at an observation point is due to straight-line propagation and reflection in the ground plane between the source and receiver. The arrangement is altered by introducing a very long barrier, which prevents both initial propagation paths but introduces four new paths (see Section 8.5). Compute the noise reduction due to the introduction of the barrier. In situation A, before alteration, the sound pressure level at the observation point is LpA and propagation loss over the path reflected in the ground plane is 5 dB. In situation B, after alteration, the losses over the four new paths are respectively 4, 6, 7 and 10 dB.
Solution
Using Equation (1.101) gives the following result: