Raymond A. Serway
Chris Vuille
Chapter Thirteen
Vibrations and Waves
1
Hooke’s Law Revisited
: Spring force, which is a restoring force
: Spring constant, a measure of the
spring’s stiffness
•
: Displacement from its equilibrium
position
• The negative sign indicates that the force
is always directed opposite to the
displacement, i.e., pushed or pulled
toward the equilibrium position
•
•
Section 13.1
2
Periodic Motion
• Periodic motion: Motion over the
same path
• Simple harmonic motion: Motion
that occurs when the net force along
the motion obeys Hooke’s Law, i.e.,
proportional to the displacement and
pointed toward equilibrium
– Not all periodic motions are simple
harmonic motions unless the restoring
force follows Hooke’s law
Section 13.1
3
Simple Harmonic Motion
• Amplitude,
 Maximum position from its equilibrium
 In the absence of friction, the object oscillates between
• Period,
 The time for the object to complete one cycle of motion, e.g.,
from
to
and back to
• Frequency,
 Number of complete cycles or vibrations per unit time
 Frequency is the reciprocal of the period,
• Acceleration,

or
: not constant and the uniformly
accelerated motion equation cannot be applied
Section 13.1
4
Motion of the Spring-Mass System
• Object is pulled to
and
released from rest
• Object moves
toward
,
and decrease,
but increases
• At
, and
are zero, but is
at its maximum
• The object will
overshoot
•
and start to
increase in the
opposite direction
to decrease
• The motion
momentarily stops
at
• It then accelerates
back toward
• The motion
continues
indefinitely
Section 13.1
5
Velocity as a Function of Position
Conservation of Energy
 Speed is at its maximum at
 Speed is zero at
 The indicates the object can be
traveling in either direction
Problem 9
Section 13.2
6
Simple Harmonic vs. Uniform Circular Motion
https://brilliant.org/wiki/circular-motion-dynamics/
7
Simple Harmonic vs. Uniform Circular Motion
• A ball is attached to the end of a rod or a string
• When the rod rotates at a constant speed, the projection
of the ball in the x-axis moves in simple harmonic motion
 Ball in the circular path:
 Note that
 The velocity of the shadow is then
 Compare to the simple harmonic motion:
Section 13.3
8
Period and Frequency
Circular motion
• Period is the time to complete one cycle,
Harmonic oscillation
• From energy conservation,
• Period
of harmonic oscillations:
• Frequency : Number of cycles per second
 SI unit: Hertz (Hz), 1/sec
• Angular frequency : Number of radians
per second (1 cycle = 2 radians)
Problems 3 – 7
Section 13.3
9
Motion Equations: Displacement
• Difficult to derive directly from
Newton’s law (calculus needed)
• Using a reference circle,
• For the circular motion,
• For the simple harmonic motion,
Section 13.4
10
Motion Equations – Velocity and
Acceleration
•
•
•
• Again, the uniformly accelerated
motion equations are not applicable
Problem 8
Section 13.4
11
Graphical Representation
• Sinusoidal motion
• When is at its maximum
or minimum, is zero
• When is zero, is at its
maximum
• When is at its maximum
in the positive direction,
is at its maximum in the
negative direction
Problems 1, 2
Section 13.4
12
Example 13.6
Compare
with
m,
•
rad/s,
•
(a)
(b)
Hz,
•
at
1s?
sec
•
m,

m/s,

m/s2

Section 13.4
13
Simple Pendulum
• Mass suspended by a string of length
fixed at the upper end
• The restoring force is the component of
the weight tangent to the motion

 For small angles
,
,
harmonic motion is a good approximation

 Compare with Hooke’s law,
Section 13.5
14
Period of Simple Pendulum
depends on the length of the
pendulum and the gravitational
acceleration
•
is independent of the
amplitude and the mass
• The simple pendulum is
analogous to the harmonic
oscillator (mass on a spring)
•
Problems 10 – 13
Section 13.5
15
Damped Oscillations
• Only ideal systems
oscillate indefinitely
• In real systems, friction
slows down the motion
• Friction reduces the total
energy of the system and
the oscillation is damped
• The amplitude decreases
with time
a. Underdamped: Vibrating motion
preserved with decreasing amplitude
b. Critically damped: Object rapidly
returns to equilibrium without
oscillation due to high viscosity
c. Overdamped: object returns without
ever passing through the equilibrium
position with a long time
Section 13.6
16
Wave Motion
• Waves in our daily life: sound, light,
radio, water wave, seismic wave…
• A wave is the motion of a disturbance
• Mechanical wave requires
 A source of disturbance
 A medium that can be disturbed
 Physical connection through which
adjacent portions of the medium
influence each other
• All waves carry energy and momentum
Section 13.7
17
Traveling Waves
• Rope under tension and fixed at the
other end
• The pulse travels to the right with a
definite speed
• Shape of the pulse stays relatively
the same
• The disturbance is a traveling wave
Section 13.7
18
Transverse vs. Longitudinal Waves
• Transverse wave: each element that is disturbed moves in
a direction perpendicular to the wave motion
• Longitudinal (compression) wave: elements of the
medium undergo displacements parallel to wave motion
Section 13.7
19
Types of Waves
• Traveling versus standing waves
 Traveling wave is not confined to a given space of the medium
 Standing wave is, on the other hand, confined
• Displacement of the medium versus wave direction
 Perpendicular: Transverse waves
 Parallel: Longitudinal waves
 Complex: Combination of transverse and longitudinal waves
• The defiant: The solitary wave front (a soliton) propagates
in isolation without changing its form (no dispersion)
Demo: Transverse and longitudinal waves
Section 13.7
20
Waveform
 Wave propagates
 Points oscillate but not propagate
• Two “snapshots” of the
wave at time and
• The high points are crests
• The low points are
troughs
• The wave moves by to
the right in an interval of
Section 13.7
21
Longitudinal Wave as a Sine Curve
• A longitudinal wave can also be represented as a sine curve
 Also called density waves or pressure waves
• Compressions correspond to crests (higher density) and
stretches correspond to troughs (lower density)
Section 13.7
22
Amplitude and Wavelength
• Amplitude ( ): maximum
displacement away from the
equilibrium position
• Wavelength ( ): distance
between two successive
points that behave identically
• Simple harmonic motion

rather than
 Amplitude
 Period
Section 13.8
23
Wave Speed
• Wave speed,
, is the speed at which a particular
part (say, the crest) moves through the medium
• Take
and
, the general wave equation is
• Note that for point P,
the oscillation velocity
points upward
Problems 14 – 17
Section 13.8
24
Producing Waves
• The blade oscillates steadily in simple
harmonic motion to produce a continuous
wave (propagating left to right)
• A small segment of the string (point P)
oscillates with simple harmonic motion
(oscillating up and down)
Section 13.8
25
Wave Speed on a Stretched String
• The wave speed depends on properties of the medium,
such as the density of the medium  and the tension F
(derivation not required)
is the mass per unit length or the linear density
• Distinguish the wave speed (independent of time) from
the oscillation velocity
,
which does depend on time
• Link between oscillation and wave
Section 13.9
26
Example
kg
• Tension in the rope
19.6 N
• Linear mass of the rope
0.005 kg/m
• Wave speed in the rope
62.6 m/s
• Time to travel to the pulley
0.08 s
Problems 19, 20 (combined with knowledge of Young’s modulus)
Section 13.9
27
Wave Interference
• Two traveling waves can meet and pass through each
other without being destroyed or altered
• Superposition Principle: When
two traveling waves encounter
each other, the resulting wave
is found by adding together the
displacements of the individual
waves point by point
Demo: Wave Superposition Model
Section 13.10
28
Constructive Interference – Pulses
• Two pulses are traveling in opposite directions
• The net displacement when they overlap increases
• Pulses are unchanged after the interference
Section 13.10
29
Destructive Interference – Pulses
• Two pulses are traveling in opposite directions
• The net displacement when they overlap decreases
• Pulses are unchanged after the interference
Section 13.10
30
Constructive Interference –
Continuous Waves
• Two waves, a and b, with
the same frequency and
amplitude (in phase),
travel in the same
direction
• The combined wave, c,
has the same frequency
and a greater amplitude
Section 13.10
31
Destructive Interference –
Continuous Waves
• Two waves, a and b, with
the same amplitude and
frequency but inverted
relative to each other (180°
out of phase), travel in the
same direction
• When they combine, the
waveforms cancel
Problem 18
Section 13.10
32
Reflected Wave – Fixed End
• Some or all of a traveling wave is reflected
when reaching a boundary
• When reflected from a fixed end, the wave
is inverted but its shape remains the same
• The wall exerts an opposite reaction force
(Newton’s 3rd law) to invert the wave
Section 13.11
33
Reflected Wave – Free End
• When a traveling wave is reflected from a free end (negligible
mass and friction), the pulse is not inverted
• The ring is accelerated by the pulse and
returns to original position due to tension
Section 13.11
34