EXAM 2
Thursday November 18 8-9:30PM
Location: PHYS112/ PHYS114
Covers Chapters 6-10 (inclusive)
Closed book exam.
Bring a number 2 pencil, your calculator and your student ID. You
may not use a communication device (such as a cell phone,
iPhone, etc.) as a calculator.
You will be provided with an equation sheet, which will include a copy
of Table 8.2, giving moments of inertia for different shapes.
There is no make up exam. If I excuse you from the exam, the
average of Exam 1 and Final Exam grades will replace your
Exam 2 grade.
Physics 218 Fall 2010
1
Lecture 21: Chapter 12
Waves
Physics 218 Fall 2010
Mechanical Waves
Require:
1. A source of disturbance
Physics 218 Fall 2010
Mechanical Waves
Require:
1. A source of disturbance
2. A medium that can be disturbed.
QuickTime™ and a
Sorenson Video 3 decompressor
are needed to see this picture.
Physics 218 Fall 2010
Mechanical Waves
Require:
1. A source of disturbance
2. A medium that can be disturbed.
DEMO 4A-03
Physics 218 Fall 2010
Mechanical Waves
Require:
1. A source of disturbance
2. A medium that can be disturbed.
3. Some connection through which adjacent portions of
the medium can influence each other
Physics 218 Fall 2010
Types of Mechanical Waves
Traveling Waves: Waves that travel from point to point
as a waves moves through a medium.
Impulse or Pulse
Harmonic Motion (Periodic)
DEMO
Physics 2181S-15
Fall 2010
Types of Mechanical Waves
Traveling Waves: Waves that travel from point to point
as a waves moves through a medium.
Longitudinal Waves (Compressional):
* particles in a medium undergo displacements in a
direction parallel to the direction of propagation.
* medium is alternately dilated and compressed.
Physics 218 Fall 2010
Types of Mechanical Waves
Traveling Waves: Waves that travel from point to point
as a waves moves through a medium.
Transverse Waves (Shear):
* particle motion is perpendicular to the direction of
propagation.
* V T < VL
Physics 218 Fall 2010
Types
of
Mechanical
Waves
Traveling Waves: Waves that travel from point to point
as a waves moves through a medium.
Rayleigh Waves:
* particle motion consists of both transverse and
longitudinal components.
VR ~ 0.92 Vs
Physics 218 Fall 2010
Example
d
d
t  t 2  t1 

VT VL
 VT VL
d  
VL  VT

t

Earth’s Crust
VL  5600m / s
VT  3500m / s
d  9333t (meters)
t
t
1
Physics 218 Fall 2010
t
t2
Harmonic (Periodic) Waves
Function that describes a wave is any function of the
form f (x  vt)
Harmonic Function: y(x,t)  A sin(
2 x

 2 ft)
Amplitude = A
Phase=(2  x/+ 2  f t)
y(t)
t
x
Physics 218 Fall 2010
Periodic Waves
Harmonic Function: y(x,t)  A sin(
2 x

 2 ft)
Wavelength,  : minimum distance between two points
on a wave that behave identically. (Units meters)
y(x,0) (arb. units)
Wavenumber, k : how the oscillations repeat in space
(spatial frequency). (Units radians/meters)
12
8
4
0
-4
-8
-12
k
0
20
40
60
x (arb. units)
Physics 218 Fall 2010

80
100
2

v

f
Periodic Waves
Harmonic Function:
y(x,t)  Asin(
2 x

 2 ft)
Frequency, f : rate at which a disturbance repeats itself.
(Units cycles/sec)
1 
f  
T 2
y(0,t) (arb. units)
  2 f
12
8
4
0
-4
-8
-12
0
20
40
60
80
100
t (arb. units)
Physics 218 Fall 2010

DEMO 4A-01
Periodic Waves
2 x
Harmonic Function: y(x,t)  A sin(   2 ft)
Velocity, v : waves propagate with a specific velocity
2 x
 2 ft  constant

2 dx

 dt
y(t,0) (arb. units)
t
12
8
4
0
-4
-8
-12
dx
dt
0
20
v
40
0

Wave speed
2
60
x (arb. units)
Physics 218 Fall 2010
80
100
Question:
Wave Motion

A harmonic wave moving in the positive x direction can be
described by the equation
y(x,t)  A sin(

2 x

 2 ft)
Which of the following equations describes a harmonic wave
moving in the negative x direction?
A.
y(x,t)  Asin(
2 x
 2 ft)

2 x
 2 ft)
B. y(x,t)  Asin(

2 x
 2 ft)
C. y(x,t)  A sin(

Physics 218 Fall 2010
Lecture 13
Solution

Recall
y(x,t)  A sin(
2 x

 2 ft)

The sign of the term containing the t determines the
direction of propagation.

We change the sign to change the direction:
y(x,t)  Asin(
y(x,t)  Asin(
Physics 218 Fall 2010
2 x

2 x

Lecture 13
 2 ft) moving toward +x
 2 ft)
moving toward -x
Perodic Waves
Velocity of particle, vparticle :
v particle
y(x,t)  Asin(
2 x

dy(x,t)
2 x

 2 fA cos(
  t)

dt
y(x,0) (arb. units)
Velocity of wave, vwave : depends on the medium
12
8
4
0
-4
-8
-12
vwave
vparticle
0
Physics 218 Fall 2010
20
40
60
x (arb. units)
80
100
 2 ft)
Periodic Waves
Velocity of wave, vwave : depends on the medium
F
For a string: v 
m/L
kT
For an ideal gas: v 

For a liquid: v 
B
For a solid: v 
E


F = tension in string
m/L = linear density
k = Boltzman constant,  = density,
T=temperature,  = Cp/Cv
B = Bulk Modulus
 = density
E = Elastic Modulus
 = density
v   / k  f   / T
Physics 218 Fall 2010
Question
Wave Motion

A heavy rope hangs from the ceiling, and a small
amplitude transverse wave is started by jiggling the
rope at the bottom.
As the wave travels up the rope, its speed will:
v
(a) increase
(b) decrease
(c) stay the same
Physics 218 Fall 2010
Lecture 13
Solution

The speed at any point will be determined by
F
v
m/ L


at that point
The tension  in the rope near the top is greater than the
tension near the bottom since it has to support the
weight of the rope beneath it!
The speed of the wave will be greater at the top!
Physics 218 Fall 2010
Lecture 13
v
Periodic Waves
Quantity
Units
Velocity
distance/time
Period
time
Angular Frequency
1/time
Frequency
1/time
Wavelength
distance
Wavenumber
Physics 218 Fall 2010
1/distance
v   / k  f   / T
T  2 /   1 / f   / v
  2 /T  2f  kv
f   / 2  1 / T  v / 
  2 / k  v / f  vT
k  2 /    / v  2  f / v
Quiz
You send a traveling wave along a
particular string by oscillating one
end. If you increase the frequency of
oscillations
(A) The speed of the wave increases
(B) The wavelength decreases
(C) The tension in the string increases
(D) All of the above
Physics 218 Fall 2010