Programmed Instruction In

APPENDIX

PROGRAMMED INSTRUCTION IN

THE DECIBEL[2]

Charles I. Berlin

GOTO SAMPLE DECIBEL PROBLEMS

In order to study the decibel, you must first review something about the logarithm. Logarithms are simply exponents and you know what an exponent is; in the expression 102=100, the 2 is an exponent.
exponent / (1)In the expression 102=100, the 2 is an .
exponent / (2)We have said that a logarithm is an .
logarithm / (3) Therefore, the number 2 in the expression 102=100 is both an exponent and a .
The exponent or logarithm above the number 10 tells us how many times to use the ten in multiplication. Thus, 102 means the same as 10*10 or 100.
multiplication / (4)The expression 103 means use the number 10 three times in .
ten multiplication / (5)The expression 104 means use the number
four times in .
6 / (6)The expression 10___ means use the number 10 six times in multiplication.
10 10 10 / (7) The expression 103 means the same as
___*___*___, or 1,000.
10*10*10*10*10 (or 100,000) / (8)The expression 105 means the same as
.
logarithm
multiplication / (9) In the expression 105 the number 10 is called the base, while the number 5 is the exponent or
which tells you how many times to use the base in .
4 / (10)In the expression 43 and 103 the exponents are the same (the number three) but the bases are _____ and 10 respectively.
4 4 4 / (11)If 103 means 10*10*10, then 43 must mean _____*______*_____.
10 6 / (12) In the expression 106 we mean use the number _____ (how many) _____ times in multiplication.
1,000,000 / (13)Thus, 106=(whole number) ______.
10,000 / (14)While 104=(whole number) ______.
3 / (15)The number 1,000 can be expressed as 10__.
4 10 / (16) The logarithm in the expression 104 is the number _____, while the base is the number ______.
100 / (17)102=______.
2 / (18)The logarithm of 100 is _____.
3 / (19)103=1,000, which can be expressed as the log of 1,000 is ____.
4 / (20)The log of 10,000 is ______.
You may have noticed that when the base is 10, the exponent tells you how many zeroes appear after the number 1. Thus, 102 is the number 1 followed by two zeroes, or 100. The expression 104 is the number 1 followed by four zeroes or 10,000.
In dealing with decibels, the base we shall be concerned with will always be 10. When we ask, "What is the logarithm of 100," for example, we mean, "What is the logarithm of 100 to the base 10?"
9
1,000,000,000 / (21)109 is the number 1 followed by ______zeroes, or the whole number ______.
10 / (22)It follows then that 101=______and 100=1.[3]
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/m2 (power) output, and
WR=watt/m2 (power) reference.
output / (36)WO=watt/m2 power ______, and
WR =watt/m2 power reference.
reference / (37)WO=watt/m2 power output while
WR=watts /m2 power ______.
watt/m2 power output / (38)WO=______.
watt/m2 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=10-8 and WR=10-12, 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=10-8 and WR=10-12, 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=10-8 and WR=10-12, the decibel output with regard to the reference is 10*logor ______dB IL.
130 / (56)   If WO is 101 and the reference is 10-12 watt/m2, then the decibel output with regard to the reference is ______dB IL.
Physicists and engineers have settled on an Intensity Level Reference of 10-12 watt/m2 when we talk about references from which to make Intensity Level measurements.
1 / (57)   If WO=10-12 and WR also=10-12 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=10-12 and WR=10-12, 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/m2, and WO or (2 words) ______were also 100 watts/m2, 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 10-12 watt/m2 or 100 watts/m2 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 10-12 watt/m2. Decibels described by the equation 10*log WO/WR are expressed as dB IL or deciBels Intensity Level.
10*log
70 /
(63)If you must calculate the number of decibels a 10-5 watt/m2 power will generate, the equation to use is: ______, and the dB IL re 10-12 watt/m2 is ______.
10-12 / (64)The most common and most likely reference point from which this measurement will be made is ______watt/m2.
30 / (65)However, if the reference were 10-8 watt/m2, a 10-5 watt/m2 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 ______.
10-12 / (67)When you see the phrase dB IL or dB Intensity Level, this usually means that the reference was ______watt/m2.
Henceforth, let us assume a reference of
10-12watt/m2 equals 0dBIL (or Intensity Level).
110 /
(68)If the power output is 10-1watt/m2, dBIL=______.
10 / (69)When WO=10-11, dB IL=______.
20 / (70)When WO=10-10, dB IL=______.
30 / (71)When WO=10-9, dB IL=______.
-20 (that is right…minus 20 dB IL) / (72)When WO=10-14, 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 10-12 watt/m2, but 106 or 1,000,000 times greater than 10-12 watt/m2. For your own use, construct a table like this:
Power Measurements
dB=10*log where WR=10-12 watt/m2
0.1 -1 -10
10,000 4 40 / watt/m2 (WO/WR)
(Wo) output Ratio Log dB IL
10-14 0.01 -2 -20
10-13 ______
10-12 1 0 0
10-11 10 1 10
10-10 100 2 20
10-9 1,000 3 30
10-8 ______
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

where is now a pressure output
where 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) ______.