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Waves and
Harmonic
Motion
AP Physics
M. Blachly
Review: SHO Equation
Consider a SHO with a mass of 14 grams:
Positions are given in mm
y (t )  25cos(16 t )  10
SHO Equation
Find the following:
T
k
At t = 3, find
y
Etotal
a
Period of a Harmonic Oscillator
For a system with a mass m, and a linear
restoring force characterized by k, the period is:
m
T  2
k
Pendulum
For small angles, gravity
acts as a linear restoring
force.

 mg 
Fs  
 x
 L 
k  mg / L
x
Pendulum
L
T  2
g
Pendulum
A 500 g. mass is suspended from a 2 m long
string. It is then pulled to a position of x=4 cm
and released. Write the equation for the position
of this pendulum at any time.
Damping
The damped Harmonic Oscillator is a system
where energy is not conserved.
Applet: http://www.lon-capa.org/~mmp/applist/damped/d.htm
In the real world, there is always some damping,
from friction, air resistance, internal deformations
of the medium, etc.
Damped SHO Examples
Car Spring and
shocks
• Lots of
damping is
good
Radio receivers
• Very little
damping is
good
Wave Motion
Types of Waves:
• Transverse
• Longitudinal
• Torsional
Characteristics of Waves
• Frequency
• Period
• Amplitude
• Wavelength
• velocity
Wave Speed
v  f 
Wave Phenomena
Superposition
Interference
Beats
Types of Waves
Transverse: The medium oscillates
perpendicular to the direction the wave is
moving.
• Water (more or less)
• Slinky demo
Longitudinal: The medium oscillates in the
same direction as the wave is moving
• Sound
• Slinky demo
8
Slinky TnR
Suppose that a longitudinal wave moves along a Slinky
at a speed of 5 m/s. Does one coil of the slinky move
through a distance of five meters in one second?
1. Yes
2. No
5m
12
Harmonic Waves
y(x,t) = A cos(wt –kx)
Wavelength: The distance  between identical points on the wave.
Amplitude: The maximum displacement A of a point on the wave.
Angular Frequency w: w = 2  f
Wave Number k: k = 2  / 
Recall: f = v / 
y
Wavelength

Amplitude A
A
x
20
Period and Velocity


Period: The time T for a point on the wave to undergo one
complete oscillation.
Speed: The wave moves one wavelength  in one period T
so its speed is v = / T.
v

T
22
TnR
Suppose a periodic wave moves through some medium. If the
period of the wave is increased, what happens to the
wavelength of the wave assuming the speed of the wave
remains the same?
correct
1. The wavelength increases
2. The wavelength remains the same
3. The wavelength decreases
=vT
26
ACT

The wavelength of microwaves generated by a microwave oven
is about 3 cm. At what frequency do these waves cause the
water molecules in my burrito to vibrate ?
(a) 1 GHz
(b) 10 GHz
(c) 100 GHz
1 GHz = 109 cycles/sec
The speed of light is c = 3x108 m/s
29
Solution

Recall that v = f.
v 3  108 m s 10
f 
 10 Hz  10 GHz
 .03m
H
H
Makes water molecules rotate
O
1 GHz = 109 cycles/sec
The speed of light is c = 3x108 m/s
30
Water and Waves
Visible
Absorption
coefficient
of water as a
function
of frequency.
f = 10 GHz
“water
hole”
31
Interference and Superposition
When too waves overlap, the
amplitudes add.
• Constructive: increases
amplitude
• Destructive: decreases
amplitude
34
Reflection
A slinky is connected to a wall at
one end. A pulse travels to the
right, hits the wall and is reflected
back to the left. The reflected
wave is
A) Inverted
B) Upright
• Fixed boundary reflected
•
wave is inverted
Free boundary reflected wave
upright
37
Standing Waves Fixed Endpoints
Fundamental
n=1
n = 2L/n
fn = n v / (2L)
Wave Speed
F
v

Standing Waves:
L  / 2
f1 = fundamental frequency
(lowest possible)
A guitar’s E-string has a length of 65 cm and is stretched to a tension of 82N. If it
vibrates with a fundamental frequency of 329.63 Hz, what is the mass of the string?
v=f
= 2 (0.65 m) (329.63 s-1)
= 428.5 m/s
v2 = F / 
 = F / v2
m= F L / v2
= 82 (0.65) / (428.5)2
= 2.9 x 10-4 kg