Frequency and Period of a Wave
The nature of a wave was discussed in Lesson 1 of this unit. In
that lesson, it was mentioned that a wave is created in a slinky by the
periodic and repeating vibration of the first coil of the slinky. This
vibration creates a disturbance that moves through the slinky and transports
energy from the first coil to the last coil. A single back-and-forth vibration
of the first coil of a slinky introduces a pulse into the slinky. But the act
of continually vibrating the first coil with a back-and-forth motion in
periodic fashion introduces a wave into the slinky.
Suppose
that a hand holding the first coil of a slinky is moved back-and-forth two
complete cycles in one second. The rate of the hand's motion would be 2
cycles/second. The first coil, being attached to the hand, in turn would
vibrate at a rate of 2 cycles/second. The second coil, being attached to the
first coil, would vibrate 
at a rate
of 2 cycles/second. The third coil, being attached to the second coil, would
vibrate at a rate of 2 cycles/second. In fact, every coil of the slinky would
vibrate at this rate of 2 cycles/second. This rate of 2 cycles/second is
referred to as the frequency of the wave. The frequency of a wave refers to how often the particles of
the medium vibrate when a wave passes through the medium. Frequency is a part
of our common, everyday language. For example, it is not uncommon to hear a
question like "How frequently do you mow the lawn during the summer
months?" Of course the question is an inquiry about how often the lawn is mowed and the answer is usually
given in the form of "1 time per week." In mathematical terms, the
frequency is the number of complete vibrational cycles of a medium per a given
amount of time. Given this definition, it is reasonable that the quantity frequency would have units of cycles/second,
waves/second, vibrations/second, or something/second. Another unit for
frequency is the Hertz (abbreviated Hz) where 1 Hz is equivalent to 1
cycle/second. If a coil of slinky makes 2 vibrational cycles in one second,
then the frequency is 2 Hz. If a coil of slinky makes 3 vibrational cycles in
one second, then the frequency is 3 Hz. And if a coil makes 8 vibrational
cycles in 4 seconds, then the frequency is 2 Hz (8 cycles/4 s = 2 cycles/s).
The
quantity frequency is often confused with the quantity period. Period refers to
the time that it takes to do something. When an event occurs repeatedly, then
we say that the event is periodic and refer to the time for the event to repeat
itself as the period. The period of a wave is the time for a particle on a
medium to make one complete vibrational cycle. Period, being a time, is
measured in units of time such as seconds, hours, days or years. The period of
orbit for the Earth around the Sun is approximately 365 days; it takes 365 days
for the Earth to complete a cycle. The period of a typical class at a high
school might be 55 minutes; every 55 minutes a class cycle begins (50 minutes
for class and 5 minutes for passing time means that a class begins every 55
minutes). The period for the minute hand on a clock is 3600 seconds (60
minutes); it takes the minute hand 3600 seconds to complete one cycle around
the clock.
Frequency and period are distinctly different, yet related,
quantities. Frequency refers to how often something happens. Period refers to
the time it takes something to happen. Frequency is a rate quantity. Period is
a time quantity. Frequency is the cycles/second. Period is the seconds/cycle.
As an example of the distinction and the relatedness of frequency and period,
consider a woodpecker that drums upon a tree at a periodic rate. If the
woodpecker drums upon a tree 2 times in one second, then the frequency is 2 Hz.
Each drum must endure for one-half a second, so the period is 0.5 s. If the
woodpecker drums upon a tree 4 times in one second, then the frequency is 4 Hz;
each drum must endure for one-fourth a second, so the period is 0.25 s. If the
woodpecker drums upon a tree 5 times in one second, then the frequency is 5 Hz;
each drum must endure for one-fifth a second, so the period is 0.2 s. Do you
observe the relationship? Mathematically, the period is the reciprocal of the
frequency and vice versa. In equation form, this is expressed as follows.
Since
the symbol f is used for frequency and the symbol T is used
for period, these equations are also expressed as:
The
quantity frequency is also confused with the quantity speed. The speed of an
object refers to how fast an object is moving and is usually expressed as the
distance traveled per time of travel. For a
wave, the speed is the distance traveled by
a given point on the wave (such as a crest) in a given period of time. So while
wave frequency refers to the number of cycles occurring per second, wave speed
refers to the meters traveled per second. A
wave can vibrate back and forth very frequently, yet have a small speed; and a
wave can vibrate back and forth with a low frequency, yet have a high speed.
Frequency and speed are distinctly different quantities.
Check Your Understanding
Throughout
this unit, internalize the meaning of terms such as period, frequency, and
wavelength. Utilize the meaning of these terms to answer conceptual questions;
avoid a formula
fixation.
1.
A wave is introduced into a thin wire held tight at each end. It has an
amplitude of 3.8 cm, a frequency of 51.2 Hz and a distance from a crest to
the neighboring trough of 12.8 cm.
Determine the period of such a wave.
Answer: 0.0195 sec
Here is an example of a problem with a lot of extraneous
information. The period is simply the reciprocal of the frequency. In this
case, the period is 1/(51.2 Hz) which is 0.0195
seconds.
Know your
physics concepts to weed through the extra information.
2.
Frieda the fly flaps its wings back and forth 121 times each second. The period
of the wing flapping is ____ sec.
Answer: 0.00826 seconds
The quantity 121 times/second is the frequency. The period is
the reciprocal of the frequency.
T=1/(121 Hz) = 0.00826 s
3. A tennis coach paces back and forth along the sideline 10
times in 2 minutes. The frequency of her pacing is ________ Hz.
|
a. 5.0 |
b. 0.20 |
c. 0.12 |
d. 0.083 |
Answer: D
Frequency refers to the number of occurrences of a periodic
event per time and is measured in cycles/second. In this case, there are 10
cycles per 2 minutes (also known as 10 cycles per 120 seconds). So the
frequency is
f =10 cycles /
120 s = 0.0833
cycles/s
4.
Non-digital clocks (which are becoming more rare)
have a second hand that rotates around in a regular and repeating fashion. The
frequency of rotation of a second hand on a clock is _______ Hz.
|
a. 1/60 |
b. 1/12 |
c. 1/2 |
|
|
d. 1 |
e. 60 |
|
|
Answer: A
Frequency refers to the number of occurrences of a periodic
event per time and is measured in cycles/second. In this case, there is 1 cycle
per 60 seconds. So the frequency is
f = 1 cycle /
(60 s) = (1 / 60) Hz
5.
Olive Udadi accompanies her father to the
park for an afternoon of fun. While there, she hops on the swing and begins a
motion characterized by a complete back-and-forth cycle every 2 seconds. The
frequency of swing is _________.
|
a. 0.5 Hz |
b. 1 Hz |
c. 2 Hz |
Answer: A
Frequency refers
to the number of occurrences of a periodic event per time and is measured in
cycles/second. In this case, there is 1 cycle per 2 seconds. So the frequency
is 1 cycles/2 s = 0.5 Hz.
6.
In problem #5, the period of swing is __________.
|
a. 0.5
second |
b. 1
second |
c. 2
second |
Answer: C
Period refers to
the time for something to happen. In this case, the period is the time for one
complete swing - given as 2 seconds.
7.
A period of 5.0 seconds corresponds to a frequency of ________ Hertz.
|
a. 0.2 |
b. 0.5 |
c. 0.02 |
|
d. 0.05 |
e. 0.002 |
|
Answer: A
Frequency is the
reciprocal of the period. The period is 5 seconds, so the frequency is 1/(5 s) = 0.20 Hz.
8.
A common physics lab involves the study of the oscillations of a pendulum. If a
pendulum makes 33 complete back-and-forth cycles of vibration in 11 seconds,
then its period is ______.
Answer: 0.33 second
Period refers to
the time for something to happen and is measured in seconds/cycle. In this
case, there are 11 seconds per 33 vibrational cycles. Thus the period is (11 s)
/ (33 cycles) = 0.33 seconds.
9.
A child in a swing makes one complete back and forth motion in 3.2 seconds.
This statement provides information about the child's
a. speed
b. frequency
c. period
Answer: B and C
We now know that
the period is 3.2 seconds and that the frequency is 0.31 Hz.
10.
The period of the sound wave produced by a 440 Hertz tuning fork is ___________.
Answer: 0.00227 seconds
GIVEN: f = 440 Hz
Find T
T = 1 / f = 1 /
(440 HZ) = 0.00227 s
11. As the frequency of a wave increases, the period of the
wave ___________.
a. decreases
b. increases
c. remains the same
Answer: A
Period is the
reciprocal of the frequency. So as f increases, 1 / f decreases.