Chapter 13
Vibrations and Waves
Periodic motion
• Periodic (harmonic) motion – self-repeating motion
• Oscillation – periodic motion in certain direction
• Period (T) – a time duration of one oscillation
• Frequency (f) – the number of oscillations per unit
time, SI unit of frequency 1/s = Hz (Hertz)
1
f 
T
Heinrich Hertz
(1857-1894)
Motion of the spring-mass system
• Hooke’s law:
F  kx
• The force always acts toward the equilibrium
position: restoring force
• The mass is initially pulled to a distance A and
released from rest
• As the object moves toward the equilibrium
position, F and a decrease, but v increases
Motion of the spring-mass system
• At x = 0, F and a are zero, but v is a maximum
• The object’s momentum causes it to overshoot the
equilibrium position
• The force and acceleration start to increase in the
opposite direction and velocity decreases
• The motion momentarily comes to a stop at x = - A
Motion of the spring-mass system
• It then accelerates back toward the equilibrium
position
• The motion continues indefinitely
• The motion of a spring mass system is an example
of simple harmonic motion
Simple harmonic motion
• Simple harmonic motion – motion that repeats itself
and the displacement is a sinusoidal function of time
x(t )  A cos(t   )
Amplitude
• Amplitude – the magnitude of the maximum
displacement (in either direction)
x(t )  A cos(t   )
Phase
x(t )  A cos(t   )
Phase constant
x(t )  A cos(t   )
Angular frequency
x(t )  A cos(t   )
 0
A cos t  A cos  (t  T )
cos   cos(  2 )
cos(t  2 )  cos  (t  T )
2  T
2

T
  2f
Period
x(t )  A cos(t   )
T
2

Velocity of simple harmonic motion
x(t )  A cos(t   )
v(t )  A sin( t   )
Acceleration of simple harmonic motion
x(t )  A cos(t   )
a(t )   A cos(t   )
2
a(t )   x(t )
2
The force law for simple harmonic
motion
• From the Newton’s Second Law:
2
F  ma  m x
• For simple harmonic motion, the force is
proportional to the displacement
• Hooke’s law:
F  kx
k  m
2
k

m
m
T  2
k
Energy in simple harmonic motion
• Potential energy of a spring:
U (t )  kx / 2  (kA / 2) cos (t   )
2
2
2
• Kinetic energy of a mass:
K (t )  mv / 2  (m A / 2) sin (t   )
2
2
 (kA / 2) sin (t   )
2
2
2
2
m  k
2
Energy in simple harmonic motion
U (t )  K (t ) 
 (kA / 2) cos (t   )  (kA / 2) sin (t   )
2
2

2
2
 (kA / 2) cos (t   )  sin (t   )
2
 (kA / 2)
2
2
2

E  U  K  (kA / 2)
2
Energy in simple harmonic motion
kA / 2  kx / 2  mv / 2
2
2
2


k 2
2
v
A  x  
m
A  x  mv / k
2
A
2
2
x
2
2

E  U  K  (kA / 2)
2
Chapter 13
Problem 11
A simple harmonic oscillator has a total energy E. (a) Determine the kinetic and
potential energies when the displacement is one-half the amplitude. (b) For
what value of the displacement does the kinetic energy equal the potential
energy?
Pendulums
• Simple pendulum:
• Restoring torque:
   L( Fg sin  )
• From the Newton’s Second Law:
I     L( Fg sin  )
• For small angles
sin   
mgL
 

I
Pendulums
• Simple pendulum:
at

L
s

L
mgL
 

I
mgL
a
s
I
• On the other hand
a(t )   x(t )
2
mgL

I
Pendulums
• Simple pendulum:
mgL

I
mgL


2
mL
2
I  mL
2
g
L
L
T
 2

g
Pendulums
• Physical pendulum:
mgh

I
2
I
T
 2

mgh
Chapter 13
Problem 32
An aluminum clock pendulum having a period of 1.00 s keeps perfect time at
20.0°C. (a) When placed in a room at a temperature of –5.0°C, will it gain time or
lose time? (b) How much time will it gain or lose every hour?
Simple harmonic motion and uniform
circular motion
• Simple harmonic motion is the projection of uniform
circular motion on the diameter of the circle in which
the circular motion occurs
Simple harmonic motion and uniform
circular motion
• Simple harmonic motion is the projection of uniform
circular motion on the diameter of the circle in which
the circular motion occurs
x(t )  A cos(t   )
vx (t )  A sin( t   )
Simple harmonic motion and uniform
circular motion
• Simple harmonic motion is the projection of uniform
circular motion on the diameter of the circle in which
the circular motion occurs
x(t )  A cos(t   )
vx (t )  A sin( t   )
Simple harmonic motion and uniform
circular motion
• Simple harmonic motion is the projection of uniform
circular motion on the diameter of the circle in which
the circular motion occurs
x(t )  A cos(t   )
a x (t )   A cos(t   )
2
Damped simple harmonic motion
Fb  bv
Damping
force
Damping
constant
Forced oscillations and resonance
• Swinging without outside help – free oscillations
• Swinging with outside help – forced oscillations
• If ωd is a frequency of a driving force, then forced
oscillations can be described by:
x(t )  A(d / , b) cos(d t   )
• Resonance:
d  
Forced oscillations and resonance
• Tacoma Narrows Bridge disaster (1940)
Wave motion
• A wave is the motion of a disturbance
• All waves carry energy and momentum
Types of waves
• Mechanical – governed by Newton’s laws and exist
in a material medium (water, air, rock, ect.)
• Electromagnetic – governed by electricity and
magnetism equations, may exist without any medium
• Matter – governed by quantum mechanical
equations
Types of waves
Depending on the direction of the displacement
relative to the direction of propagation, we can define
wave motion as:
• Transverse – if the direction of displacement is
perpendicular to the direction of propagation
• Longitudinal – if the direction of displacement is
parallel to the direction of propagation
Types of waves
Depending on the direction of the displacement
relative to the direction of propagation, we can define
wave motion as:
• Transverse – if the direction of displacement is
perpendicular to the direction of propagation
• Longitudinal – if the direction of displacement is
parallel to the direction of propagation
Superposition of waves
• Superposition principle – overlapping waves
algebraically add to produce a resultant (net) wave
• Overlapping solutions of the linear wave equation
do not in any way alter the travel of each other
Sinusoidal waves
• One of the most characteristic solutions of the
linear wave equation is a sinusoidal wave:
y ( x  vt)  A sin( k ( x  vt)   )
 A cos( k ( x  vt)     / 2)
• A – amplitude, φ – phase constant
Wavelength
y ( x, t )  A cos( k ( x  vt)   )
• “Freezing” the solution at t = 0 we obtain a
sinusoidal function of x:
y ( x,0)  A cos( kx   )
• Wavelength λ – smallest distance (parallel to the
direction of wave’s travel) between repetitions of the
wave shape
Wave number
y ( x,0)  A cos( kx   )  A cos( k ( x   )   )
 A cos( kx  k   )
• On the other hand:
cos( kx   )  cos( kx  2   )
• Angular wave number: k = 2π / λ
k  2 / 
Angular frequency
y ( x, t )  A cos( k ( x  vt)   )
• Considering motion of the point at x = 0
we observe a simple harmonic motion (oscillation) :
y (0, t )  A cos(  kvt   )  A cos( kvt   )
• For simple harmonic motion:
y (t )  A cos(t   )
  kv  2v / 
• Angular frequency ω
Frequency, period
• Definitions of frequency and period are the same as
for the case of rotational motion or simple harmonic
motion:
f  1 / T   / 2
T  2 / 
• Therefore, for the wave velocity
v   / k   / T  f
y ( x, t )  A cos( kx  t   )
Wave velocity
• v is a constant and is determined by the properties
of the medium
• E.g., for a stretched string with linear density μ
m/l under tension T
v
T

=
Chapter 13
Problem 41
A harmonic wave is traveling along a rope. It is observed that the oscillator that
generates the wave completes 40.0 vibrations in 30.0 s. Also, a given maximum
travels 425 cm along the rope in 10.0 s. What is the wavelength?
Interference of waves
• Interference – a phenomenon of combining waves,
which follows from the superposition principle
• Considering two sinusoidal waves of the same
amplitude, wavelength, and direction of propagation
y1 ( x, t )  A cos(kx  t )
• The resultant wave:
y2 ( x, t )  A cos(kx  t   )
y( x, t )  y1 ( x, t )  y2 ( x, t )
 A cos( kx  t )  A cos( kx  t   )
 2 A cos( / 2) cos( kx  t   / 2)
  
cos   cos   2 cos
 2
    
 cos

2
 

Interference of waves
y ( x, t )  2 A cos( / 2) cos( kx  t   / 2)
• If φ = 0 (Fully constructive)
y ( x, t )  2 A cos( kx  t )
• If φ = π (Fully destructive)
y ( x, t )  0
• If φ = 2π/3 (Intermediate)
y ( x, t )  2 A cos( / 3) 
 cos( kx  t   / 3)
 A cos( kx  t   / 3)
Reflection of waves at boundaries
• Within media with boundaries, solutions to the wave
equation should satisfy boundary conditions. As a
results, waves may be reflected from boundaries
• Hard reflection – a fixed zero value of deformation at
the boundary – a reflected wave is inverted
• Soft reflection – a free value of deformation at the
boundary – a reflected wave is not inverted
Questions?
Answers to the even-numbered problems
Chapter 13
Problem 2
(a) 1.1 × 102 N
(b) The graph is a straight line passing
through the origin with slope equal to k
= 1.0 × 103 N/m.
Answers to the even-numbered problems
Chapter 13
Problem 8
(a) 575 N/m
(b) 46.0 J
Answers to the even-numbered problems
Chapter 13
Problem 12
(a) 2.61 m/s
(b) 2.38 m/s
Answers to the even-numbered problems
Chapter 13
Problem 16
(a) 0.15 J
(b) 0.78 m/s
(c) 18 m/s2