1
Waves
All substantive material is from Wave Motion and Sound by James Dann. http://www.ck12.org/flexr/ unless
otherwise noted. Illustrations are copyright free.
Objects in motion that return to the same position after a fixed period of time
are said to be in harmonic motion. Objects in harmonic motion have the
ability to transfer some of their energy over large distances. They do so by
creating waves in a medium.
Mediums
A medium is the substance through which the wave travels. For example,
water acts as the medium for ocean waves, while air molecules act as the
medium for sound waves. When a wave passes through a medium, the
medium is only temporarily disturbed. When an ocean wave travels from one
side of the Mediterranean Sea to the other, no actual water molecules move
this great distance. Only the disturbance propagates (moves) through the
medium.
2 Types of Waves
In longitudinal waves, the vibrations of the medium are in the same direction
as the wave motion. A classic example is a wave traveling down a line of
standing dominoes: each domino will fall in the same direction as the motion
of the wave. A more commonly-seen example is a sound wave. For sound
waves, high and low pressure zones move both forward and backward as the
wave moves through them.
In transverse waves, the vibrations of the medium are perpendicular to the
direction of motion. A classic example is a wave created in a long rope: the
wave travels from one end of the rope to the other, but the actual rope moves
up and down, and not from left to right as the wave does.
Parts of Waves
The speed of a wave on a string depends on the material the string is made
of, as well as the tension in the string. This fact is why tightening a string on
your violin or guitar will change the sound it produces.
The frequency,
frequency f, is the number of cycles an object goes through in 1 second.
Frequency is measured in Hertz (Hz). I Hz = 1cycle per sec. An object
oscillating with frequency f will create waves in a medium which oscillate
with the same frequency f.
2
The amplitude,
amplitude A, is the furthest distance moved by a particle from the
medium as the wave passes over the particle. The amplitude, therefore, is
half of the total distance covered by the oscillating particle.
The period of a wave is the time it takes for one complete cycle to occur. For
a transverse wave, a period includes one crest and one trough. For a
longitudinal wave, it includes a compression and a rarefaction. Period is a
measure of time so the SI units for measuring it are seconds.
wave type
transverse
longitudinal
period includes
crest + trough
compression + rarefaction
Parts of Transverse Waves
The crest is the high point of movement for a particle of
medium. The trough is the low point of movement for a particle
of medium. Wavelength is the distance covered by one complete
crest and one complete trough. Or, another way to view it is the
distance between two side-by-side crests.
Parts of Longitudinal Waves
A compression
compression is the part of a wave where the particles of the
medium have been shoved closer together by the energy of the
wave. The rarefaction is the part of a wave where the particles of
the medium have been pulled farther apart by the energy of the
wave. Wavelength is the distance of one complete rarefaction and
one complete compression:
wavelength = compression + rarefaction.
Energy Transmission
Waves are energy moving through a medium. The energy is transmitted in
the direction the wave moves. However, the motion of the particles in the
medium may not be the same as the motion of the wave. In longitudinal
waves, the medium particles oscillate in the same direction (parallel) as the
wave moves. In transverse waves, the particles move at a 90˚ (perpendicular)
to the direction the wave moves—think how the molecules in a rope move
compared to the motion of the wave sent down the rope.
wave type
movement of
medium particles
movement of
wave
transverse
left to right
longitudinal
left to right
movement of
energy
3
Interference
When two waves bump into each other it is called interference. Constructive
interference occurs when two waves combine to create a larger wave. This
occurs when the parts of the waves
line up with each other: crest-to-crest
and trough-to-trough or compressionto-compression and rarefaction-torarefaction. When constructive
interference occurs, the result is a
single wave whose amplitude is greater than the sum of the two original
waves.
Destructive interference occurs when two
waves combine and cancel each other out. This
occurs when the “opposite” parts of a wave line
up with each other: crest-to-trough or
compression-to-rarefaction. When destructive
interference occurs, the result is a cancellation
of the wave and no movement of the medium occurs, the wave disappears.
The cancellation of the wave is because each wave tugs equally strongly on
the particle of medium, but the waves pull in opposite directions. This stops
the medium particle from moving. The wave can go no further.
The previous examples assumed the waves were identical to each other. But
if the waves are not of the same amplitude or frequency, they do not give
perfect results. For example, two waves undergoing destructive interference
(if they are not identical) will only cancel out part way. Some wave will be
left.
When two waves move in opposite directions, through each other,
interference takes place. If the two waves have the same frequency and
wavelength then standing waves are
generated. Standing waves are so-called
because they appear to be standing still.
Twirling a jump rope is a way of forming a
standing wave (the crest goes up and the
trough goes down). If you twirl it fast enough,
the wave appears to be a solid sphere which
isn’t moving. In comparison, snap a rope so that a wave travels down the
length of the rope, this is not a standing wave.
Nodes, Antinodes, and Rest Positions
Before a wave travels through a medium, the particles of the medium are at
rest position,
position sitting still. As the wave moves through, the particles of the
4
medium will be moved by the passing wave of energy. Remember that the
amplitude is the measure of how far the particles are moved from the rest
position. Once the wave passes by, the medium’s particles return to their
rest positions. When two transverse waves meet each other they will either
add or subtract their energy from each other. If a crest and a trough meet, it
is called destructive interference. The smaller of the two waves will be
subtracted from the larger wave. If they are the same size (amplitude) they
will cancel each other out. The point where they cancel each other out is
called a node.
node
If two waves meet, and the crests meet each other or troughs meet each
other, they will add their energy together and create a larger crest or trough.
This is constructive interference. The result of constructive interference is an
antianti-node.
node
A standing wave can be understood as a series of nodes and anti-nodes.
Reflection, Refraction, and Diffraction
Reflection of a wave occurs when a wave bounces off of a barrier. An echo is
an example of a sound wave being reflected. Your reflection in a mirror is
formed by light waves bouncing off of the surface of the mirror.
Diffraction is the bending of a wave around a barrier. Sound waves bending
around a barrier explain why you are able hear noises in a different room.
The wave will go around the edge of the barrier and then spread out to fill the
space.
Refraction occurs when waves enter a new medium. Each medium will
transmit waves at a different speed. As the wave enters the new medium it
will change speeds and consequently change direction. For example: a straw
placed in a glass of water will appear to be in two pieces. Refraction of the
light coming off of the straw accounts for this. The light traveling in only air
is going at a different speed then the light traveling through the air and
water. The difference in speeds makes the straw appear to be broken.
Velocity of A Wave
The speed (V) and wavelength (λ) of a wave depend upon the nature of the
medium through which the wave travels. But regardless of the medium, the
relationship between velocity, wavelength, and frequency remains the same:
V = λ f or sometimes it appears as V = λ ν
f = frequency (Hz) sometimes f is written as the Greek letter nu (ν)
V = velocity (m/s)
λ = wavelength (m)
5
Example Problem 1
Question
When a particular string is vibrated at a frequency of 10Hz, a transverse
wave of wavelength 0.25m is produced. Determine the speed of the wave as it
travels along the string.
Answer
Answer
Step 1 : Determine what is given and what is required
frequency of wave: f =10Hz
wavelength of wave: λ =0.25m
We are required to calculate the speed of the wave as it travels along the
string. All quantities are in SI units.
Step 2 : Determine how to approach the problem
We know that the speed of a wave is:
v = λ・f
and we are given all the necessary quantities.
Step 3 : Substituting in the values
v=f・λ
= (10 Hz)(0.25m)
= 2.5m/s
Step 4 : Write the final answer
The wave travels at 2.5m/s in the string.
[from The Free High School Science Texts: Textbooks for High School Students Studying the Sciences Physics Grades
10 – 12. www.fhsst.org]
6
Example Problem 2
Question
A cork on the surface of a swimming pool bobs up and down once per second
on some ripples. The ripples have a wavelength of 20 cm. If the cork is 2m
from the edge of the pool, how long does it take a ripple passing the cork to
reach the shore?
Answer
Step 1 : Determine what is given and what is required
We are given:
frequency of wave: f = 1Hz
wavelength of wave: λ = 20 cm
distance of leaf from edge of pool: d = 2m
We are required to determine the time it takes for a ripple to travel between
the cork and the edge of the pool. The wavelength is not in SI units and
should be converted.
Step 2 : Determine how to approach the problem
The time taken for the ripple to reach the edge of the pool is obtained from:
t = d/v(from v = d/t)
We know that
v=λ・ f
Therefore, t = d/ (f ・ λ)
Step 3 : Convert wavelength to SI units
20 cm = 0.2m
Step 4 : Solve the problem
t = d/ (f ・ λ)
=2m/[(1 Hz)(0.2m)]
= 10 s
Step 5 : Write the final answer
A ripple passing the leaf will take 10 s to reach the edge of the pool.
[from The Free High School Science Texts: Textbooks for High School Students Studying the Sciences Physics Grades
10 – 12. www.fhsst.org]
7
Wave Lab
Each group needs one slinky, four pieces of tape, one stopwatch. Each group
will contain the following people:
2 node people
2 tapers
1 timer
The node people will each hold an end of the slinky against the floor. Stretch
the slinky, but not too tight. On the end node each node person should place
a piece of tape, mark the rest position with a piece of tape. The goal is to
produce a standing wave. Once the standing wave is produced, each taper
(they should be on the same side of the slinky) will use their piece of tape to
mark adjacent (side-by-side) crests. The timer should time how long it takes
for 10 oscillations (back-and-forth movement of the slinky between a crest
and trough: in other words, the crest would appear 10 times in 10
oscillations) to occur. This might be easiest if one of the node people counts
out each time (for 10 times) their hand moves all the way to the right. One of
the node people should count the number of nodes between the end nodes
(inclusive of the end nodes). Do this 3 times. Now PUT THE SLINKY
AWAY, FAR FAR AWAY!!!! Fill in the following table.
# of
nodes
Time for
10
oscillations
(s)
# of
oscillations
per second
(frequency)
2 x distance
between 2
crests
(wavelength)
(m)
Velocity
of 1 wave
(v=λf)
Period
(period=1/f)
# of
antinodes
(area
between 2
nodes)
4
5
6
Compare the wavelength that was measured and the one calculated.
What happens to frequency as period gets larger?
Calculated
wavelength (divide
the distance between
the end nodes by the
number of antinodes
and multiply by 2)
8
Review Questions:
Review questions are from the SD Achievement Series.
1.
The bending of waves around the edges of barriers is called
A.
diffraction
B. refraction
C. reflection
D. dispersion
2.
Destructive interference occurs when what happens?
A.
the crests of two waves overlap
B. two waves of the same color overlap
C. two waves of the same wavelength meet
D. the crest of one wave meets the trough of another wave
3.
Which of the following may be produced during destructive
interference of waves?
A.
a node
B. a reflection
C. a higher crest
D. a lower trough
4.
The change in wave direction at the boundary of two different
media is
A.
incidence
B. refraction
C. reflection
D. diffraction
9
5.
The spreading of waves around the edge of a barrier is
A.
incidence
B. refraction
C. reflection
D. diffraction
6.
When does constructive interference occur?
A.
the crests of two waves overlap
B. two waves of the same color overlap
C. two waves of the same wavelength meet
D. the crest of one wave meets the trough of another wave
7.
At which point will destructive interference occur on the
concentric waves below?
A.
1
B. 2
C. 3
D. 4
8.
A sound wave is an example of which type of wave?
A.
surface wave
B. inverted wave
C. transverse wave
D. longitudinal wave
10
9.
Both transverse and longitudinal waves
A.
transfer energy through a medium
B. have compressions and rarefactions
C. are capable of moving the medium a long distance
D. move at right angles to the vibrations of the medium
10.
The frequency of longitudinal waves may be measured by
counting the number of successive compression zones that
pass a point in a given time interval. What is the similar part of
a transverse wave that may also be used to measure
frequency
A.
crests
B. wavelengths
C. amplitudes
D. rarefactions
11.
If the frequency of a given wave doubles, what happens to its
wavelength?
A.
it doubles
B. it is halved
C. it quadruples
D. it stays the same
12.
If 300 waves pass a point in 60 seconds, what is their
frequency?
A.
5 Hz
B. 30 Hz
C. 300 Hz
D. 18000 Hz
11
13.
A wave has a frequency of 4.0 hertz and a wavelength of 15
mm. What is its speed?
A.
.27 mm/s
B. 3.8 mm/s
C. 60 mm/s
D. 60 Hertz/s
14.
A light wave is an example of which type?
A.
surface wave
B. inverted wave
C. transverse wave
D. compressional wave
15.
The types of waves produced by a piano are classified as
A.
radio waves
B. micro waves
C. transverse waves
D. longitudinal waves
16.
You are creating a wave on a spring. If you start shaking the
spring more quickly, what happens to its wavelength?
A.
it increases
B. it decreases
C. it remains the same
D. it depends on its amplitude
17.
What is the frequency of a wave if 250 waves pass a point in
0.5 seconds?
A.
125 Hz
B. 250 Hz
C. 500 Hz
D. 5000 Hz
12
18.
Which of the following wave interactions will result in constructive interference?
A.
B.
C.
D.
19.
Which of the following occurs as a result of constructive interference?
A. a wave that has a smaller amplitude
B. a wave that has a larger amplitude
C. a wave that has a larger wavelength
D. a wave that has a smaller wavelength
13
20.
Which of the following diagrams correctly shows constructive interference?
A.
B.
C.
D.
21.
Why does the straw in the glass of water shown below look different above water than it
does underwater?
A. The light from the straw is being refracted because the light scatters as it hits the
surface of the water, causing some of the light to reflect and leaving only part of
the light to be seen underwater.
B. The light from the straw is being refracted because as the light travels from air
into water, it slows down and changes its direction.
C. The light is scattered as it enters the water, so it spreads out underwater,
causing the straw to look larger and bent.
D. The light from the straw above the surface of the water reflects off the curved
bottom of the glass, bending the light slightly and making the straw seem bent.
14
22.
Use the diagram below to answer the question.
Which diagram illustrates the direction of a particle's motion in the transverse wave
above?
A.
B.
C.
D.
23.
Which term describes the area of the longitudinal wave indicated by the arrow?
A. compression
B. rarefaction
C. node
D. antinode
24.
What part of a longitudinal wave is under the least amount of compression?
A. the area that is experiencing compression
B. the crest of the wave
C. the trough of the wave
D. the area that is experiencing rarefaction
15
25.
The frequency of a wave traveling at 15 m/s with an amplitude of 0.030 m is 6.0 Hz.
What is its wavelength?
A. 2.5 m
B. 90 m
C. 0.40 m
D. 83 m
26.
What is the velocity of a wave if its frequency is 250 Hz, its wavelength is 20.0 m, and
its amplitude is 0.150 m?
A. 37.5 m/s
B. 0.080 m/s
C. 1667 m/s
D. 5000 m/s
27.
How would a wave be affected if its wavelength increased and its frequency remained
constant as it traveled through a new medium?
A. Its velocity would increase.
B. Its velocity would decrease.
C. Its amplitude would increase.
D. Its amplitude would decrease.
28.
How would a wave be affected if its frequency increased and its wavelength remained
constant as the wave entered a new medium?
A. Its amplitude would decrease.
B. Its velocity would increase.
C. Its velocity would decrease.
D. Its amplitude would increase.
29.
How would a wave's frequency be affected if its velocity and its wavelength were
doubled?
A. the frequency would decrease
B. the frequency would double
C. the frequency would remain the same
D. the frequency would increase slightly
16
Advanced Questions:
1) Summarize the motion of a particle in a medium during the transmission
of a wave.
2) Compare/contrast longitudinal and transverse waves.
3) If v=λf is true and if the speed of light is 3 x 108 m/s, predict what will
happen to the frequency of a wave of light as its wavelength increases.
4) If two pebbles are tossed near each other in a pond, describe the
interaction of the resulting waves.
17
Waves Teaux
C
A
D
B
E
Choose the letter which best identifies the specified part of a wave.
1) Wavelength
2) Amplitude
3) Crest
4) Trough
Choose the best answer.
5) If a wave has a wavelength of 650 nm and a frequency of 77 cycles per second, what
is its velocity?
a) 5005 m/s b) 50000 m/s
c) 0.00005 m/s
6) Two beams of light have different wavelengths. Wave A has a wavelength of 33000
nm while wave B has a wavelength of 499 m. Which has the greater frequency?
a) A b) B
7) What is the velocity of an electromagnetic wave whose frequency is 88.5 hertz.
a) 3x108 m/s b) 88.5 m/s c) unsolvable d) 9.8 m/s/s
8) What is the wavelength of a wave whose frequency is 8000000 hz and whose velocity
is 216 m/s?
a) 1728000000 m
b) 2.7x10-5 m
c) 3704 m
9) What is the frequency of a ray of light whose wavelength is 997 nm?
a) 3x108 hz b) 3x1014 hz
c) unsolvable
10) What is the wavelength of light whose frequency is 7.45 x 1013 hz?
a) 4x10-6 m
b) 2.2x1022 m/s
c) 1.3x10-14
11) For any given wave traveling at a constant velocity, as its wavelength lengthens, its
frequency
a) increases
b) decreases
12) As a wave's frequency decreases, the total energy of the wave
a) increases
b) decreases
13) As a wave's wavelength increases, the total energy of the wave
a) increases
b) decreases
18
19
Bye, Bye, Waves
Choose the best answer.
1) If a radio wave has a frequency of 4.8 khz, what is its wavelength in m?
a) 7.9x10-5
b) 6.25x104
c) 1.8x104
d) 6.25x103
2) If an EM wave has a wavelength of 2.3 mm, how much energy does it carry in Joules?
a) 1.3x108
b) 8.6x10-26
c) 8.6x10-29
d) all of the above
3) Planck's equation allows what to be calculated, assuming you can find values for the
proper variables?
a) energy of an EM wave
b) frequency of an EM wave
c) the value of Planck's constant
d) all of the above
4) If a photon has a wavelength of 330km, what is its energy in J?
a) 6x10-31
b) 6x10-28
c) 6x10-19
d) 9.1x105
5) What is the frequency of a compression wave traveling at 600 m/s and having a
wavelength of 220m
a) 2.7
b) 132000
c) 0.37
d) none of the above
20
6) For which of the following do the particles of its medium oscillate parallel to the
transmission of energy?
a) transverse wave
b) sound wave
c) compression wave
d) both a and b
e) both b and c
7) For which of the following do the particles of its medium oscillate vertically to the
transmission of energy?
a) transverse wave
b) sound wave
c) compression wave
d) both a and b
e) both b and c
8) Which of the following is the same type of wave as a light wave?
a) transverse wave
b) sound wave
c) compression wave
d) both a and b
e) both b and c
9) As wavelength for a photon increases, frequency
a) stays the same
b) increases
c) decreases
10) For a sound wave, as frequency increases, wavelength
a) stays the same
b) increases
c) decreases
11) For a photon, as frequency increases, velocity
a) stays the same
b) increases
c) decreases