Introduction to Vibrations and
Waves
Experiments with Slinky Springs
What is a wave?
A wave is a disturbance or oscillation
that propagates (or travels) through
space and time.
A wave can transport energy without
carrying any mass with it. The
energy passes through the material
carrying the wave.
A. TRANSVERSE PULSE:
1. A transverse pulse or wave is:
A transverse wave is a pulse
or oscillation that vibrates
perpendicular to the
direction the wave travels.
a. Diagram a transverse pulse.
b. Make a statement about a transverse pulse relating the motion of
the separate coils of the spring to the path traversed by the pulse.
As the energy of the wave passes through the material,
the material is made to oscillate in a direction
perpendicular to the wave’s direction of motion.
c. How does the shape of a short pulse change as it moves along the
spring?
The amplitude or size of the wave gets smaller.
d. Can you suggest a reason for the loss?
Sound, heat energy and absorption from the holding
objects, sap mechanical energy from the spring.
e. Upon what does the initial amplitude depend?
The amplitude depends on the size of the initial
disturbance that creates the wave.
f. Does the speed of the pulse appear to change with its shape?
The speed of the wave stays constant.
g. Does the speed of the pulse appear to depend on the size or shape
of the pulse?
No.
2. Speed of the pulse.
a. Measure the length of the stretched slinky and the travel time of a
pulse generated at one end.
length = _________________________
time = _________________________
b. Compute the speed of the traveling pulse.
length

speed =
time
c. Change the tension of the spring and repeat steps (a) and (b).
length = _________________________
time = _________________________
length

speed =
time
d. Is the speed of propagation dependent on or affected by the
tension in the spring? If yes, is the speed greater or less with
increased tension?
When coils from the end of the spring were pulled out
from the oscillations, the tension was increased and the
speed of the wave increased.
e. Does the slinky, under different tensions, represent the same
transmitting media? Explain.
For a stretch spring, no it does not. As a model for other
media, changing tension could represent different
density materials.
3. Interference.
a. What is interference?
Interference occurs when one
wave passes through another
wave.
b. How does the pulse amplitude during interference compare with the
individual amplitudes before and after superposition when...
(1) the pulses are on the same side of the spring?
The pulses add constructively to produce a larger
amplitude wave.
(2) the pulses are on opposite sides of the spring?
The pulses add destructively to produce a smaller
amplitude wave.
c. What conclusions can you draw about the displacement of the
medium at a point where two pulses interfere?
At any instant, the medium is the vector resultant for all
points. The medium is shaped by the combination of
both passing waves.
Figure 11-38
Interference
4. Reflection From Fixed and Free
End Terminations.
a. How does the amplitude of a
reflected single pulse compare to its
original pulse?
The pulse keeps the same
amplitude. No energy is lost
to the “collision”.
b. What is the phase of the reflected
pulse relative to the transmitted pulse
when the spring has a fixed end
termination?
From drawing (a), the wave
switches sides, corresponding
to a 180 degree phase change.
c. What is the phase of the reflected
pulse relative to the transmitted pulse
when the spring has a free end
termination?
From drawing (b), the wave
does not switch sides,
corresponding to a 0 degree
phase change.
5. Wave Behavior Between Two Media.
a. What happens to the pulse when it reaches the junction between
the two springs?
Some of the pulse will pass through to the other spring,
but some will reflect back from the junction.
b. How does the speed of propagation in the slinky compare with that
in the heavier spring?
The pulses travels faster in the light weight slinky and
more slowly in the heavy spring.
c. Describe what happens when a pulse is transmitted from the slinky
to the heavier spring.
The transmitted pulses remains on the same side of the
spring, but the reflected pulse travels back on the
opposite side of the spring. Behaves like reflection from
a fixed end.
d. Describe what happens when a pulse is transmitted from the
heavier spring to the lighter spring (physics demo 9-19 wave coupling.)
The transmitted pulses remains on the same side of the
spring as before, but the reflected pulse travels back on
the same side of the spring. Behaves like reflection from
a free end.
B. LONGITUDINAL WAVES.
a. Why is this called a
longitudinal wave?
A longitudinal wave is a
pulse or oscillation that
vibrates parallel to the
direction the wave travels.
b. Make a statement about the longitudinal pulse relating the motion
of the separate coils of the spring to the path traversed by the pulse.
Omit. Same answer as a.
1
Day #2: Standing Waves
Notes
{continued}
Vibrations
• Professor Julius Sumner Miller
Characteristics of a single-frequency continuous wave.
C. STANDING WAVES. A standing wave is produced by the
interference of two periodic waves of the same amplitude and
wavelength traveling in opposite directions.
a. How does the motion of a standing wave compare to that of a
transverse wave?
The standing wave seems to “stand still”, oscillating in
the same place on the spring.
b. What is the effect of frequency on a standing wave?
The higher the frequency of the oscillations, the more
“loop” patterns appear in the spring.
c. Draw a diagram of a standing wave with
low and high frequency.
low frequency
high frequency
d. Do all of the parts of the spring move equally? Describe any
variation, if any.
Some parts of the wave have large amplitude, other
points have zero amplitude.
e. Compare the motion of antinodes & nodes.
node = zero amplitude
antinode = maximum amplitude
constructive
f. Loops are caused by _________________ interference while nodes
completely __________________
destructive
are caused by ________________
interference.
in phase, nodes when
g. Loop, are produced by waves that are _____
o
180 out of phase.
they are ______________
h. What always occurs at the ends of the spring?
Each end is fixed, so the end has a node. This is where
destructive interference occurs.
nodes than ____________.
antinodes How
i. There are always more _________
many more of one are there than the other?
one more node than antinode
j. Label the antinodes {A} and the nodes {N} on each of the drawings
of standing waves. Give the length of each wave form in wavelengths.
A
N
N
A
N
N
l.
2L
ln 
n
The n’s are called the
harmonics
_________________
A
N
nl
L
2
A
N
k. Generate a formula relating
wavelength l to the length of the
string L, for any number of
nodes, n.
A
N
A
N
and n = 1 is referred to as the
N
fundamental
__________________.
D. STANDING WAVES ON A STRING. The main purpose of this
section is to find a relationship between the wavelength of the standing
wave on a string and the frequency of standing wave on the string.
1. Diagram a representative standing wave and define each of the
following terms:
a. standing wave
b. wavelength
l
Figure 11-46
The characteristics of a single-frequency wave at t = 0
c. period and frequency:
The period (T) is the time for the wave to complete one
cycle. The frequency (f) is the number of cycles a wave
completes per unit of time, specifically one second.
1
T
f
1
f 
T
1
Hz  hertz 
second
d. wave speed:
A traveling wave will move a distance equal to one
wavelength in a time of one period.
v
l
T
l f
e. frequency of the nth harmonic standing wave:
2L
ln 
n
v  ln f n
fn = frequency of the nth harmonic standing wave, corresponding to
wavelength ln.
v
nv
fn 

ln 2 L
2. Factors to influence the wave speed:
In general, the speed of a wave through a string will be
independent of the wavelength or frequency of the waveform. What
does contribute to the velocity of a wave through a string is the tension
force in the string and the inertia of the string. As shown earlier,
raising the tension in the string increases the speed of the wave.
Increasing the density of the string has the effect of slowing the wave.
In general, the wave speed in any medium is given as:
v
elastic  property

inertial  property
F

F = force of tension in the string
μ = mass per unit length of the string
= “linear mass density”
mass

length
E. Examples.
Ex. #1: When a single pulse (wave) travels through a string, the pulse
covers 2.75 m in a time of 50.0 ms. What would be the frequency of a
standing wave with a wavelength of 27.0 cm on this string?
dist
v
time
2.75 m

50.0 10 3 s
vl f
55.0 m s
f 
0.270 m
f 
 55.0 m s
v
l
 204 Hz
Ex. #2: A string is tied between two fixed ends 90.0 cm apart and
vibrates as show in the diagram. The speed of the wave along the string
is 425 m/s. Determine the wavelength and frequency of this wave.
3rd harmonic
2L
ln 
n
290.0 cm 
l3 
 60.0 cm
3
2L
l3 
3
425 m s
f 
 708 Hz
0.600 m
Ex. #3: a. What is the speed of a transverse wave in a rope of length
2.00 m and mass 60.0 g under a tension of 500 N?
m 0.0600 kg
 
 0.0300 kgm
L
2.00 m
v
F
500 N
m


129
s

0.0300 kgm
b. What is the speed of a transverse wave in a rope of equal length and
same material, but twice the diameter as the previous rope?
A rope with twice the diameter and same material would
have four times the volume and four times the mass per
unit length as the original rope.
If the mass per length was four times bigger, the speed
of the wave in the new rope would be half of the speed in
the original rope.
v
F

Ex. #4: Two pieces of steel wire with identical cross section have
lengths L and 2L. The wires are each fixed at both ends and stretched
so that the tension in the long wire is four times greater than in the
shorter wire. If the fundamental frequency in the shorter wire is 60
Hz, what is the frequency of the second harmonic in the long wire?
Same cross section means same mass per unit length.
v
F

Longer wire has four times the tension, so
the longer wire has twice the wave speed
as the shorter wire.
Fundamental frequency of short wire:
nvshort 1vshort
vshort
f n 1 


 60 Hz
2 Lshort 2 Lshort 2 Lshort
Frequency of second harmonic of long wire:
f n  2, long 
nvlong
2 Llong

2 2vshort 
vshort

2
 120 Hz
22 Lshort 
2 Lshort
Ex. #5: A 12-kg object hangs in equilibrium from a string of total
length L = 5.0 m and linear mass density μ = 0.001 0 kg/m. The string
is wrapped around two light, frictionless pulleys that are separated by
the distance d = 2.0 m (diagram, below left). (a) Determine the
tension in the string. (b) At what frequency must the string between
the pulleys vibrate in order to form the standing-wave pattern shown in
figure below right?
1.00 m
cos  
1.50 m
2.00 m

1.50 m
1.50 m
T

  48.2
T
2T sin   mg

mg
T
2 sin 
T
12.0 kg 9.80 m s 

2 sin 48.19

2
T  78.9 N
(b) At what frequency must the string between the pulleys vibrate in
order to form the standing-wave pattern shown in figure below right?
3rd harmonic
v
nv
fn 

ln 2 L
3v
f3 
2L
v
F

78.89 N

kg
0.0010 m
 281 m s
3v
f3 
2L
3281 m s 

22.00 m 
f 3  211 Hz