Engineering Acoustics. Malcolm J. Crocker

tones are present. However, the method is somewhat time‐consuming and involves measuring the area under a curve. It has been well described elsewhere [13].

      Stevens [14] at about the same time (1957–1961) developed quite different procedures for calculating loudness. Stevens' method, which he named Mark VI, is simpler than Zwicker's method but is only designed to be used for diffuse sound fields and when the spectrum is relatively flat and does not contain any prominent pure tones.

      The procedure used in the Stevens Mark VI method is to plot the noise spectrum in either octave or one‐third‐octave bands onto the loudness index contours. The loudness index (in sones) is determined for each octave (or one‐third‐octave) band and the total loudness S is then given by

(4.2)

where S_max is the maximum loudness index and ∑S is the sum of all the loudness indices. The 0.3 constant (used for octave bands) in Eq. (4.2) is replaced by 0.15 for one‐third‐octave bands. The Zwicker method is based on the critical band concept and, although more complicated than the Stevens method, can be used either with diffuse or frontal sound fields. Complete details of the procedures are given in the ISO standard [15] and Kryter [16] has discussed the critical band concept in his book.

      Example 4.3

The octave band sound pressure levels (see column 2 of Table 4.1) of the noise from machines in a factory are given for the center frequencies (see column 1 of Table 4.1). Calculate the loudness level.

      Solution

From Figure 4.8 the loudness indices, S, given in column 3 of Table 4.1 are found.

      We obtain that S_max = 26.5 and ∑S = 134.2. Thus from Eq. (4.2) the loudness is: S = 26.5 + 0.3 (134.2 − 26.5) = 59 sones (OD, Octave Diffuse) and from Eq. (4.1), or Figure 4.7, the loudness level is: P = 99 phons (OD).

Table 4.1 Octave band levels of factory noise and Stevens' loudness indices.

Octave band center frequency, Hz	Octave band level, dB	Band loudness index S
31.5	75	3.0
63	79	6.2
125	82	10.5
250	85	15.3
500	85	18.7
1000	87	26.5
2000	82	23.0
4000	75	17.5
8000	68	13.5

Figure 4.8 Contours of equal loudness index.

      There are several other aspects of loudness which we have not had room to discuss in this book, e.g. impulsive noise, and monaural and binaural loudness. Readers will find these well covered in Kryter's book [16].

      4.3.3 Masking

      The masking phenomenon is well known to most people. A loud sound at one frequency can cause another quieter sound at the same frequency or a sound close in frequency to become inaudible. This effect is known as masking. Broadband sounds can have an even more complicated masking effect and can mask louder sounds over a much wider frequency range than narrow‐band sounds. These effects are important in human assessment of product sound quality.

Noise that contains pure tones is generally annoying and unpleasant. In the case of automobile interior noise, engine and exhaust system noise consists mostly of fundamental pure tones together with integer multiplies and broadband engine combustion noise. Similarly cooling fan noise consists of pure tones and broadband aerodynamic fan noise. In the case of automobiles, exhaust and cooling fan noise have some strong pure tone components. Wind noise and tire noise are mostly broadband in nature and can mask some of the engine, exhaust, and cooling fan noise, particularly at high vehicle speed. Figure 4.9 shows the masking effect of broadband white noise. The solid lines represent the sound pressure level of a pure test tone (given in the ordinate) that can just be heard when the masking sound is white noise at the level shown on each curve. For example, it is seen that white noise at 40 dB masks pure tones at about 55 dB at 100 Hz, 60 dB at 1000 Hz, and 65 dB at 5000 Hz.

Figure 4.9 Contours joining sound pressure levels of pure tones at different frequencies that are masked by white noise at the spectral density level L_WN shown on each contour [17, 18].

The masking of noise has been studied extensively for more than 70 years. The curves in Figure 4.9 [17, 18] are very similar to those of much earlier research work by Fletcher and Munson in 1950 but are shown here since they have been extended to lower and higher frequencies than these earlier published results [19]. One interesting fact is that the curves shown in Figure 4.9 are all almost parallel and are separated by about 10‐dB intervals, which is also the interval between the masker levels. This suggests that the masking of noise is almost independent of level at any given frequency. This fact was also established by Fletcher and Munson in 1950 as is shown in Figure 4.10 [20]. Here the amount of masking is defined to be the upward shift in decibels in the baseline hearing threshold caused by the masking noise [19]. Figure