The noise measured in the vicinity of a jet engine is shown in Table C3, expressed as dB re. 20 μPa in standard third-octave bands.
(a) Find the overall sound pressure level (OASPL) of the noise in dB re. 20 μPa.
(b) Express the spectrum given in Table C3 in the form of power spectral density, (PSD), in Pa²/Hz, suitable for use in a structural response calculation.

Question

Accepted Answer

Part (a)
To add dB levels, they must first be expressed as mean square levels. In Table C4, the RMS level, p, in each band is given in the third column, using Eq. (C4) in the inverted form:

[latex]N\mathrm{(dB)} = 10\log_{10} \left(\frac{p^{2}}{p_{REF}^{2}} ight) = 20\log_{10} \left(\frac{p}{p_{REF}} ight) (C4) \ p = p_{REF} 10^{\frac{N}{20}}[/latex]

where N is the dB figure in the second column and [latex]p_{REF} = 20 μPa[/latex]. Mean square values, [latex]p^{2}[/latex], are given in the fourth column. These are summed, giving an overall mean square value of [latex]8.596 × 10^{6} Pa^{2}[/latex], and, taking the square root, an overall RMS value of 2932 Pa. Equation (C4) then gives

[latex]\mathrm{OASPL(dB)} = 20\log_{10} \frac{p}{p_{REF}} = 20\log_{10} \frac{2932}{20 imes 10^{-6}} = 163.3 \mathrm{dB re.} 20 \mu Pa[/latex]

Part (b)
The bandwidth of each third octave band is given by Eq. (C5), i.e. [latex]0.2315 f_{c}[/latex], and is entered in the fifth column. The PSD in each frequency band is then given by dividing the mean square pressure in the fourth column by the bandwidth in the fifth column.
The last column gives the spectral density in Pa²/Hz, as required for a structural response calculation.

[latex]\mathrm{Bandwidth} = \left(2^{\frac{1}{6}} f_{c} - 2^{-\frac{1}{6}} f_{c} ight) = 0.2315 f_{c} (C5)[/latex]

Table C3
500	400	320	250	200	160	125	100	80	64	50	Band center (Hz)
152	153	154	156	156	155	153	150	148	146	144	SPL in band (dB re.20 μPa)

Table C4
PSD (Pa²/Hz)	Bandwidth (Hz)	Mean SQ. pressure p² (Pa²)	RMS pressure p (Pa)	S.P.L. in band (dB re.20 μPa)	Band center $f_{c}$ (Hz)
8680	11.58	100×10³	317	144	50
10750	14.82	159×10³	399	146	64
13630	18.52	252×10³	502	148	80
17280	23.15	400×10³	632	150	100
27580	28.93	798×10³	893	153	125
34150	37.04	1265×10³	1125	155	160
34390	46.30	1592×10³	1262	156	200
27510	57.88	1592×10³	1262	156	250
13560	74.08	1005×10³	1002	154	320
8620	92.60	798×10³	893	153	400
5480	115.80	634×10³	796	152	500
		$8.596×10^{6}$	2932	163.3	Overall

Question C.1: The noise measured in the vicinity of a jet engine is shown ......