Chapter 15 - Waves
• Traveling Waves
–
–
–
–
–
Types
Classification
Harmonic Waves
Definitions
Direction of Travel
• Speed of Waves
• Energy of a Wave
• Standing Waves
– Reflection and Transmission
– Superposition and Interference
• (Refraction and Refraction)
Types of Waves
• Mechanical Waves - Those waves resulting from
the physical displacement of part of the medium from
equilibrium.
• Electromagnetic Waves - Those wave resulting
from the exchange of energy between an electric and
magnetic field.
• Matter Waves - Those associated with the wave-like
properties of elementary particles.
Requirements for Mechanical Waves
• Some sort of disturbance
• A Medium that can be disturbed
• Physical connection or mechanism through
which adjacent portions of the medium can
influence each other.
Classification of Waves
• Transverse Waves - The
particles of the medium
undergo displacements in a
direction perpendicular to the
wave velocity
– Polarization - The orientation
of the displacement of a
transverse wave.
• Longitudinal
(Compression) Waves The particles of the medium
undergo displacements in a
direction parallel to the
direction of wave motion.
– Condensation/Rarefraction
Waves on the surface of a liquid
3D Waves
Sound Waves
http://www.kettering.edu/~drussell/Demos/waves/wavemotion.html
Harmonic Waves
• Transverse displacement looks like:
y (m)
1.5
1
0.5
0
-0.5 0
-1
-1.5
At t = 0
A
l
2
4
x (m)
 2 
y  x   A sin 
x
 l 
6
Let the wave move
Traveling Wave
1.5
vt
y (m)
1
0.5
0
-0.5 0
2
4
6
-1
-1.5
x (m)
 2

y  x, t   A sin   x  vt  
l

8
Standing at the origin
y (m)
• Transverse displacement looks like:
1.5
1
0.5
0
-0.5 0
-1
-1.5
At x = 0
T
Dm 2
4
6
t (sec)
2v 
2 
 2

 2
 2
y  x, t   A sin   0  vt    A sin  0 
t   A sin  0 
t
l 
T 
l

l
 l
Phase Velocity
distance moved in one cycle l
v
  fl
time required for one cycle T
• Wave velocity is a function of the properties
of the medium transporting the wave
That negative sign
• Wave moving
right
• Wave moving
left
2 
 2
y  x, t   A sin  x 
t
T 
l
2 
 2
y  x, t   A sin  x 
t
T 
l
Alternate notation
2 
 2
y  x, t   A sin  x 
t
T 
l
y  x, t   Asin  kx  t 
Wave number
2
k
l
2
Angular frequency  
T
l l 2 
v 

T 2 T k
Definitions
• Amplitude - (A, ym) Maximum value of the displacement of a particle in a
medium (radius of circular motion).
• Wavelength - (l) The spatial distance between any two points that behave
identically, i.e. have the same amplitude, move in the same direction (spatial period)
• Wave Number - (k) Amount the phase changes per unit length of wave
travel. (spatial frequency, angular wavenumber)
• Period - (T) Time for a particle/system to complete one cycle.
• Frequency - (f) The number of cycles or oscillations completed in a period of
time
• Angular Frequency -  Time rate of change of the phase.
• Phase - kx - t Time varying argument of the trigonometric function.
• Phase Velocity - (v) The velocity at which the disturbance is moving
through the medium
Velocity of transverse wave in a cord
Fy t  p
FT v

Fy v
 v 
 FT  t   vt  v
 v
v
FT

General rule for wave speeds
Elastic Property
v
Inertial Property
Longitudinal wave
in a long bar
Young's modulus
E
v

density

Longitudinal wave
in a fluid
Bulk modulus
B

density

v
Superposition
• Waves in the same
medium will add
displacement when at the
same position in the
medium at the same time.
• Overlapping waves do not
in any way alter the travel
of each other (only the
medium is effected)
Superposition in reverse
• Fourier’s Theorem – any complex wave can be
constructed from a sum of pure sinusoidal
waves of different amplitudes and frequencies
Interference
(Superposition of equal amplitude waves)
Constructive
Destructive
http://www.kettering.edu/~drussell/Demos/superposition/superposition.html
Interference of harmonic waves
• Constructive - Waves are
in phase. Amplitude
doubling occurs
• Destructive - Waves are
180 degrees out of phase.
Amplitude cancellation
occurs
Reflection
Fixed Boundary
“Flips”
Free Boundary
Doesn’t flip”
http://www.kettering.edu/~drussell/Demos/reflect/reflect.html
Standing Waves - Resonance
yr  x, t   Asin  kx  t 
yl  x   Asin kx  t 
y  x, t   Asin kx  t   Asin kx  t 
1  2
1  2
sin 1  sin 2  2sin
cos
2
2
y  2Asin  kx  cos  t 
Nodes and Antinodes
• Node – position of no
displacement
• Antinode – position of
maximum displacement
y  2Asin  kx  cos  t 
2
kx 
x  0, , 2,3...
l
l
3l
x  0, , l, ,.....
2
2
kx 
2
 3 5 7 
x  , , , ...
l
2 2 2 2
x
l 3l 5l 7l
, , , .....
4 4 4 4
Natural frequencies
y  2Asin  kx  cos  t 
2
kL 
L  0, , 2,3, 4,....
l
2L
ln 
n
n=1,2,3,....
v nv
fn 

l n 2L
v
n=1,2,3,4,....
FT

Energy in a Wave
P  2 vAf s
2
2 2
max
Intensity
P
I
 22vf 2s 2max
Area
Two dimensional wave reflection
i  r