```WAVE LESSON 3
The Wave Equation
Lets recap:
A wave is produced when a vibrating source periodically disturbs the first particle of a
medium.
This creates a wave pattern which begins to travel along the medium from particle to
particle.
The frequency at which each individual particle vibrates is equal to the frequency at
which the source vibrates.
Similarly, the period of vibration of each individual particle in the medium is equal to the
period of vibration of the source. In one period, the source is able to displace the first
particle upwards from rest, back to rest, downwards from rest, and finally back to rest.
This complete back-and-forth movement constitutes one complete wave
cycle.
The diagrams at the right show several "snapshots" of the production of a
wave within a rope. The motion of the disturbance along the medium after
every one-fourth of a period is depicted.
Observe that in the time it takes from the first to the last snapshot, the hand
has made one complete back-and-forth motion. A period has elapsed.
Observe that during this same amount of time, the leading edge of the
disturbance has moved a distance equal to one complete wavelength. So in a
time of one period, the wave has moved a distance of one wavelength.
Combining this information with the equation for speed (speed =
distance/time), it can be said that the speed of a wave is also the
wavelength/period.
Since the period is the reciprocal of the frequency, the expression 1/f can be substituted
into the above equation for period. Rearranging the equation yields a new equation of the
form:
Speed = Wavelength • Frequency
The above equation is known as the wave equation. It states the mathematical
relationship between the speed (v) of a wave and its wavelength ( ) and frequency (f).
Using the symbols v, , and f, the equation can be rewritten as
v=f•
As a test of your understanding of the wave equation and its mathematical use in
analyzing wave motion, consider the following three-part question:
Stan and Anna are conducting a slinky experiment. They are studying the possible affect
of several variables upon the speed of a wave in a slinky. Their data table is shown
below. Fill in the blanks in the table, analyze the data, and answer the following
questions.
Medium
Wavelength
Frequency
Speed
1.75 m
2.0 Hz
______
0.90 m
3.9 Hz
______
1.19 m
2.1 Hz
______
0.60 m
4.2 Hz
______
0.95 m
2.2 Hz
______
1.82 m
1.2 Hz
______
Zinc,
1-in. dia. coils
Zinc,
1-in. dia. coils
Copper,
1-in. dia. coils
Copper,
1-in. dia. coils
Zinc,
3-in. dia. coils
Zinc,
3-in. dia. coils
Multiply the frequency by the wavelength to determine the speed.
Row 1: speed = 3.5 m/s
Row 2: speed = 3.5 m/s
Row 3: speed = 2.5 m/s
Row 4: speed = 2.5 m/s
Row 5: speed = 2.1 m/s
Row 6: speed = 2.2 m/s (any difference with row 5 is just experimental
error)
a. As the wavelength of a wave in a uniform medium increases, its speed will _____.
a. decrease
b. increase
c. remain the same
In rows 1 and 2, the wavelength was altered but the speed remained the same. The same
can be said about rows 3 and 4 and rows 5 and 6. The speed of a wave is not affected by
the wavelength of the wave.
b. As the wavelength of a wave in a uniform medium increases, its frequency will _____.
a. decrease
b. increase
c. remain the same
In rows 1 and 2, the wavelength was increased and the frequency was decreased.
Wavelength and frequency are inversely proportional to each other.
c. The speed of a wave depends upon (i.e., is causally affected by) ...
a. the properties of the medium through which the wave
travels
b. the wavelength of the wave.
c. the frequency of the wave.
d. both the wavelength and the frequency of the wave.
Whenever the medium is the same, the speed of the wave is the same. However, when the
medium changes, the speed changes. The speed of these waves were dependent upon the
properties of the medium.
Wave speed is dependent upon medium properties and independent of wave
properties.
Even though the wave speed is calculated by multiplying wavelength by
frequency, an alteration in wavelength does not affect wave speed. Rather, an
alteration in wavelength affects the frequency in an inverse manner. A doubling
of the wavelength results in a halving of the frequency; yet the wave speed is not
changed.
What if you put a lot of energy into the original disturbance, what does it do? (whip the
slinky up and down with a lot of energy)?
Answer: it does not affect the speed of the wave or the wavelength or the frequency, only
the amplitude.
What if you whip the slinky up and down very quickly?
Answer: This increases the frequency and decreases the wavelength and does nothing to
the speed of the wave.
What changes the speed of the wave then?
Answer: Only the properties of the medium influence the speed of the wave (what is it
Behavior of Waves
Boundary Behavior
A sound wave is known to reflect off canyon walls and other obstacles to produce an
echo. A sound wave traveling through air within a canyon reflects off the canyon wall
and returns to its original source.
What affect does reflection have upon a wave?
Does reflection of a wave affect the speed of the wave?
Does reflection of a wave affect the wavelength and frequency of the wave?
Does reflection of a wave affect the amplitude of the wave?
Or does reflection affect other properties and characteristics of a wave's motion?
When one medium ends, another medium begins; the interface of the two media is
referred to as the boundary and the behavior of a wave at that boundary is described as
its boundary behavior.
Fixed End Reflection
This end of the rope is referred to as a fixed end because last particle of the rope will be
unable to move when a disturbance reaches it
If a pulse is introduced at the left end of the rope, it will travel through the rope towards
the right end of the medium. When the incident pulse reaches the boundary, two things
occur:


A portion of the energy carried by the pulse is reflected and returns towards the
left end of the rope. The disturbance which returns to the left after bouncing off
the pole is known as the reflected pulse.
A portion of the energy carried by the pulse is transmitted to the pole, causing
the pole to vibrate.
Because the vibrations of the pole are not visibly obvious, the energy transmitted to it is
not typically discussed.
What characteristics and properties could describe the reflected pulse’s motion?
First the reflected pulse is inverted.
When a crest reaches the end of a medium ("medium A"), the last particle of the medium
A receives an upward displacement. This particle is attached to the first particle of the
other medium ("medium B") on the other side of the boundary. As the last particle of
medium A pulls upwards on the first particle of medium B, the first particle of medium B
pulls downwards on the last particle of medium A. This is merely Newton's third law of
action-reaction.
The upward pull on the first particle of medium B has little affect upon this particle due
to the large mass of the pole and the lab bench to which it is attached. The affect of the
downward pull on the last particle of medium A (a pull which is in turn transmitted to the
other particles) results in causing the upward displacement to become a downward
displacement. The upward displaced incident pulse thus returns as a downward displaced
reflected pulse.
Other notable characteristics of the reflected pulse include:

The speed of the reflected pulse is the same as the speed of the incident pulse.
WHY?
Since the speed of a wave (or pulse) is dependent upon the medium through which it
travels, two pulses in the same medium will have the same speed.

The wavelength of the reflected pulse is the same as the wavelength of the
incident pulse.
WHY?
Every particle within the rope will have the same frequency. Being connected to one
another, they must vibrate at the same frequency. Since the wavelength of a wave
depends upon the frequency and the speed, two waves having the same frequency and the
same speed must also have the same wavelength.

The amplitude of the reflected pulse is less than the amplitude of the incident
pulse.
WHY?
The amplitude of the reflected pulse is less than the amplitude of the incident pulse since
some of the energy of the pulse was transmitted into the pole at the boundary. Amplitude
is proportionate to energy in pulse.
ANIMATION - Reflection of a Pulse at a Fixed End @
http://www.physicsclassroom.com/mmedia/
Free End Reflection
Now consider what would happen if the end of the rope were free to move.
When the incident pulse reaches the end of the medium, the last particle of the rope can
no longer interact with the first particle of the pole, they will slide past each other.
So when a crest reaches the end of the rope, the last particle of the rope receives
the same upward displacement; only now there is no adjoining particle to pull
downward upon the last particle of the rope to cause it to be inverted.
The result is that the reflected pulse is not inverted. When an upward displaced
pulse is incident upon a free end, it returns as an upward displaced pulse after
reflection.
ANIMATION - Reflection of a Pulse at a Free End @
http://www.physicsclassroom.com/mmedia/
What if the original medium were attached to another rope with different properties?
Transmission of a Pulse Across a Boundary from Less to More Dense
Let's consider a thin rope attached to a thick rope, with each rope held at opposite ends by
people. And suppose that a pulse is introduced by the person holding the end of the thin
rope.
Upon reaching the boundary, the usual two behaviors will occur.


A portion of the energy carried by the incident pulse is reflected and returns
towards the left end of the thin rope. This disturbance is the reflected pulse.
A portion of the energy carried by the incident pulse is transmitted into the thick
rope. The disturbance is the transmitted pulse.
The reflected pulse will be found to be inverted in situations such as this.
During the interaction between the two media at the boundary, the first particle of
the more dense medium overpowers (pulls down on) the smaller mass of the last
particle of the less dense medium. This causes an upward pulse to become a
downward reflected pulse in the less dense medium.
The more dense medium receives an upward pull when the incident pulse reaches
the boundary causing an upward displacement. For this reason, the transmitted
pulse is not inverted.
In fact, transmitted pulses can never be inverted. WHY?
Since the particles in this medium are originally at rest, any change in their
state of motion would be in the same direction as the displacement of the
particles of the incident pulse.
Comparisons can also be made between the characteristics of the transmitted pulse and
those of the reflected pulse. Once more there are several noteworthy characteristics.

The transmitted pulse (in the more dense medium) is traveling slower than the
reflected pulse (in the less dense medium).
WHY?
Waves always travel fastest in the least dense medium.

The transmitted pulse (in the more dense medium) has a smaller wavelength than
the reflected pulse (in the less dense medium).
WHY?
Particles in the more dense medium will be vibrating with the same frequency as
particles in the less dense medium. Since the transmitted pulse was introduced into
the more dense medium by the vibrations of particles in the less dense medium, they
must be vibrating at the same frequency. So the reflected and transmitted pulses have
the different speeds but the same frequency. Since the wavelength of a wave depends
upon the frequency and the speed, the wave with the greatest speed must also have
the greatest wavelength.

The speed and the wavelength of the reflected pulse are the same as the speed and
the wavelength of the incident pulse.
WHY?
The incident and the reflected pulse share the same medium. Since the two pulses are in
the same medium, they will have the same speed. Since the reflected pulse was created
by the vibrations of the incident pulse, they will have the same frequency. And two
waves with the same speed and the same frequency, must also have the same wavelength.
ANIMATION - Characteristics of a Transmitted Pulse
A Less Dense to a More Dense Medium @
http://www.physicsclassroom.com/mmedia/
Transmission of a Pulse Across a Boundary from More to Less Dense
Consider a thick rope attached to a thin rope, with the incident pulse originating in the
thick rope.
There will be partial reflection and partial transmission at the boundary.
Since the incident pulse is in a heavier medium, when it reaches the boundary, the
first particle of the less dense medium does not have sufficient mass to overpower
the last particle of the more dense medium. The result is that an upward displaced
pulse incident towards the boundary will reflect as an upward displaced pulse
The Before and After snapshots of the two media are shown in the diagram below.
Comparisons between the characteristics of the transmitted pulse and the reflected pulse

The transmitted pulse (in the less dense medium) is traveling faster than the
reflected pulse (in the more dense medium).
WHY?
Waves always travel fastest in the least dense medium.

The transmitted pulse (in the less dense medium) has a larger wavelength than the
reflected pulse (in the more dense medium).
WHY?
Particles in the more dense medium will be vibrating with the same frequency as
particles in the less dense medium. Since the transmitted pulse was introduced into
the less dense medium by the vibrations of particles in the more dense medium, they
must be vibrating at the same frequency. So the reflected and transmitted pulses have
the different speeds but the same frequency. Since the wavelength of a wave depends
upon the frequency and the speed, the wave with the greatest speed must also have
the greatest wavelength.

The speed and the wavelength of the reflected pulse are the same as the speed and
the wavelength of the incident pulse.
WHY?
The incident and the reflected pulse share the same medium. Since the two pulses are
in the same medium, they will have the same speed. Since the reflected pulse was
created by the vibrations of the incident pulse, they will have the same frequency.
And two waves with the same speed and the same frequency, must also have the same
wavelength.
The boundary behavior of waves in ropes can be summarized by the following principles:





The wave speed is always greatest in the least dense rope.
The wavelength is always greatest in the least dense rope.
The frequency of a wave is not altered by crossing a boundary.
The reflected pulse becomes inverted when a wave in a less dense rope is heading
towards a boundary with a more dense rope.
The amplitude of the incident pulse is always greater than the amplitude of the
reflected pulse.
WAVE LESSON 3 HOMEWORK
1. Two waves on identical strings have frequencies in a ratio of 2 to 1. If their wave
speeds are the same, then how do their wavelengths compare?
a. 2:1
b. 1:2
c. 4:1
d. 1:4
2. Mac and Tosh stand 8 meters apart and demonstrate the motion of a transverse wave
on a snakey. The wave e can be described as having a vertical distance of 32 cm from a
trough to a crest, a frequency of 2.4 Hz, and a horizontal distance of 48 cm from a crest to
the nearest trough. Determine the amplitude, period, and wavelength and speed of such a
wave.
3. Dawn and Aram have stretched a slinky between them and begin experimenting with
waves. As the frequency of the waves is doubled,
a. the wavelength is halved and the speed remains constant
b. the wavelength remains constant and the speed is doubled
c. both the wavelength and the speed are halved.
d. both the wavelength and the speed remain constant.
4. A ruby-throated hummingbird beats its wings at a rate of about 70 wing beats per
second.
a. What is the frequency in Hertz of the sound wave?
b. Assuming the sound wave moves with a velocity of 350 m/s, what is the
wavelength of the wave?
5. Ocean waves are observed to travel along the water surface during a developing storm.
A Coast Guard weather station observes that there is a vertical distance from high point
to low point of 4.6 meters and a horizontal distance of 8.6 meters between adjacent crests.
The waves splash into the station once every 6.2 seconds. Determine the frequency and
the speed of these waves.
v = f • wavelength = (0.161 Hz) • (8.6 m)
6. Two boats are anchored 4.0 meters apart. They bob up
and down, returning to the same up position every 3.0
seconds. When one is up the other is down. There are never
any wave crests between the boats. Calculate the speed of
the waves.
Case 1: A pulse in a more dense medium is traveling
towards the boundary with a less dense medium.
7. The reflected pulse in medium 1 ________ (will, will not) be inverted because
_______.
8. The speed of the transmitted pulse will be ___________ (greater than, less than, the
same as) the speed of the incident pulse.
9. The speed of the reflected pulse will be ______________ (greater than, less than, the
same as) the speed of the incident pulse.
10. The wavelength of the transmitted pulse will be ___________ (greater than, less than,
the same as) the wavelength of the incident pulse.
11. The frequency of the transmitted pulse will be ___________ (greater than, less than,
the same as) the frequency of the incident pulse.
Case 2: A pulse in a less dense medium is traveling towards the boundary with a more
dense medium.
12. The reflected pulse in medium 2 ________ (will, will not) be inverted because
_____________.
13. The speed of the transmitted pulse will be ___________ (greater than, less than, the
same as) the speed of the incident pulse.
14. The speed of the reflected pulse will be ______________ (greater than, less than, the
same as) the speed of the incident pulse.
15. The wavelength of the transmitted pulse will be ___________ (greater than, less than,
the same as) the wavelength of the incident pulse.
16. The frequency of the transmitted pulse will be ___________ (greater than, less than,
the same as) the frequency of the incident pulse.
17. The diagram 5-62 shows a transverse pulse traveling along a heavy rope toward its
junction with a lighter rope. Which of the diagrams 5-63 best illustrate the ropes at the
instant that the reflected pulse again passes through its orignal position marked X?
a. A
b. B
c. C
d. D
e. E
18. The line X represents the boundary between two dissimilar springs. A pulse is shown
approaching the boundary in diagram 5-7. Which one of the sketches in diagram 5-8
shows a possible configuration of the system shortly after the pulse reaches the
boundary?
a. A
b. B
c. C
d. E
HOMEWORK KEY WAVE LESSON 3
1. B
2. 16 m, 96 cm, 0.42 s, 230 cm/s
3. A
4. 7.0 x 101 Hz, 5.0 m
5. 8.6 m, 6.2 s, 0.16 Hz, 1.4 m/s
6. 2.7 m/s
7. will not
8. faster
9. the same as
10. greater than
11. the same as
12. will
13. less than
14. same as
15. less than
16. E
17. C
WAVE LESSON 3 HOMEWORK
1. Two waves on identical strings have frequencies in a ratio of 2 to 1. If their wave
speeds are the same, then how do their wavelengths compare?
a. 2:1
b. 1:2
c. 4:1
d. 1:4
Frequency and wavelength are inversely proportional to each other. The wave with the
greatest frequency has the shortest wavelength. Twice the frequency means one-half the
wavelength. For this reason, the wavelength ratio is the inverse of the frequency ratio.
2. Mac and Tosh stand 8 meters apart and demonstrate the motion of a transverse wave
on a snakey. The wave e can be described as having a vertical distance of 32 cm from a
trough to a crest, a frequency of 2.4 Hz, and a horizontal distance of 48 cm from a crest to
the nearest trough. Determine the amplitude, period, and wavelength and speed of such a
wave.
Amplitude = 16 cm
(Amplitude is the distance from the rest position to the crest position
which is half the vertical distance from a trough to a crest.)
Wavelength = 96 cm
(Wavelength is the distance from crest to crest, which is twice the
horizontal distance from crest to nearest trough.)
Period = 0.42 s
(The period is the reciprocal of the frequency. T = 1 / f)
Speed = 230 cm/s
(The speed of a wave is calculated as the product of the frequency times
the wavelength.)
3. Dawn and Aram have stretched a slinky between them and begin experimenting with
waves. As the frequency of the waves is doubled,
a. the wavelength is halved and the speed remains constant
b. the wavelength remains constant and the speed is doubled
c. both the wavelength and the speed are halved.
d. both the wavelength and the speed remain constant.
Doubling the frequency will not alter the wave speed. Rather, it will halve the
wavelength. Wavelength and frequency are inversely related.
4. A ruby-throated hummingbird beats its wings at a rate of about 70 wing beats per
second.
a. What is the frequency in Hertz of the sound wave?
b. Assuming the sound wave moves with a velocity of 350 m/s, what is the
wavelength of the wave?
Answer: f = 70 Hz and wavelength = 5.0 m
The frequency is given and the wavelength is the v/f ratio.
5. Ocean waves are observed to travel along the water surface during a developing storm.
A Coast Guard weather station observes that there is a vertical distance from high point
to low point of 4.6 meters and a horizontal distance of 8.6 meters between adjacent crests.
The waves splash into the station once every 6.2 seconds. Determine the frequency and
the speed of these waves.
Answer: The wavelength is 8.6 meters and the period is 6.2 seconds.
The frequency can be determined from the period. If T = 6.2 s, then
f =1 /T = 1 / (6.2 s)
f = 0.161 Hz
Now find speed using the v = f • wavelength equation.
v = f • wavelength = (0.161 Hz) • (8.6 m)
v = 1.4 m/s
6. Two boats are anchored 4 meters apart. They bob up and
down, returning to the same up position every 3 seconds.
When one is up the other is down. There are never any
wave crests between the boats. Calculate the speed of the
waves.
Answer: The diagram is helpful. The wavelength must be 8 meters (see diagram).
The period is 3 seconds so the frequency is 1 / T or 0.333 Hz.
Now use speed = f • wavelength Substituting and solving for v, you will get 2.67 m/s.
Case 1: A pulse in a more dense medium is traveling towards the boundary with a less
dense medium.
7. The reflected pulse in medium 1 ________ (will, will not) be inverted because
_______.
8. The speed of the transmitted pulse will be ___________ (greater than, less than, the
same as) the speed of the incident pulse.
9. The speed of the reflected pulse will be ______________ (greater than, less than, the
same as) the speed of the incident pulse.
10. The wavelength of the transmitted pulse will be ___________ (greater than, less than,
the same as) the wavelength of the incident pulse.
11. The frequency of the transmitted pulse will be ___________ (greater than, less than,
the same as) the frequency of the incident pulse.
Answer: 7. will not... because the reflection occurs for a wave in a more dense medium
heading towards a less dense medium.
8. faster
9. the same as
10. greater than
11. the same as
Case 2: A pulse in a less dense medium is traveling towards the boundary with a more
dense medium.
12. The reflected pulse in medium 2 ________ (will, will not) be inverted because
_____________.
13. The speed of the transmitted pulse will be ___________ (greater than, less than, the
same as) the speed of the incident pulse.
14. The speed of the reflected pulse will be ______________ (greater than, less than, the
same as) the speed of the incident pulse.
15. The wavelength of the transmitted pulse will be ___________ (greater than, less than,
the same as) the wavelength of the incident pulse.
16. The frequency of the transmitted pulse will be ___________ (greater than, less than,
the same as) the frequency of the incident pulse.
12. will... because the reflection occurs for a wave in a less dense medium heading
towards a more dense medium.
13. less than
14. the same as
15. less than
16. the same as
17. The diagram 5-62 shows a transverse pulse traveling along a heavy rope toward its
junction with a lighter rope. Which of the diagrams 5-63 best illustrate the ropes at the
instant that the reflected pulse again passes through its orignal position marked X?
a. A
b. B
c. C
d. D
e. E
A, B, and C can be ruled out since it shows the amplitude of the reflected and incident
pulse to be the same size. An incident pulse would give up some of tis energy to the
transmitted pulse at the boundary thus making the amplitude of the reflected pulse less
than that of the incident pulse. Rule out D since it shows the reflected pulse moving
faster than the transmitted pulse. This would not happen unless moving from less dense
to more dense. This leaves E as the answer.
18. The line X represents the boundary between two dissimilar springs. A pulse is shown
approaching the boundary in diagram 5-7. Which one of the sketches in diagram 5-8
shows a possible configuration of the system shortly after the pulse reaches the
boundary?
a. A
b. B
c. C
d. E
The speed in the two media can be deduced by the distance of the pulses from the
boundary. In A and E, the speed is shown as fastest on the right, which makes the
transmitted medium the less dense. Rule out A and E since a reflected pulse should not
invert when moving from more dense to less dense. Rule out B for just the opposite
reasons, the wave is moving from less to more dense and should invert upon reflection.
Rule out D because the transmitted pulse never inverts. That leaves C as the answer.
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