The Physics of Sound



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Measured Ratio Ratio in Exponent Decibel

Sound Intensity (Im) (Im/Ir) Exp. Not. (log10) (10 x log10)
Threshold 10-12 w/m2 1 100 0 0

@ 1 kHz
Whisper 10-8 w/m2 10,000 104 4 40


Conversational 10-6 w/m2 1,000,000 106 6 60

Speech
City Traffic 10-4 w/m2 100,000,000 108 8 80


Rock & Roll 10-2 w/m2 10,000,000,000 1010 10 100
Jet Engine 100 w/m2 1,000,000,000,000 1012 12 120

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Legitimate or not, the bel finds its sole application in textbooks attempting to explain the decibel. For reasons that are purely historical, the log10 of the intensity ratio is multiplied by 10, changing bel into the decibel (dB). As shown in the last column of Table 3-1, this has the very simple effect of turning 4 bels into 40 decibels, 8 bels into 80 decibels, etc. The formula for the decibel, then, is:
dBIL = 10 log10 Im/Ir, where:
Im = a measured intensity

Ir = a reference intensity


The designation "IL" stands for intensity level, and it indicates that the underlying measurements are of sound intensity and not sound pressure. As will be seen below, a different version of this formula is needed if sound pressure measurements are used. The multiplication by 10 in the dBIL formula is a simple operation, but it can sometimes have the unfortunate effect of making the formula appear more obscure that it is. The decibel values that are calculated, however, should be readily interpretable. For example, 30 dBIL means 3 factors of 10 more intense than Ir, 60 dBIL means 6 factors of 10 more intense than Ir, and 90 dBIL means 9 factors of 10 more intense than Ir.
Deriving a Pressure Version of the dB Formula
In a simple world, we would be finished with the decibel scale. The problem is that the formula is based on measurements of sound intensity, but as a purely practical matter sound intensity is difficult to measure. Sound pressure, on the other hand, is quite easy to measure. An ordinary microphone, for example, is a pressure sensitive device. The problem, then, is that the decibel is defined in terms of intensity measurements, but the measurements that are actually used will nearly always be measures of sound pressure. This problem can be addressed since there is a predictable relationship between intensity (I) and pressure (E): intensity is proportional to pressure squared:
I 
Knowing this relationship allows us to create a completely equivalent version of the decibel formula that will work when sound pressure measurements are used instead of sound intensity measurements. All we need to do is substitute squared pressure measurements in place of the intensity measurements:
dBIL = 10 log10 Im/Ir (intensity version of formula)

dBSPL= 10 log10 E2m/E2 (pressure version of formula)


The designation "SPL" stands for sound pressure level, and it indicates that measures of sound pressure have been used and not measures of sound intensity. Although the dBSPL formula shown here will work fine, it will almost never be seen in this form. The reason is that the formula is algebraically rearranged so that the squaring operation is not needed. The algebra is shown below:
(1) dBIL = 10 log10 Im/Ir (the intensity version of the formula)

(2) dBSPL = 10 log10 E2m/E2 (measures of E2 replace measures of I because I  E2)


(3) dBSPL = 10 log10 (Em/Er)2 (a2/b2 = (a/b)2)
(4) dBSPL = 10 2 log10 Em/Er (this is the only tricky step: log ab = b log a)
(5) dBSPL = 20 log10 Em/Er (2 10 = 20)
With the possible exception of the fourth step,4 the algebra is straightforward, but the details of the derivation are less important than the following general points:
1. The decibel formula is defined in terms of intensity ratios. The basic formula is;
dBIL = 10 log10 Im/Ir.
2. While sound intensity is difficult to measure, sound pressure is easy to measure. It is therefore necessary to derive a version of the decibel formula that works when measures of sound pressure are used instead of sound intensity.
3. The derivation of the pressure version of the formula is based entirely on the fact that intensity is proportional to pressure squared (I ). This allows measures of E2 to replace measures of I, turning: dBIL = 10 log10 Im/Ir into dBSPL = 10 log10 E2m/Er2. A few algebra tricks are applied to turn this formula into the more aesthetically pleasing final version: dBSPL = 20 log10 Em/Er .


  1. The two versions of the formula are fully equivalent to one another (see Box 3-3).

This last point about the equivalence of the intensity and sound pressure versions of the formula is explained in some detail in Box 3-3, but the basic point is quite simple. The pressure version of the dB formula was derived from the intensity version of the formula through algebraic manipulations (based on this relationship: I ). The whole


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Box 3-2
Harmonics, Octaves, Linear Scales, and Logarithmic Scales
As we will see when the decibel scale is introduced, there is an important distinction to be made between linear scales, which are quite common, and logarithmic scales, which are less common but quite important. This distinction can be illustrated by examining the difference between a harmonic progression and an octave progression. Notice that in a harmonic progression, the spacing between the harmonics is always the same; that is, the difference between H1 and H2 is the same as the difference between H2 and H3, and so on. This is because increases in frequency between one harmonic and the next involve adding a constant, with the constant being the fundamental frequency. For example:
H1 500

H2 1000 (add 500)

H3 1500 (add 500)

H4 2000 (add 500)

. .

. .


. .
To get from one scale value to another on an octave progression involves multiplying by a constant rather than adding a constant. For example, an octave progression starting at 500 Hz looks like this:
O1 500

O2 1000 (multiply by 2)

O3 2000 (multiply by 2)

O4 4000 (multiply by 2)

. .

. .


. .
As a result of the fact that we are multiplying by a constant rather than adding a constant, the spacing is no longer even (i.e., the spacing between O1 and O2 is 500 Hz, the spacing between O2 and O3 is 1000 Hz, and so on). The point to be made of this is that there are two fundamentally different kinds of scales: (1) scales like harmonic progressions that are created by adding a constant, which are by far the more common, and (2) scales like octave progressions that are created by multiplying by a constant. Scales that are created by adding a constant are called linear scales, while scales that are created by multiplying by a constant are called logarithmic scales. Note that for an octave progression, the multiplier happens to be 2, meaning that progressing from one frequency to an octave above that frequency involves multiplication by 2. However, a logarithmic scale can be built using any multiplier. We will return to the distinction between linear and logarithmic scales when we talk about the decibel scale, and there we will see that a logarithmic scale is built around multiplication by a constant value of 10 rather than 2.

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point of algebra, of course, is to keep the expression on the left equal to the expression on the right. The simple and useful point that emerges from this is this: If an intensity meter shows that a given sound measures 60 dBIL, for example, a pressure meter will show that the same sound measures exactly 60 dBSPL. (This may seem counterintuitive due to the differences in the formulas, but see Box 3-3 for the explanation.) The equivalence of the two versions of the dB formula greatly simplifies the interpretation of sound levels that are expressed in decibels.


References
The reference that is used for the Mach scale is always the speed of sound. One of the virtues of the decibel scale is that any reference can be used as long as it is clearly specified. The only reference that has been mentioned

so far is 10-12 w/m2, which is roughly the audibility threshold for a 1,000 Hz pure tone. This is a standard reference intensity, and unless otherwise stated it should be assumed that this is used when a signal level is reported in dBIL.

The standard reference that is used for dBSPL is 20 Pa, so when a signal level is reported in dBSPL it should be assumed that this reference is used unless otherwise stated.5
Many references besides these two standard references can be used. For example, suppose that a speech signal is presented to a listener at an average level of 3500 Pa in the presence of a noise signal whose average sound pressure is 1400 Pa. The speech-to-noise ratio (S/N) can be represented on a decibel scale, using the level of the speech as Em and the level of the noise as Er:
dBs/n = 20 log10 Em/ Er

= 20 log10 3500/1400

= 20 log10 2.5

= 20 (0.39794)

= 7.96 dB


To take one more example, assume that a voice patient prior to treatment produces sustained vowels that average 2300 Pa. Following treatment the average sound pressures increase to 8890 Pa. The improvement in sound pressure (post-treatment relative to pre-treatment) can be represented on a decibel scale:
dBImprovement = 20 log10 Epost/Epre

= 20 log10 8890/2300

= 20 log10 (3.86522)

= 20 (0.58717)

= 11.74 dB
A final example can be used to make the point that the decibel scale can be used to represent intensity ratios for any type of energy, not just sound. Bright sunlight has a luminance measuring 100,000 cd/m2 (candela per square meter). Light from a barely visible star, on the other hand, has a luminance measuring 0.0001 cd/m2. We can now ask how much more luminous bright sunlight is in relation to barely visible star light, and the dB scale can be used to represent this value. Since the underlying physical quanities here are measures of electromagnetic intensity, we want the intensity version of the formula rather than the pressure version.
dB = 10 log10 Isunlight/Istarlight

= 10 log10 100000/0.0001

= 10 log10 105/10-4

= 10 log10 109 (division is done by subtracting exponents: 5 – (-4) = 9)

= 10 (9)

= 90 dB
The fact that we are measuring light rather than sound makes no difference: a decibel is 10 log10 Im/Ir (or, equivalently, 20 log10 Em/Er), regardless of whether the energy comes from sound, light, electrical current, or any other type of energy.


dB Hearing Level (dBHL)
The dB Hearing Level (dBHL) scale was developed specifically for testing hearing sensitivity for pure tones of different frequencies. The sound-level dials on clinical audiometers,6 for example, are calibrated in dBHL rather than dBSPL. To understand the motivation for the dBHL scale examine Figure 3-24, which shows the sound level (in dBSPL) required for the average, normal-hearing listener to barely detect pure tones at frequencies between 125 and 8000 Hz. This is called the audibility curve and the simple but very important point to notice about this graph is that the curve is not a flat line; that is, the ear is clearly more sensitive at some frequencies than others. The differences in sensitivity are quite large in some cases. For example, the average normal-hearing listener will barely detect a 1000 Hz pure tone at 7 dBSPL, but at 125 Hz the sound level needs to be cranked all the way up to 45 dBSPL, an increase in intensity of nearly 4000:1. Now suppose we were to test pure-tone sensitivity using an audiometer that is calibrated in dBSPL. Imagine that a listener barely detects a 1000 Hz pure tone at 25 dBSPL. Does this listener have a hearing loss, and if so how large? The only way to answer this question is to consult the data in Figure 3-24, which shows that the threshold of audibility for the average normal hearing listener at 1000 Hz is 7 dBSPL. This means that the hypothetical listener in this example has a hearing loss of 25-7 = 18 dB. Suppose further that the same listener detects a 250 Hz tone at 20 dBSPL. The table in Figure 3-24 shows that normal hearing sensitivity at 250 Hz is 25.5 dBSPL, meaning that the listener has slightly better than normal hearing at this frequency. As a final example, imagine that this listener barely detects a 500 Hz tone at 30 dBSPL. Since the table shows that normal hearing sensitivity at 500 Hz is 11.5 dBSPL, the listener has a hearing loss of 30.0-11.5 = 18.5 dB. The simple point to be made about these examples is that, with an audiometer dial that is calibrated in dBSPL, it is not possible to determine whether a listener has a hearing loss, or to measure the size of that loss, without doing some arithmetic involving the normative data in Figure 3-24. The dBHL scale, however, provides a simple solution to this problem that avoids this arithmetic entirely. The solution involves calibrating the audiometer in such a way that, when the level dial is set to 0 dBHL, sound level is set to the threshold of audibility for the average normal-hearing listener for that signal frequency. For example, when the level dial is set to 0 dBHL at 125 Hz the level of tone will be 45 dBSPL – the threshold of audibility for the average normal hearing listener at this frequency. Now if a listener barely detects the 125 Hz tone at 0 dBHL, no arithmetic is needed; the listener has normal hearing at this frequency. Further, if the listener barely detects this 125 Hz tone at 40 dBHL, for example, the listener must have a 40 dB loss at this frequency – and again it is not necessary to consult the data in Figure 3-24. Similarly, when the level dial is set to 0 dBHL at 250 Hz the level of the tone will be 25.5 dBSPL, which is the audibility threshold at 250 Hz. If this tone is barely detected at 0 dBHL, the listener has normal hearing at this frequency. However, if the tone is not heard until the dial is increased to 50 dB dBHL, for example, the listener has a 50 dB hearing loss at this frequency. The same system is used for all signal frequencies: in all cases, the 0 dBHL reference is not a fixed number as it is for dBSPL (a constant value of 20 Pa, no matter what the signal frequency is) or dBIL (a constant value of 10-12 watts/m2, again independent of signal frequency), but rather a family of numbers. In each case the reference for the dBHL scale is the threshold of audibility for an average, normal-hearing listener at a particular signal frequency. What this means is that values in dBHL are a fixed distance above the audibility curve, although they may be very different levels in dBSPL. For illustration, Figure 3-25 shows the audibility curve (the filled symbols) and, above that in the unfilled symbols, a collection of values that all measure 30 dBHL. Although the sound levels on the 30 dBHL curve vary considerably in dBSPL (i.e. measured using 20 Pa as the reference), every data point on this curve is a constant 3 factors of 10, or 30 dB, above the audibility curve. The value of 30 dB in this figure is just an example. All values in dBHL and dBSPL are interpreted in the same way: 50 dBSPL means that the signal being measured is 100,000 times (i.e., 5 factors of 10) more intense than the fixed reference of 20 Pa, independent of frequency; 50 dBHL, on the other hand, means that the signal being measured is 100,000 times (again, 5 factors of 10) more intense than a tone that is barely audible to a normal-hearing listener at that signal frequency. Similarly, 20 dBSPL means that the signal is 20 dB (2 factors of 10) more intense than the fixed reference of 20 Pa, while 20 dBHL means that the signal is 20 dB (again, 2 factors of 10) above the audibility curve.
Summary
The decibel is a powerful scale for representing signal amplitude. The scale has two important properties: (1) similar to the Mach scale, it represents signal level not in absolute terms but as a measured level divided by a reference level; and (2) like the Richter scale, the dB scale is logarithmic rather than linear, meaning that it is based on equal multiplicative distances rather than equal additive distances. While the decibel is defined in terms of intensity ratios, for practical reasons, measures of sound pressure are far more common than measures of sound intensity. Consequently, a version of the decibel formula was derived that makes use of pressure ratios rather than intensity ratios. The derivation was based on the fact that intensity is proportional to pressure squared. The two versions of the decibel formula (dBIL = 10 log 10 Im/Ir and dBSPL = 20 log 10 Em/Er) are fully equivalent, meaning that if a sound measures 60 dBIL that same sound will measure 60 dBSPL. Unlike the Mach scale, which always uses the speed of sound as a reference, any number of references can be used with the decibel scale. The standard reference for the dBIL scale is 10-12 w/m2 and the standard reference for the dBSPL scale is 20 Pa. However, any level can be used as a reference as long as it is specified. The dBHL scale, widely used in audiological assessment, was developed specifically for measuring sensitivity to pure tones of difference frequencies. The reference that is used for the dBHL scale is the threshold of audibility at a particular signal frequency for the average, normal-hearing listener. Sound levels in dBSPL and dBHL are interpreted quite differently. For example, a pure tone measuring 40 dBSPL is 4 factors of 10 (i.e., 40 dB) greater than the fixed SPL reference of 20 Pa, while a pure tone measuring 40 dBHL is 4 factors of 10 (again, 40 dB) greater than a tone of that same frequency that is barely audible to an average, normal-hearing listener.


Frequency Threshold

125 45.0

250 25.5

500 11.5

750 8.0

1000 7.0

1500 6.5

2000 9.0

3000 10.0

4000 9.5

6000 15.5

8000 13.0


Figure 3-24. The threshold of audibility for the average, normal-hearing listener for pure tones varying between 125 and 8000 Hz. The audibility threshold is the sound level in dBSPL that is required for a listener to barely detect a tone. Values on this curve are shown in the table to the right. The most important point to note about this graph is that the curve is not flat, meaning that the ear is more sensitive at some frequencies than others. In particular, the ear is more sensitive in a range of mid-frequencies between about 1000 and 4000 Hz than it is at lower and higher frequencies. The complex shape of this curve provides the underlying motivation for the dBHL scale. See text for details.




Figure 3-25. The lower function is the audibility curve – the sound level in dBSPL that is required for an average normal hearing listener to barely detect pure tones of different frequencies. The upper function shows sound levels for a set of tones that all measure 30 dBHL. These tones vary quite a bit in dBSPL (i.e., relative to the constant value of 20 Pa) but in all cases the tones are a constant 3 factors of 10 in intensity (i.e., 30 dB) above the audibility curve.

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Box 3-3
THE EQUIVALENCE OF THE INTENSITY AND PRESSURE

VERSIONS OF THE DECIBEL FORMULA
One fact about the two versions of the dB formula that is not always well understood is that the dBIL and dBSPL formulas are fully equivalent. By "fully equivalent" we mean the following: suppose that a sound intensity meter is used to measure the level of some sound, and we find that this sound is 1,000 times more intense than the standard intensity reference of 10-12 w/m2. The sound would then measure 30 dBIL (10 log10 1,000 = 10 (3) = 30 dBIL). Now suppose that we put the sound intensity meter away and use a sound pressure meter to measure the same sound. You might think that the sound would measure 60 dBSPL since now we are multiplying by 20 instead of 10, but the trick is that the ratio is no longer 1,000. Recall that intensity is proportional to pressure squared, which means that pressure is proportional to the square root of intensity. This means that if the intensity ratio is 1,000, the pressure ratio must be the square root of 1,000, or 31.6. So, the formula now becomes 20 log 31.6 = 20 (1.5) = 30 dBSPL, which is exactly what we obtained originally. It will always work out this way: if a sound measures 50 dBIL, that same sound will measure 50 dBSPL.
Table 3-2 might help to make this more clear. The first column shows an intensity ratio, the second column shows the corresponding pressure ratio (this is always the square root of the intensity ratio), the third column shows the dBIL value (10 log of the intensity ratio), and the fourth column shows dBSPL value (20 log of the pressure ratio). As you can see, they are always the same.

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Table 3-2. Intensity ratios, equivalent pressure ratios,  dBIL values and

dBSPL values showing the equivalence of the intensity and pressure versions

of the dB formula.

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Intensity Pressure dBIL dBSPL

Ratio  Ratio (10 log10 Im/Ir) (20 log10 Em/Er)


10 3.16 10.00 10.00

20 4.47 13.01 13.01

40 6.32 16.02 16.02

50 7.07 16.99 16.99

60 7.75 17.78 17.78

70 8.37 18.45 18.45

80 8.94 19.03 19.03

90 9.49 19.54 19.54

100 10.00 20.00 20.00

200 14.14 23.01 23.01

300 17.32 24.77 24.77

400 20.00 26.02 26.02

500 22.36 26.99 26.99

1000 31.62 30.00 30.00

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Study Questions: Physical Acoustics
1. Explain the basic processes that are involved in the propagation of a sound wave.
2. Draw time- and frequency-domain representations of simple periodic, complex periodic, complex aperiodic, and transient sounds.
3. Draw time- and frequency-domain representations of two complex periodic sounds with different fundamental frequencies.
4. Draw time-domain representations of two simple periodic sounds with the same frequency and phase, but different amplitudes.
5. Draw time-domain representations of two simple periodic sounds with the same frequency and different amplitudes but different phases.
6. Draw amplitude spectra of two sounds with the same fundamental frequencies but different spectrum envelopes.
7. Draw amplitude spectra of two sounds with different fundamental frequencies but similar spectrum envelopes.
8. Calculate signal frequencies for sinusoids with the following values:
a. period = 0.34 s

b. period = 2 s

c. period = 10 ms

d. period = 2 ms

e. wavelength = 20 cm

f. wavelength = 100 cm


Answers:
a. f = 1/0.34 = 2.94 Hz

b. f = 1/2 = 0.5 Hz

c. f = 1/0.01 = 100 Hz

d. f = 1/.002 = 500 Hz

e. f = c/WL (speed of sound/wavelength) = 35000/20 = 1750 Hz

f. f = c/WL (speed of sound/wavelength) = 35000/100 = 350 Hz


9. Calculate the three lowest resonant frequencies of the following uniform tubes that are closed at one end and open at the other end:
a. 10 cm

b. 30 cm


c. 40 cm
Answers:
a. wavelength of lowest resonance = 40 cm (10 x 4)

f = 35000/40 = 875

R1 = 875 (R1 = frequency of resonance number 1)

R2 = 2625

R3 = 4375
b. wavelength of lowest resonance = 120 cm (30 x 4)

f = 35000/120 = 291.7

R1 = 291.7

R2 = 875.0

R3 = 1458.3
c. wavelength of lowest resonance = 160 cm (40 x 4)

f = 35000/160 = 218.75

R1 = 218.75

R2 = 656.25

R3 = 1093.75
10. Show what the frequency-response curves look like for the tubes in the problem above.
11. A complex periodic signal has a fundamental period of 4 msec. What is the fundamental frequency of the signal? At what frequencies would we expect to find energy?
12. How are the terms octave and harmonic different?
13. Give examples of the following kinds of graphs, being sure to label both axes:
a. amplitude spectrum

b. phase spectrum

c. frequency-response curve

d. time-domain representation


14. Give a brief explanation of the basic idea behind Fourier analysis. What is the input to Fourier analysis and what kind of output(s) does it produce?
15. Draw and label frequency-response curves for low-pass, high-pass, and band-pass filters.
16. What parameters control the frequency of vibration of a spring and mass system?
17. Draw the time domain representation of one cycle of a sinusoid as variations in instantaneous air pressure over time and one cycle of that same sinusoid as variations in instantaneous velocity over time.
18. How, if at all, are the terms resonant frequency and harmonic different?
19. How, if at all, are the terms resonant frequency and formant different?
20. A harmonic is a peak in: (a) a frequency response curve, (b) an amplitude spectrum, or (c) either a frequency response curve or an amplitude spectrum.
21. A resonance is a peak in: (a) a frequency response curve, (b) an amplitude spectrum, or (c) either a frequency response curve or an amplitude spectrum.
22. A formant is a peak in: (a) a frequency response curve, (b) an amplitude spectrum, or (c) either a frequency response curve or an amplitude spectrum.
23. A frequency response curve describes a _________________________.
24. An amplitude spectrum describes a _________________________.
Frequency Response Problems

Answers to Frequency Response Problems

Decibel Study Questions



  1. What reference is used for the dBIL scale?




  1. What reference is used for the dBSPL scale?




  1. What reference is used for the dBHL scale?




  1. What reference is used for the dBSL scale?




  1. A listener barely detects a 125 Hz pure tone at 55 dBSPL. Does this listener have a hearing loss at 125 Hz, and if so, what is the size of the hearing loss?




  1. A listener barely detects a 1,000 Hz pure tone at 55 dBSPL. Does this listener have a hearing loss at 1,000 Hz, and if so, what is the size of the hearing loss?




  1. A listener barely detects a 125 Hz pure tone at 55 dBHL. Does this listener have a hearing loss at 125 Hz, and if so, what is the size of the hearing loss?




  1. A listener barely detects a 1,000 Hz pure tone at 55 dBHL. Does this listener have a hearing loss at 1,000 Hz, and if so, what is the size of the hearing loss?




  1. 60 dBSPL at 1,000 Hz means ___________________ more intense than ___________________.




  1. 60 dBIL at 1,000 Hz means ___________________ more intense than ___________________.




  1. 60 dBHL at 1,000 Hz means ___________________ more intense than ___________________.




  1. The reference that is used for the dBSPL scale is:




  1. a number

  2. a sentence




  1. If the answer to the question above is a number, give the number; if it’s a sentence, give the sentence.




  1. The reference that is used for the dBHL scale is:




  1. a number

  2. a sentence




  1. If the answer to the question above is a number, give the number; if it’s a sentence, give the sentence.




  1. A specific individual has a 70 dB hearing loss in the left ear at 1,000 Hz. A 90 dBHL, 1,000 Hz tone that is presented to this listener’s left ear would measure ______ dBSL.




  1. A sound measures 42 dBIL. On the dBSPL scale, that same sound will measure:




  1. 84 dBSPL because with the dBSPL formula we are now are multiplying the ratio by 20 instead of 10.

  2. 42 dBSPL because the two versions of the formula are equivalent

18. A sound measures 60 dBIL. (a) The measured intensity (IM) must therefore be _________ times

greater than the reference intensity (IR). (b) What would the pressure ratio (EM/ER) be for this same sound? (c) Do the arithmetic to show what this sound would measure in dBSPL.
19. A sound measures 40 dBIL. (a) The measured intensity (IM) must therefore be _________ times

greater than the reference intensity (IR). (b) What would the pressure ratio (EM/ER) be for this same sound? (c) Do the arithmetic to show what this sound would measure in dBSPL.


20. On the graph below, put a mark at: (a) 3,000 Hz, 20 dBSPL, and (b) 3,000 Hz, 20 dBHL (the

grid lines on the y axis are spaced at 2 dB intervals).





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