Waves
• Traveling Waves
–
–
–
–
–
Types
Classification
Harmonic Waves
Definitions
Direction of Travel
• Speed of Waves
• Energy of a Wave
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
Harmonic Waves
• Transverse displacement looks like:
s (m)
1.5
1
0.5
0
-0.5 0
-1
-1.5
At t = 0
s0
l
2
4
x (m)
 2 
s  x   s 0 sin 
x
 l 
6
Let the wave move
s (m)
Traveling Wave
1.5
1
0.5
0
-0.5 0
-1
ct
2
4
6
-1.5
x (m)
 2

s  x, t   s 0 sin   x  ct  
l

8
Standing at the origin
s (m)
• Transverse displacement looks like:
1.5
1
0.5
0
-0.5 0
-1
-1.5
At x = 0
T
s0 2
4
6
t (sec)
 2

 2c 
 2 
s  0, t   s o sin   0  ct    s o sin  
t   s o sin  t 
l

 l 
 T 
Phase Velocity
distance moved in one cycle l
c
  fl
time required for one cycle T
• Wave velocity is a function of the properties
of the medium transporting the wave
so
-so
ct
That negative sign
2 
 2
s  x, t   s o sin  x 
t
T 
l
• Wave moving
right
2 
 2
s  x, t   s o sin  x 
t
T 
l
• Wave moving
left
so
-so
ct
Alternate notation
2 
 2
s  x, t   s 0 sin  x 
t
T 
l
s  x, t   so sin kx  t 
Wave number
2
k
l
2
Angular frequency  
T
l l 2 
c 

T 2 T k
Definitions
• Amplitude - (so) 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
Two dimensional wave motion
Spherical Wave
Plane Wave
i  r
Acoustic Variables
s  x, t   so sin kx  t 
static
pressure
acoustic
pressure
ptotal  pstatic  pa
•
•
•
•
Displacement
s
ParticleVelocity u 
t
Pressure
  x, t   o
Density
o
Condensation = Compression
Rarefaction = Expansion
A microscopic picture of a fluid
Initial Position – VI
p (x1)
A
VI
x1
• Assumptions:
p(x2)
– Adiabatic
– Small displacements
– No shear deformation
x2
Later Position - VF
p (x1)
A
VI
p(x2)
A
VF
s2
x1
• Physics Laws:
x2
s1
– Newton’s Second Law
– Equation of State
– Conservation of mass
The Wave Equation
Newton’s Second Law/
Conservation of Mass
 2 s 
 pa
   2 
x
t 
Equation of State/
Conservation of Mass
s
pa  B
x
PDE – Wave Equation
 2s     2s
  2
2
x  B  t
Solutions to differential equations
• Guess a solution
• Plug the guess into the differential equation
– You will have to take a derivative or two
• Check to see if your solution works.
• Determine if there are any restrictions (required
conditions).
• If the guess works, your guess is a solution, but it
might not be the only one.
• Look at your constants and evaluate them using
initial conditions or boundary conditions.
The Plane Wave Solution
s  x, t   so sin  kx t 
 2s     2s
  2
2
x  B  t

2
s o k sin  kx  t      s o  sin  kx  t 
 B
2
 2
k   
B
2

B
c
k

General rule for wave speeds
Elastic Property
c
Inertial Property
Longitudinal wave
in a long bar
Young's modulus
Y
c

density

Longitudinal wave
in a fluid
Bulk modulus
B
c

density

Bulk modulus
B
c

density

Sound Speed
Air
Sea Water
Bulk Modulus 1.4(1.01 x 105) Pa 2.28 x 109 Pa
Density
1.21 kg/m3
1026 kg/m3
Speed
343 m/s
1500 m/s
Variation with Temperature:
Air
Seawater
m
v   331  0.60T 
s
m
v  1449.05  4.57T  .0521T  .00023T 
s
2
3
Example
•
•
•
•
•
A plane acoustic wave is propagating in a
medium of density =1000 kg/m3. The equation
for a particle displacement in the medium due to
the wave is given by:
s  1x10 6 cos8x  12000t 
where distances are in meters and time is in
seconds.
What is the rms particle displacement?
What is the wavelength of the sound wave?
What is the frequency?
What is the speed of sound in the medium?
Alternate Solutions
s  x, t   so cos  kx  t 
i kx t 
s(x, t)  soe
s(x, t)  s o e
 kx t 


  
2
s  x, t   so sin  nkx  mt 
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
• Fourier’s Theorem – any complex wave can be
constructed from a sum of pure sinusoidal
waves of different amplitudes and frequencies
Alternate Views
Particle Displacement
Particle Velocity
s  x, t   so sin  kx  t 
s
u   s 0 cos  kx  t 
t
Pressure
s
pa  x, t  = - B
 -  c 2 s 0 kcos  kx- wt 
x
Density
  x, t   o pa

 so k cos  kx  t 
0
B
Pitch is frequency
Audible
20 Hz – 20000 Hz
Infrasonic
< 20 Hz
Ultrasonic
>20000 Hz
Middle C on the piano has a frequency of 262 Hz.
What is the wavelength (in air)?
1.3 m
Specific Acoustic Impedance
• Like electrical impedance
• Acoustic analogy
– Pressure is like voltage
– Particle velocity is like
current
• Specific acoustic
Impedance:
• For a plane wave:
Zelectric
V

I
p(x, t)
z
u(x, t)
2
p c s0 k cos  kx  t 
z 
 c
u
s0ck cos  kx  t 
Energy Density in a Plane Wave
2
1 2 1
1
 K  u   s o  cos  kx  t    2s o2 cos 2  kx  t 
2
2
2
2
1
1
1
 P  2s 2  2 s o sin  kx  t    2s o2 sin 2  kx  t 
2
2
2
1 2
1 pa2 max
u max 
2
2 c2
K
P
Average Energy Density
K
P
1 2 2
1 2 2 1 2
2
  s o cos  kx  t    s o  u max
2
4
4
1 2 2
1 2 2 1 2
2
  s o sin  kx  t    s o  u max
2
4
4
1 2
1 pa2 max
u max 
2
2 c2
  K  P
Or
1 2
 u max
2
1 pa2 max
 
2 c2
K
P
Average Power and Intensity
dE   Acdt
A
cdt
dE
P 
  Ac
dt
2
P
p
1
1 a max 1
2
I 
  c  cu max 
 pa max u max
A
2
2 c
2
Instantaneous Intensity
 pa  x, t  
2
I  x, t   p a  x, t  u  x, t  
 z  u  x, t  
z
2
1   Force  displacement 
 Power  Work
I 






Area  time
 Area   time Area  
I   Pressure  velocity 
V2
P
 ZI 2  VI
Z
Root Mean Square (rms) Quantities
p
2
a
 p rms
p max

2
therefore:
2
max
2
rms
p
p
I 

2 c  c