Preliminaries
2
Preliminaries
Understanding the basic principles of hearing science does not require an extensive background in mathematics
or physics. There are, however, a few basic concepts in mathematical notation and elementary physics that will help
pave the way for some of the material that will be encountered later in the text. The two concepts that will be
reviewed in this chapter are exponential notation and the representation of physical quantities.
Exponential Notation
In dealing with very large or very small numbers it is often more convenient to express numbers using
exponential notation. As will be seen in the next chapter, some familiarity with exponents is needed to understand
the decibel scale, which is in widespread use as a measure of sound intensity. The most commonly used exponential
notation system is centered around the familiar base 10 number system, although the same idea can be applied to any
number system. In all cases, exponential notation involves raising a number called the base to some power, also
called the exponent. The exponent indicates how many times the base is used as a factor. Table 2-1 below shows
how exponential notation works when using 10 as the base. The numbers in the first column are in "ordinary"
notation, the numbers in the center column are in exponential notation, and the factors of 10 that are associated with
these numbers are shown in the third column.
__________________________________________________________________________________________
__________________________________________________________________________________________
Table 2-1. Exponential notation for even powers of 10.
Exponential
Number
Notation
Factors of 10
__________________________________________________________________________________________
0
1
10
(any number raised to the zero power = 1)
1
10
10
10 x 1 = 10
2
100
10
10 x 10 = 100
3
1,000
10
10 x 10 x 10 = 1,000
4
10,000
10
10 x 10 x 10 x 10 = 10,000
5
100,000
10
10 x 10 x 10 x 10 x 10 = 100,000
6
1,000,000
10
10 x 10 x 10 x 10 x 10 x 10 = 1,000,000
__________________________________________________________________________________________
__________________________________________________________________________________________
As can be seen in the last column, the exponent simply indicates how many times the number ten is used as a
factor. It can also be seen that, for these whole-number powers of ten (i.e., 10 raised to an integer power), the
exponent is equal to the number of zeros that are used when the number is written in ordinary notation. The
0
factorizations shown in the last column make intuitive sense, with the possible exception of 10 . Although a longer
explanation would show that this example makes sense as well, for our purposes it is important simply to note that
any number raised to the zero power equals one.
Exponential Notation for Numbers Smaller than One
Positive numbers smaller than one can also be represented using exponential notation. The number 0.1 (1/10) is
-1
1
-2
2
represented as 10 because 0.1 = 1/10 . Similarly, 0.01 (1/100) is represented as 10 because 0.01 = 1/10 . Using the
-5
5
same logic, 10 = 1/10 = 1/100,000 = 0.00001. In all cases, 10-x = 1/10x. Table 2-2 below shows how the system
works. Notice that for these even powers of ten, the exponent (with a negative sign) is equal the number of zeros that
appear after the decimal point plus one; e.g., for 0.0001, there are 3 zeros after the decimal point, giving an exponent
of –(3+1) = -4.
Preliminaries
3
_____________________________________________________________________________________________
_____________________________________________________________________________________________
Table 2-2. Exponential notation for numbers less than 1.
Exponential
Number
Notation
Factors of 10
_____________________________________________________________________________________________
-1
1
0.1
10
1/(10x1) = 1/10
-2
2
0.01
10
1/(10x10) = 1/10
-3
3
0.001
10
1/(10x10x10) = 1/10
-4
4
0.0001
10
1/(10x10x10x10) = 1/10
-5
5
0.00001
10
1/(10x10x10x10x10) = 1/10
-6
6
0.000001
10
1/(10x10x10x10x10x10) = 1/10
_____________________________________________________________________________________________
_____________________________________________________________________________________________
Logarithms
2
In a number such as 10 , the number 10 is referred to as the base, and the number 2 can be referred to either as
a power, an exponent, or a logarithm. These three terms – power, exponent, and logarithm – are completely
interchangeable. When you take the logarithm of a number, you are simply finding the exponent that is used when
the number is written in exponential notation. For example, the base-10 logarithm (log10 for short) of 100 is 2
2
because 100 = 10 . Similarly:
3
log10 1,000 = 3
(because 1,000 = 10 )
log10 1,000,000 = 6
(because 1,000,000 = 10 )
log10 1 = 0
(because 1 = 10 )
log10 0.1 = -1
(because 0.1 = 10 )
log10 0.0001 = -4
(because 0.0001 = 10 )
6
0
-1
-4
All of the examples above show logarithms for whole-number powers of ten such as 10, 100, 1,000, etc.
However, logarithms can be found for any positive number. For example, it stands to reason that the logarithm of
200 has to be somewhere between 2 and 3 because the number 200 is greater than 100 (which has a log 10 of 2) but
less than 1,000 (which has a log10 of 3). If you have a calculator with a log button, you can confirm that the log 10 of
2.3
200 is approximately 2.3, which means that 10 = 200. There are several methods that can be used to obtain the
logarithm of a number that is not a whole number power of ten, but the simplest by far is to use a hand calculator
that is equipped with a log function.
Arithmetic Using Exponential Notation
We have seen that exponential notation provides a convenient method for representing very large or very small
9
numbers (e.g., 10 instead of 1,000,000,000). An additional advantage is that certain arithmetic operations become
very simple when numbers are expressed in exponential notation. For example:
1. Multiplication (add exponents)
2
3
5
10 x 10 = 10
2
-5
-3
10 x 10 = 10
Preliminaries
4
2. Division (subtract exponents)
5
3
2
10  10 = 10
7
-2
9
10  10 = 10
3. Raising a number to a power (multiply exponents)
3 2
6
(10 ) = 10
5 3
15
(10 ) = 10
3
6
(10 raised to the second power = 10 )
5
15
(10 raised to the third power = 10 )
Representation of Physical Quantities
As we will see soon, understanding how sound is produced and measured requires some understanding of
physical units such as force, pressure, intensity, etc. Our understanding of these physical quantities need not be very
detailed, but a general understanding will make it easier to grasp certain concepts that are introduced later on.
Basic Versus Derived Properties
There are many quantities that are used in physics, but only three of these are considered to be basic or
fundamental. The three fundamental quantities are:



distance (also length)
mass
time
All other quantities, such as velocity, acceleration, work, force, pressure, etc., are derived by combining these
basic quantities in various ways. For example, velocity is simply the change in the position of an object (a measure
of distance) divided by the time taken to travel that distance (e.g., 60 miles per hour, 300 meters per second, etc.).
So, we got the derived property of velocity by combining the basic properties of distance and time. There are several
derived properties that are needed to understand the production and measurement of sound.
Velocity and Acceleration
As mentioned above, velocity is the change in the position of an object (Δx) divided by the time over which
that position change occurred:
v = Δx/t, where:
v = velocity (more precisely, average velocity)
Δx = change in position (the Δ symbol means “change in”)
t = time over which the position change occurred
Acceleration is the change in velocity over time. You have probably heard it said that a certain sports car can do
"0 to 60 in 6 seconds." The "0 to 60 in 6 seconds" is a measure of acceleration: the change in velocity is 60 mph and
the time over which the change occurred is 6 seconds. The formula looks like this:
a = Δv/t, where:
a = acceleration (more precisely, average acceleration)
Δv = the change in velocity
t = time over which velocity change occurred
The formula may make acceleration look more obscure than it is. Acceleration is simply the change in velocity
divided by the time over which the velocity change occurred. In the example above, the change in velocity is 60 mph
and the time over which the velocity change occurred is 6 s, so the acceleration is 10 mph/s. The example above is
an instance of speeding up, or positive acceleration; slamming the brakes on would cause the car to slow down. This
can be referred to either as negative acceleration or deceleration.
Preliminaries
5
Force
Imagine a billiard ball sitting motionless on a perfectly level pool table. You know from every day experience –
and from Newton's first law of motion that you probably learned in seventh-grade science class – that the ball will
remain motionless unless acted upon by an outside influence. What is needed to move the ball is a force. Force can
be thought of as the amount of "push" or "pull" on an object. Force is the mass of the object multiplied by its
acceleration:
F = Ma, where:
F = force
M = mass
a = acceleration
The formula tells us that force can be increased either by increasing mass or by increasing acceleration (or, of
course, both). So, our billiard ball will be highly disturbed if it is struck by a massive, highly accelerated object. On
the other hand, the ball will be disturbed by a small amount if it is struck by a light object that is moving with a low
rate of acceleration. Notice that it is acceleration and not velocity that is a determinant of force. The old saying, ‘it is
not the fall that hurts, it is the sudden stop at the end’ expresses exactly this fact about the role acceleration in
generating force. No force is generated by an object that is traveling at a constant velocity (and, therefore, zero
acceleration), no matter how great the velocity. Force requires a change in velocity; that is, force requires
acceleration. This explains why airline passengers feel the force of the airplane primarily during takeoff and landing
but not in the middle of the flight when the plane is cruising at a high but relatively constant velocity. This also
explains why we have no sensation of the exceedingly high velocity with which the earth is simultaneously
revolving and its axis and hurtling through space.
Figure 2-1. Two objects with identical masses that differ only in orientation. If the two objects
were dropped from the same height, they would strike a surface below with identical force.
However, the pressure generated by the object on the right would be much greater because the
area over which the force is delivered would be considerably smaller; that is, the pressure
would be greater.
Pressure
As will be discussed in later chapters, the distinction between force and pressure is quite important for
understanding how the middle ear works. Force, as we have seen, is the amount of push or pull on an object.
Pressure, on the other hand, is force per unit area, or the amount of force that is "focused" on a given area. The
formula is:
E = F/A, where:
1
E = pressure1
F = force
A = area over which force is delivered
The symbol 'E' will be used for pressure because 'P' will be needed later for power.
Preliminaries
6
The difference between force and pressure can be illustrated with the objects shown in Figure 2-1, which we can
assume are identical in mass and differ only with respect to orientation. If each of these objects were dropped on a
napkin from the same height, they would strike the napkin with exactly the same force since both mass and
acceleration would be identical (F = Ma). It is obvious that the object on the right would be more likely to tear the
napkin, and the reason is that the force is being delivered over a much smaller area, producing much greater pressure
on impact. So, in the object on the right, pressure has been amplified without an increase in force by decreasing the
area over which the force is delivered. Exactly the same principle explains: (1) why a sharp knife cuts better than a
dull knife, (2) why thumb tacks are inserted into a bulletin board pointy-end first rather than vice versa, and (3) why
a woman would be more likely to sink into a wet lawn if she is wearing stiletto heels rather than tennis shoes. As we
will see later, the middle ear exploits this principle to amplify the pressure wave that is delivered to the inner ear.
Work
Work is accomplished when a force that is applied to an object causes the object to move. The amount of work
that is done increases when the size of the force increases. Also, more work is done if the object is moved a greater
distance. The formula is:
W = Fx, where:
W = work
F = force
x = distance
The formula indicates that a large amount of work is done when large forces cause objects to move great distances.
The principle can be illustrated with a simple example. Imagine that one laborer moves 100 kilograms of bricks a
distance of 30 meters, and a second laborer applies an equal force to move the same 100 kilograms of bricks a
distance of 60 meters. Both the formula and intuition tell us that the second laborer has accomplished twice as much
work as the first laborer.
Power
Power is just one step more complicated than work. Power is work per unit time, or the rate at which work is
accomplished. The formula is:
P = W/t, where: P = power
W = work
t = time taken to complete the work
Power basically tells us how rapidly work is being accomplished. We can return to our example of the two laborers.
Suppose that two laborers apply identical forces to move 100 kilograms of bricks a distance of 30 meters. Since both
the force and the distance are identical, the two laborers have done the same amount of work (W = Fx). However,
suppose that one laborer completes the job in 30 minutes while the other completes the same job in just 15 minutes.
Since power is the rate at which work is accomplished, twice as much power was generated by the second laborer
than the first.
Intensity
Intensity, which is one step more complicated than power, is simply power per unit area; that is, intensity is the
amount of power that is "focused" on a given area. The formula is:
I = P/A, where:
I = intensity
P = power
A = area over which the power is measured
Notice that the relationship between power and intensity is analogous to the relationship we saw earlier between
force and pressure; that is, pressure is force per unit area, and intensity is power per unit area. There is, in fact, a
close relationship between intensity and pressure, and this relationship is important for understanding how the
decibel scale works.
Preliminaries
Table 2-3 presents a summary of the physical quantities discussed above, along with a non-exhaustive list of
some of the units that are used to measure quantities such as acceleration, pressure, work, and so on.
___________________________________________________________________________________________________
___________________________________________________________________________________________________
Table 2-3. Summary of physical quantities.
Quantity
Formula
Units
Concept
___________________________________________________________________________________________________
velocity
v= Δx/t
meters per sec
miles per hour
Δx= change in position
t=time
high velocity means large changes in
position occurring in a short period of time
acceleration
a = Δv/t
meters/sec squared
miles per hour per second
speeding up or slowing down; i.e.,
a change in velocity over time
Newton
dyne
amount of "push" or "pull" on an object
Newton/sq meter
dyne/sq centimeter
Pascal
pressure is high when a large force
is delivered over a small area
joule
erg
foot-pound
a large amount of work is done when a large
force moves an object a great distance
watt
horsepower
the rate at which work is done
watt/square meter
watt/square centimeter
intensity is high when a large amount of
power is focused on a small area
Δv = change in vel.
t = time
force
F = Ma
M = mass
a = acceleration
pressure
E = F/A
F = force
A = area
work
W = Fx
F = force
x = distance
power
P = W/t
W = work
t = time
intensity
I = P/A
P = power
A = area
___________________________________________________________________________________________________
___________________________________________________________________________________________________
7
Preliminaries
1. In the expression 104, the number 10 is called the _________. Three different terms can be
used to refer to the number 4 in this expression: ____________, ____________, or
____________.
2. Any number raised to the zero power equals: ___________.
3. 10-7 is equivalent to 1 divided by _________. Written in ordinary, non-exponential notation,
this is equal to _________.
4. Since 105 = 100,000, the log10 of 100,000 = _________.
5. 108 / 102 = ______. Written in ordinary, non-exponential notation, this is equal to
_________.
6. 104 / 109 = ______. Written in ordinary, non-exponential notation, this is equal to
_________.
7. 105 x 103 = ______. Written in ordinary, non-exponential notation, this is equal to
_________.
8. 10-3 = ________.
9. (102)3 = ______. Written in ordinary, non-exponential notation, this is equal to _________.
10. The log10 of 435,973 is greater than ______ but less than ______.
11. These three physical quantities are considered to be basic or fundamental: ____________,
____________, and ____________. Quantities that are arrived at by combining these basic
quantities in various ways are referred to as ____________ quantities.
12. Supply the physical quantity for each of the descriptions below.
a. Amount of push or pull on an object (or the product of mass and acceleration):
____________.
b. Distance traveled divided by the time required to travel that distance: ____________.
c. The product of force and distance: ____________.
8
Preliminaries
9
d. The rate at which work is accomplished: ____________.
e. Power per unit area: ____________.
f. Force per unit area: ____________.
g. Change in velocity over time: ____________.
13. Laborer A moves 50 kilograms of cottage cheese a distance of 2 kilometers in 30 minutes.
Laborer B moves the same 50 kilograms the same distance but takes 60 minutes to finish the job.
Measured in terms of ____________ the two laborers are therefore identical. However, because
of the difference in the amount of time required to complete the job, measures of ____________
will be twice as large for laborer A than laborer B.
14. The difference between force and pressure is analogous to the difference between power and
____________.
15. The objects in Figure 2-1 are dropped from the same height. When they strike the surface
below, ____________ will be identical for the two objects but ____________ will be greater
for the object with the pointy tip facing down.
16. The purpose of sharpening a knife is to:
a.
b.
c.
d.
e.
f.
g.
amplify force
amplify pressure
amplify acceleration
decrease velocity
all of the above
a and b only
a, b, and c
Preliminaries
10
Study Question Answers: Preliminaries
1. In the expression 104, the number 10 is called the base. Three different terms can be used to
refer to the number 4 in this expression: power, exponent, or logarithm.
2. Any number raised to the zero power equals: 1.
3. 10-7 is equivalent to 1 divided by 107. Written in ordinary, non-exponential notation, this is
equal to 0.0000001.
4. Since 105 = 100,000, the log10 of 100,000 = 5.
5. 108 / 102 = 106. Written in ordinary, non-exponential notation, this is equal to 1,000,000.
6. 104 / 109 = 10-5. Written in ordinary, non-exponential notation, this is equal to 0.00001.
7. 105 x 103 = 108. Written in ordinary, non-exponential notation, this is equal to 100,000,000.
8. 10-3 = 1/103 or 0.001
9. (102)3 = 106. Written in ordinary, non-exponential notation, this is equal to 1,000,000.
10. The log10 of 435,973 is greater than 5 but less than 6.
11. These three physical quantities are considered to be basic or fundamental: time, mass,
distance. Quantities that are arrived at by combining these basic quantities in various ways
are referred to as derived quantities.
*** See text for answers to the questions below ***.
12. Supply the physical quantity for each of the descriptions below:
a.
Amount of push or pull on an object (or the product of mass and acceleration):
____________.
b.
Distance traveled divided by the time required to travel that distance: ____________.
c.
The product of force and distance: ____________.
d.
The rate at which work is accomplished: ____________.
e.
Power per unit area: ____________.
f.
Force per unit area: ____________.
Preliminaries
g.
11
Change in velocity over time: ____________.
13. Laborer A moves 50 kilograms of cottage cheese a distance of 2 kilometers in 30 minutes.
Laborer B moves the same 50 kilograms the same distance but takes 60 minutes to finish the job.
Measured in terms of ____________ the two laborers are therefore identical. However, because
of the difference in the amount of time required to complete the job, measures of ____________
will be twice as large for laborer A than laborer B.
14. The difference between force and pressure is analogous to the difference between power and
____________.
15. The objects in Figure 2-1 are dropped from the same height. When they strike the surface
below, ____________ will be identical for the two objects but ____________ will be greater
for the object with the pointy tip facing down.
16. The purpose of sharpening a knife is to:
a.
b.
c.
d.
e.
f.
amplify force
amplify pressure
amplify acceleration
decrease velocity
all of the above
a and b only