|
10
|
(22) It follows then that 101 = _______
and 100 = 1.
|
|
|
It is very important to the concept of the decibel to
remember that the logarithm of 1 is zero.
|
|
1 0 0
|
(23) 100 =________. 10___ = 1. The log of 1 is ________.
|
|
logarithm
|
(24) The ______ of 1 is zero.
|
|
3
3
|
(25) The log of 1,000 is _______. This is the same as 10___ = 1,000.
|
|
2
2
|
(26) The log of 100 is ______. This is the same as 10___ = 100.
|
|
1
1
|
(27) The log of 10 is ________. This is the same as 10___ = 10.
|
|
0
0
|
(28) The log of 1 is _____. This is the same as 10___ = 1.
|
|
1
|
(29) The log of ______ is zero.
|
|
|
We must also review a bit about ratios. If we divide a number by itself, as in 198/198,
we get a ratio of 1. If we divide
3,456 by 3,456, we still get a ratio of 1.
If we divide 0.05 by 0.05, we still get a ratio of 1.
|
|
1
|
(30) If we divide 20 by 20, we get a ratio
of _________.
|
|
1
|
(31) Regardless of the magnitude of the numbers
chosen, any number divided by itself equals _______.
|
|
10 1
|
(32)  = _______ or 10___.
|
|
100 2
|
(33)  = _______ or 10___.
|
|
1,000 3
|
(34)  = _______ or 10___.
|
|
1 0
|
(35)  = _______ or 10___.
|
|
|
THE DECIBEL IN MEASUREMENTS FROM A POWER REFERENCE
|
|
|
Study the following equation:
#
dB IL = 10 * log 
WO = watt/cm2
(power) output, and
WR = watt/cm2 (power)
reference.
|
|
output
|
(36) WO = watt/cm2 power ______,
and
WR = watt/cm2 power
reference.
|
|
reference
|
(37) WO = watt/cm2 power output
while
WR = watts /cm2 power _______.
|
|
watt/cm2 power output
|
(38) WO = ________.
|
|
watt/cm2 power reference
|
(39) WR = __________________.
|
|
|
To solve the equation, one must first find the
numerical value of  . This is a ratio.
|
|
1,000
|
(40) If WO = 100 and
WR = 0.1, then the ratio = ________.
|
|
1
|
(41) If WO = 1,000
and WR = 1,000, then the ratio = ______.
|
|
|
If your mathematics is sufficiently advanced to
permit you to answer Item 42, you may then skip to Item 54.
|
|
104
|
(42) If WO = 10-12 and WR = 10-16,
then the ratio of
 is 10,000 or
_______.
If you could not answer Item 42, you must read the next section before you
can go on; otherwise, go to Item 54
The numbers such as the 3 in 103 which you
saw were called logarithms, or exponents, and told you how many times to use the
base 10 as a factor in multiplication.
The exponents with a minus
sign before them, such as the -3 in 10-3 tell you how many times
to use the base in division, like
this:
If 103 = 10 * 10 * 10 then 10-3 = 1/(10 * 10 * 10)
or 1/1,000 or 0.001.
|
|
 =  = 0.0001
|
(43) If 104 means
10 * 10 * 10 * 10, then 10-4 means
_________________.
|
|
|
These exponents are basic to what is called
scientific notation, a language which will be clarified in a later
program. If 104 means
10,000, then 10-4 means  .
Now you recall from the footnote on page 3 that when
you multiply numbers such as 104 * 108, you add the exponents like this: 104+8, or 1012. If you were to divide 108 by 103
as in this expression:  , you subtract the
exponents like this: 108-3,
or 105.
Try these:
|
|
1010
|
(44) 107 * 103 = _______
If you got that correct, you successfully multiplied
10 million by 1,000 to get an answer of 10 billion or 10,000,000,000, which is
more simply expressed as 1010
|
|
108
|
(45) 106 * 102 = ________.
|
|
105
|
(46) 108/103 = _________.
Now the next is a critical item:
|
|
10-3
|
(47)  = ________.
That was a difficult frame. It teaches that the subtraction of the exponents
requires that the lower figure (the denominator) be subtracted from the upper
figure (the numerator) regardless of
which number is larger.
|
|
10-2
|
(48)  = _______.
|
|
10-13
|
(49)  = _______.
|
|
10-1
|
(50)  = _______.
|
|
101
|
(51)  = _______.
|
|
|
If you correctly answered Item 51, you knew that when
you subtract numbers which are preceded by a minus sign, you essentially do
this:
(-4) - (-5) = (-4) + 5 = 1.
|
|
104
|
(52)  = _________.
|
|
104
|
(53)  = _________.
If you answered this item correctly, you have learned
Item 42, which had stopped you before.
You may recall the item said:
If WO = 10-12 and WR = 10-16,
then the ratio is 104 or 10,000.
If you did not answer Item 53 correctly, go back to
Item 43 and begin again.
If you
missed Item 53 a second time, stop working on the program and see your
mentor. Be sure this mathematical
hurdle is cleared before you try to go any further.
You recall we said that  =  , or 10,000.
|
|
10
|
(54) Then we multiply the log of 10,000 by _______ as in 10 * log
10,000.
Remember that when we use numbers such as 104
or 102, the exponents are the logarithms; but if we used the
numbers 10,000 or 100, we would have to find their logarithms before we could
multiply them by 10.
|
|
40
|
(55) Thus, when WO = 10-12
and WR = 10-16, the decibel output with
regard to the reference is 10 * log  or ________ dB IL.
|
|
130
|
(56) If WO is 10-3 and the reference is 10-16
watt/cm2, then the decibel output with regard to the reference is
_________ dB IL.
Physicists and engineers have settled on an Intensity
Level Reference of 10-16 watt/cm2 when we talk about
references from which to make Intensity Level measurements.
|
|
1
|
(57) If WO = 10-16 and WR also = 10-16
then the ratio of WO over WR is ________.
We are now on the verge of one of the most critical
parts of this program.
|
|
0
|
(58) When WO = WR
as in the case of WO = 10-16 and WR = 10-16,
the ratio of the reference to the output is 1; but the logarithm of 1 is zero,
therefore, the decibel output with regard to the reference is _________ dB
IL.
|
|
0
0
|
(59) If the ratio between WO
and WR equals 1, and we know that the logarithm of one equals ______,
then the decibel output with regard to the reference is also ______ dB IL.
|
|
equal
|
Thus, 0 dB does not mean silence, or absence of sound
or absence of power, nor does it mean very faint sound or power, either. It simply means that the power output of
the system is exactly _______ to the reference from which the decibel
measurement is started.
|
|
1
0 0 0
|
(60) When WO = WR,
the ratio is _______, the logarithm of 1 is _____, and
10 * _____ = _____ dB IL.
|
|
power output
1
0
0
|
(61) For example, if we chose as our
reference point (or WR) the value 100 watts/cm2, and WO
or (2 words) ________ ________ were
also 100 watts/cm2, the ratio of  would still equal
_______, the logarithm of that ratio would still be _______, and the
resultant decibel output with regard to
this new and different reference would still be _______ dB IL.
|
|
|
Thus, either 10-16 watt/cm2 or
100 watts/cm2 can equal 0 dB, if they are chosen as references
from which to make other measurements.
It is quite permissible to choose any reference point from which to
make a dB measurement. In fact,
strictly speaking, every
decibel measurement is a decibel
difference from 0 or decibel difference with regard to the reference from
which the measurement was made.
|
|
reference
|
(62) The word decibel alone implies no fixed dimension of its own since the
_______ from which it is measured can be any value the experimenter chooses.
It is critical, therefore, to know the references from
which various decibel measurements are made.
The most common reference for measuring acoustic intensity
differences, when the variable is power,
is 10-16 watt/cm2. Decibels
described by the equation
10 * log WO/WR are expressed as dB IL or deciBels
Intensity
Level.
|
|
70
|
(63) If you must calculate the number of
decibels a 10-9 watt/cm2 power will generate, the
equation to use is: ________, and the dB IL re 10-16 watt/cm2
is __________.
|
|
10-16
|
(64) The most common and most likely
reference point from which this measurement will be made is
________ watt/cm2.
|
|
30
|
(65) However, if the reference were 10-12
watt/cm2, a 10-9 watt/cm2 signal will be
only ________ dB IL.
|
|
reference
|
(66) All that is required of the reporter
in describing his decibel is that he always
specify the _________.
|
|
10-16
|
(67) When you see the phrase dB IL or dB Intensity
Level,
this usually means that the reference was ________ watt/cm2.
Henceforth, let us assume a reference of
10-16 watt/cm2 equals 0 dB IL (or Intensity
Level).
|
|
120
|
(68) If the power output is 10-4 watt/cm2,
dB IL = _________.
|
|
10
|
(69) When WO = 10-15,
dB IL = _________.
|
|
20
|
(70) When WO = 10-14,
dB IL = _________.
|
|
30
|
(71) When WO = 10-13,
dB IL = _________.
|
|
-20 (that is
right…minus 20 dB IL)
|
(72) When WO = 10-18,
dB IL = _________.
|
|
|
Notice that as the power is multiplied by 10, the dB output simply increases additively by units of 10. Thus, the power required to move from 0 dB
to 60 dB is not sixty units greater
than 10-16 watt/cm2, but 106 or 1,000,000
times greater than 10-16 watt/cm2. For your own use, construct a table like
this:
Power Measurements
 where WR = 10-16
watt/cm2
|
|
0.1 -1 -10
10,000 4 40
|
watt/cm2 (WO/WR)
(Wo) output Ratio Log dB IL
10-18 0.01 -2 -20
10-17 ___ ___ ___
10-16 1 0 0
10-15 10 1 10
10-14 100 2 20
10-13 1,000 3 30
10-12 ___ ___ ___
|
|
|
THE DECIBEL IN MEASUREMENTS FROM A PRESSURE
REFERENCE
|
|
|
In acoustics we make pressure measurements more often
than power measurements so we should know how to convert powers to
pressures. Scientists have known for
many years that powers (watts) and pressures (mPa) have a special relationship. Sound
pressure ratios are usually proportional to the square root of
corresponding power ratios, or power: pressure2.
|
|
base
|
(73) The exponent is a number which tells
you how many times to use the ________ in multiplication.
|
|
10
|
(74) If we start with dB
(power) = _____ * log  , and we say that to make power figures proportional to
pressure, we must square the pressure
|
|
|
dB = 10*log
|
|
|
where PO
is now a pressure output
where PR is now a pressure
reference
|
|
|
When we square a number we multiply its logarithm by
2, and we can rewrite the equation in this way:
|
|
|
dB
SPL = 10 * 2 * log
|
|
|
therefore
|
|
20
|
(75) # dB SPL = * log 
|
|
|
Now we
have obtained the equation for the decibel when the reference is in terms of
sound pressures, instead of powers.
|
|
log
|
(76) # dB = 20 * _______  ,
where PO = pressure output and PR = pressure
reference.
|
|
20
|
(77) # dB SPL (pressure) =
______ * log PO/PR
|
|
reference
|
(78) In
this equation PO = pressure
output from an earphone or speaker, while PR = pressure
_______.
|
|
reference
|
(79) PO = pressure output, while PR = pressure
________.
|
|
pressure
pressure reference
|
(80) PO = ________ output, while PR = (2
words) _________ ________.
|
|
PR
|
(81) To
solve the equation dB = 20 * log  , we must first find the numerical value of the ratio
expressed by PO divided by
________.
|
|
logarithm
|
(82) Then
we find the _______ of that value.
|
|
ratio
|
(83) Note that first we
must find the numerical value of the _______ expressed by  .
|
|
logarithm
|
(84) Then we find the _______ of that
value.
|
|
multiply
|
(85) The
next step is to _______ the logarithm by 20.
|
|
PR
0
|
(86) If
the ratio between PO and PR is one, that is, where PO = _______, we obtain a
logarithm of _______.
|
|
0
|
(87) If
the logarithm then is 0, the entire equation yields a value of _______ dB SPL
using the pressure reference.
|
|
equal
|
(88) Note
again that 0 dB SPL does not mean
silence, or absence of sound, or the faintest level at which a sound can be
heard, or any modification of such verbal conveniences; 0 dB simply means that
the output of our speaker or earphone is exactly _______ to the reference
pressure we have chosen.
|
|
PO
2
40
|
(89) Should
we change the ratio of ______ over PR to a ratio of 100, then
since the log of 100 is ______, we would get
20 * 2 = ________ dB SPL.
|
|
3
20
|
(90) Suppose
the output pressure then became 1,000 times as great as the reference
pressure; the logarithm of 1,000 is ______ and thus we have
______ * 3 = 60 dB SPL.
|
|
log
6
|
(91) Suppose
the output pressure is set at 1,000,000 times the reference pressure; the
______ of 1,000,000 is ______ and the resultant decibel value is
120 dB SPL.
|
|
|
For
various practical reasons, acoustical scientists most often use 20 mPa (20 micro Pascals) as a reference from
which to measure sound pressure levels.
Historically the value used was 0.0002 dyne/cm2 . This value (0.0002 dyne/cm2)
has been replaced with 20 mPa. They are
actually the same thing. From now on
we will use 20 mPa as our reference for pressure measurements and the units to report
are dB SPL.
|
|
20
|
(92) Sound
pressure level measurements in decibels are often based on ___ mPa.
|
|
L
|
(93) Sound Pressure Level is logically abbreviated SP _____.
|
|
20 mPa
|
(94) Thus, if you read a research paper in which the
signals were given at 65 dB SPL, it means that the reference pressure was
______.
|
|
S
|
(95) A
signal presented at 40 dB _______ PL has as its reference 20 mPa
|
|
SPL
20 mPa
|
(96) Thus,
the abbreviation (3 letters) ___ ___
___ means specifically Sound Pressure Level, and tells you that the reference from which the dB was
specified was ________.
|
|
watt
|
(97) Remember
that 60 dB SPL has as its reference 20 mPa, while 60 dB IL has as its reference 10-16
________ per cm2.
|
|
|
You
have already constructed a table for power ratios and decibel measurements
using the equation 10 * log  . Now do the same
thing for pressure ratios using 20 mPa as a reference.
We will start for you:
|
|
20
20 mPa
|
Pressure
Measurements
(98) # dB SPL = _______ * log  , where
PR = (numbers) _______.
|
|
Pressure
mPa Ratio
Output (Po) (Po/PR) Log dB SPL
20 1 0 0
200 10 1 20
2000 100 2 40
20000 1000 3 60
|
|
|
and
so forth. You may add figures up to
140 dB. When you compare the table for
power measurements (see page 9) with the table for pressure measurements
above, note that power and pressure
are in a constant ratio to one another.
When power is increased by a factor of 100, there is a 20 dB increase;
however, a 100 times increase in power only generates a 10 times increase in
pressure since power : pressure2. Increasing
pressure by 10 still generates the
20 dB decibel difference, just as a 100 times increase in power generates the
same 20 dB difference due to the relationship of power to pressure.
Now
let us review.
Decibels are ratios expressed as logarithmic numbers,
so we must understand the logarithm to understand the decibel.
|
|
3
|
(99) The
log of 1,000 is ___________.
|
|
3
|
(100) This
means that in order to obtain a numerical value of 1,000 you use the base
(which is the number 10) ___________. times in multiplication.
|
|
1
|
(101) The
log of 10 is _________.
|
|
1,000
|
(102) The
log of ______ is 3
|
|
100
|
(103) The
log of ______ is 2.
|
|
10
|
(104) The
log of ______ is 1.
|
|
1
0.01
|
(105) The log of ______ is 0.
The log of ______ is -2.
|
|
|
(106) When
we use a power reference, our equation
for determining the number of decibels reads:
________.
|
|
|
(107) When
we use a pressure reference for determining decibels, our equation
reads:
_____________________.
|
|
0
|
(108) When
 is 1 (that is, when
the output pressure is the same as the reference pressure), we have a log
value of _________.
|
|
Reference
|
(109) Therefore,
0 dB is obtained when the pressure (or power) output equals the pressure (or
power) _______.
|
|
10-16
|
(110) In
the measurement of acoustic powers, the reference point used is ______
watt/cm2.
|
|
20
|
(111) In
measuring sound pressure levels, the PR chosen is _______ micro Pascals (mPa).
|
|
|
THE DECIBEL AND ITS USE IN
CLINICAL AUDIOMETRY
The decibel,
as you have learned, is always measured from an arbitrary reference. Occasionally in the literature you will see
as reference points dB re 1 microbar, or 1 Pascal,
or dB re 1 volt or
1 millivolt. These are all
different from 20 mPa or 10-16 watt/cm2,
and must be so interpreted.
We shall now learn about still another kind of decibel reference.
|
|
10-16 watt/cm2
|
(112) When
you see dB IL, you know the reference was __________.
|
|
20 mPa
|
(113) When
you see dB SPL, you know that the reference was _________.
|
|
|
You
will now learn what the dB Hearing Level or HL uses as its reference.
|
|
Hearing Level
|
(114) dB
HL means decibel H_________ L_________.
|
|
|
The
decibel hearing level uses as its reference the sensitivity of the normal human
ear at various frequencies. As you
know, the human ear needs more sound pressure to hear a 250 Hertz (Hz) tone
than it needs to hear a 1000 Hz tone.
|
|
20
|
(115) The
following table shows the sound pressure re
_______ mPa necessary to reach the normal human ear's threshold (according to
ANSI S3.6-2004 standard).
|
|
|
Frequency and
Sound Pressure Level Combinations, Considered to be the Reference Equivalent Threshold Sound Pressure Levels (RETSPLs) (dB re 20 mPa ) for
supra-aural earphones. These data are
from Table 6 of the ANSI-S3.6-2004 standard “SPECIFICATION FOR AUDIOMETERS”
Supra-aural
Earphone
Frequency TDH Type TDH
39 TDH 49/50
Hz IEC 318* NBS
9A* NBS 9A*
125 45.0 45.0 47.5
250 27.0 25.5 26.5
500 13.5 11.5 13.5
750 9.0 8.0 8.5
1000 7.5 7.0 7.5
1500 7.5 6.5 7.5
2000 9.0 9.0 11.0
3000 11.5 10.0 9.5
4000 12.0 9.5 10.5
6000 16.0 15.5 13.5
8000 15.5 13.0 13.0
Speech 20.0 19.5 20.0
*Coupler Type:
The IEC 318 type coupler
approximates the impedance of the human external ear and therefore uses the
same corrections for all TDH type earphones.
The NBS 9A 6cc coupler does
not approximate the impedance of the human external ear and therefore
uses different corrections for different earphone types.
NOTE: FOR THE PURPOSES OF THIS CLASS
YOU NEED TO KNOW THE CORRECTIONS FOR THE IEC 318 TYPE COUPLER.
|
|
45.0
|
(116) This table tells you that the normal listener needs ________
dB SPL to report hearing a 125 Hz tone.
|
|
1,000 or 1,500
|
(117) But
this normal listener needs only 7.5 dB SPL to report hearing a
_______ Hz tone.
|
|
27.0
12.0
|
(118) The
table says that _______ dB SPL is required to reach our hypothetical normal
listener's threshold at 250 Hz, but only _______ dB SPL is needed to hear
4000 Hz.
|
|
more
|
(119) Thus,
you need (more or less?)________ sound pressure to make
a 250 Hz tone audible than is needed to make a 1000 Hz tone audible.
|
|
19.5
|
(120) Obviously,
you need more sound pressure to make the 250 Hz tone audible; in fact, you
need _______ dB more pressure than is needed to make the 1000 Hz tone
audible.
|
|
|
If you had believed that 0 dB HL (that is, 0 dB setting on the
audiometer) meant that at all the frequency settings that audiometer
generated the same sound pressure, you can see now why you were
mistaken. The table above shows that
different sound pressures are needed at each frequency to reach human
threshold. The figures on this table
are the pressure references for 0 dB HL or 0 dB Hearing Level. Thus,
each figure given on the table equals 0 dB HL at the specified
frequency. If you increase the dB HL
output by 10 dB, each pressure output increases by 10 dB.
|
|
9.0
|
(121) If
you had a clinical audiometer in front of you that was calibrated to the ANSI
S3.6-2004 standard, and it was set to 2000 Hz, 0 dB HL, you would be generating
_________ dB SPL.
|
|
SPL (or re 20 mPa)
|
(122) Now
leaving the frequency at 2000 Hz, if you turned the HL dial to 40 dB you
would be generating 49 dB ________.
|
|
94.0
|
(123) With
the frequency selector still at 2000 Hz, turn the Hearing Level dial to
85 dB. You are now generating
__________ dB SPL.
|
|
98.5
|
(124) Now
leave the Hearing Level dial at 85 dB and change the frequency to 500
Hz. You are now generating _________
dB SPL.
|
|
normal
|
(125) With
the settings of 500 Hz and 85 dB HL, both signals are the same level above
normal human hearing. Thus, the
reference for calculating dB HL is ________ human hearing.
|
|
10-16 watt/cm2
|
(126) You
have now learned three notations which tell about the reference from which
decibels are often calculated: dB IL
means that the reference was ___________.
|
|
20 mPa
|
(127) dB
SPL means that the reference was _______.
|
|
normal
frequency
|
(128) dB
HL means that the reference was the sound pressure necessary for a _______
human listener just to perceive tones of various frequencies. The sound pressure reference for 0 dB HL varies with _______.
|
|
|
Now you
will learn about one more set of letters which often follow the term dB.
|
|
20
|
(129) If a subject has a 40 dB hearing
level, and you present a tone to him at 60 dB Hearing Level, he is getting a
tone _______ dB above his threshold, or 20 dB Sensation Level (SL).
|
|
SL
|
(130) Thus,
if a man has normal hearing, that is, 0 dB HL at all frequencies, all tones
at 20 dB HL are also at 20 dB _________ to him.
|
|
SL
|
(131) If
a man has normal hearing, that is, 0 dB HL at all frequencies, all tones at
20 dB HL are also at 20 dB ________ to him.
|
|
30
-40 dB
|
(132) But
if he has -10 dB HL hearing, then the same 20 dB HL tones are at _______ dB
SL to him. If he had a 60 dB HL, the
20 dB HL tone would be at _______ SL.
|
|
|
So you
see that there is also a decibel, the reference for which is the subject's
own hearing. This is the decibel sensation level, or dB SL.
|
|
|
(133) Now
we have four modifying abbreviations which follow the letters "dB"
which must be reviewed:
|
|
10-16 watt/cm2
|
dB IL
means that the reference was ________.
|
|
20 mPa
|
dB SPL
means that the reference was ______.
|
|
normal
frequency
|
dB HL means
that the reference was the sound pressure necessary for a ________ human
listener to hear tones of various frequencies. This reference changes with the _______ of
the tone.
|
|
patient's or subject's
|
dB SL
means that the reference was the ________ own threshold.
|
|
|
The ANSI Standard for ZERO Hearing Level
As
was said, 0 hearing level for
normal human listeners was specified according to the table that you used to
complete the preceding section.
Researchers in other countries, as well as in the United States, had previously
published studies of the sound pressures necessary to reach human
threshold. The more recent studies
have presented strong evidence to show that the original ASA-1951 standard
was too lenient and that ears of young sophisticated listeners respond more
sensitively to sound than the ASA standard would indicate. The ANSI S3.6-2004 scale is now the
medico-legal standard in the United
States.
The ASA-1951 and the ISO-1964 standard are now rarely heard of.
The
ANSI, ISO and ASA standards for 0 hearing
level are presented alongside each
other.
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Reference Equivalent Sound Pressure
Levels
(0 dB
Hearing Level)
Frequency
Hz ANSI-2004 ISO-1964 ASA-1951
125 45.0 45.5 54.5
250 27.0 24.5 39.6
500 13.5 11.0 24.8
750 9.0
1000 7.5 6.5 16.7
1500 7.5
2000 9.0 8.5 17.0
3000 11.5
4000 12.0 9.0 15.1
6000 16.0
8000 15.5 9.5 21.0
NOTE:
The above table is for your information only.
It is very rare that you will ever see any reference to the older ISO
and ASA
standards.
Thus, for example what was 10 dB
HL at 1 kHz according to the ASA 1951 standard will be noted essentially
as 21 dB HL according to the ANSI-2004 standard and 20 dB HL according to the
ISO-1964 standard. You should always know to what standard
your audiometer was calibrated and make some notation in the audiogram to
that effect.
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A
NOTE ON LOUDNESS vs. INTENSITY
Notice
that you have also clarified the difference between intensity and
loudness. A 125 Hz tone of 25 dB SPL
has notable intensity re 20 mPa, but no loudness to the average normal
listener. On the other hand, a 1000 Hz
tone of 25 dB SPL would have both intensity and loudness to a normal
listener. However, do not speak of the decibel as an index of loudness. Loudness grows in quite a different way from
sound pressure other means of
measuring loudness (magnitude estimation, sones, phons, noys, etc.) are
needed, and the decibel is not appropriate.
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ACKNOWLEDGMENTS
The author's
appreciation goes to his students and colleagues who helped with previous
versions, especially to Dr. Dixon Ward, who deleted critical errors and made
clarifications. Thanks also goes to
Drs. Richard Riedman, Jack Katz, S. Thomas Elder, John Black, Victor Garwood,
Joseph Chaiklin, William G. Hardy, Moise Goldstein, Henry Mark, and to Jack
Cullen and Robert Daly. Special thanks
are due to Lois Lunin and Drs. John Bordley and Francis Catlin and the staff
of the information center for their kind assistance and encouragement in
developing the decibel program.
Editorial and production matters related to the decibel program were
handled by Dr. Carl Thompson, who deserves special appreciation for his
diligence. He and Gae Decker were
responsible for this format and improved readability.
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