Chapter 11
Vibrations and Waves
Simple Harmonic Motion
A restoring force is one that moves a system back to an
equilibrium position.
Example: mass
on frictionless
table, attached
to spring.
Example: gravity
acting on a mass
hanging from a string.
Example: gravity
acting on a mass
hanging from a spring.
Hooke’s Law
Felastic  kx
When the restoring force is linearly proportional to the
amount of the displacement from equilibrium, the force is
said to be a Hooke’s Law force.
General definitions of vibrations and
waves
 Vibration: in a general sense, anything that switches back
and forth, to and fro, side to side, in and out, off and on,
loud and soft, or up and down is vibrating. A vibration is a
wiggle in time.
 Wave: a wiggle in both space and time is a wave. A wave
extends from one place to another.
 Vibrations and waves: the source of all waves is something
that is vibrating. Waves are propagations of vibrations
throughout space.
 When oscillations are small, the motion is called simple
harmonic motion (shm) and can be described by a
simple sine curve.
Wave Properties
Wavelength
Wavelength, l, is the distance between two
consecutive peaks.
Wave Properties
Amplitude
Amplitude is the height of the wave above
or below the equilibrium point.
Wave Properties
Period
The wave period, P, this the time it take one wave to pass
the observer.
Wave Properties
Frequency
Frequency, f, is the number of waves passing a particular
point in one second.
Examples of Frequency
 What is the frequency of the second hand of a
clock? Frequency = 1cycle/60 sec
Period = 60 sec
What is the frequency of US Presidential
elections?
Frequency = 1 election/4 yrs
Period = 4 yrs
In symbolic form
or
T
T 1T
Tff  T T
f
f f ff T
Wave Motion, Speed, Type
Waves to transfer energy, not matter, from one place to
another
A Vibrating source transfers a disturbance
Speed depends on type of vibrating source and medium
through which it travels
Wave speed = f x l
The same type of wave moves at the same speed
regardless of f or l
For any wave, f is inversely proportional to l
VIBRATION OF A PENDULUM

What does the period (T)
depend upon?
 Length
of the pendulum (l).
 Acceleration due to gravity (g).

Period does not depend upon
the bob mass or the amplitude
of the swing.
T  2 l g
Vibration of a
pendulum. The to-andfro vibratory motion is
also called oscillatory
motion (or oscillation).
Wave Type
Transverse waves vibrate across from direction of travel
Longitudinal waves vibrate along the direction of travel (as in a
spring)
Sound Waves
Molecules in the air vibrate about some average position
creating the compressions and rarefactions. We call the
frequency of sound the pitch.
Wave Interference
When two wave pass each other their
superposition
causes reinforcement or cancellation.
Constructive interference
Reinforcement when the crest of one wave overlaps the crest
of another
Their individual effects adds together, resulting in a wave
increased in amplitude
Destructive Interference
Cancellation when crest of one wave overlaps trough of another
reducing their individual effects
Water waves show these best
Out of phase- the crest of one wave arrives at a point at the
same time as a trough of the second wave arrives, effects
cancel each other
In phase- two waves crests and troughs arrive at a place at
the same time, effects reinforce each other
Sound Wave Interference
•Interference occurs when two sounds of difference frequency
are heard superposed.
•Constructive interference causes louder sound and destructive
inference cause fainter sound.
•This alternating pattern produces a beat.
A piano tuners listens for beats to disappear.
Water Wave Interference
 Left side is theoretical drawing of an
interference pattern.
 Right side is the actual interference pattern.
Standing Waves
Occurs when a wave reflects upon itself and interference
causes the pattern
Nodes remain stationary
Anti nodes-occur half way between nodes
Standing Waves
Change the frequency in a standing wave and more
nodes/antinodes appear in the event
Wave Behavior
We know that waves travel through
mediums.
But what happens when that
medium runs out?
25
Boundary Behavior
 The behavior of a wave when it reaches
the end of its medium is called the
wave’s BOUNDARY BEHAVIOR.
 When one medium ends and another
begins, that is called a boundary.
26
Fixed End
 One type of boundary that a wave may encounter is
that it may be attached to a fixed end.
 In this case, the end of the medium will not be able to
move.
 What is going to happen if a wave pulse goes down
this string and encounters the fixed end?
27
Fixed End
 Here the incident pulse is an upward pulse.
 The reflected pulse is upside-down. It is inverted.
 The reflected pulse has the same speed, wavelength,
and amplitude as the incident pulse.
28
Fixed End Animation
29
Free End
 Another boundary type is when a wave’s medium is
attached to a stationary object as a free end.
 In this situation, the end of the medium is allowed to
slide up and down.
 What would happen in this case?
30
Free End
 Here the reflected pulse is not inverted.
 It is identical to the incident pulse, except it is moving
in the opposite direction.
 The speed, wavelength, and amplitude are the same
as the incident pulse.
31
Free End Animation
32
Change in Medium
 Our third boundary condition is when the medium of
a wave changes.
 Think of a thin rope attached to a thin rope. The
point where the two ropes are attached is the
boundary.
 At this point, a wave pulse will transfer from one
medium to another.
 What will happen here?
33
Change in Medium
 In this situation part of the wave is reflected,
and part of the wave is transmitted.
 Part of the wave energy is transferred to the
more dense medium, and part is reflected.
 The transmitted pulse is upright, while the
reflected pulse is inverted.
34
Change in Medium
 The speed and wavelength of the reflected wave
remain the same, but the amplitude decreases.
 The speed, wavelength, and amplitude of the
transmitted pulse are all smaller than in the incident
pulse.
35
Change in Medium Animation
36