Ch11 Waves
Measures of a Wave
• Period (T): The shortest time interval
during which motion repeats.
1
2
3
4
Time (s)
5
6
7
8
Measures of a Wave
• Frequency (f): The number of complete
vibrations per second.
Number of vibration (cycles)
1
0.2
2
3
4
5
0.4
0.6
0.8
1.0
Time (s)
1.2
1.6
1.8
Waves
•
When these oscillations between two extremes are graphed wrt time, we
see the following profile emerge.
•
The Wavelength () is the distance from the “same” point on two
consecutive oscillations.
•
The Amplitude (A) is the maximum displacement from zero.
•
The Period (T) is the time between the same position on consecutive
“humps.”
•
The Frequency (f) describes how often an oscillation occurs.
•
The high points on the wave are known as “crests.”
•
The low points on the wave are known as “troughs.”
+A
0
-A
Wave Examples
• Can two waves can have the same
wavelength and frequency, but different
amplitudes?
The greater the amplitude
the greater the energy.
.01 .02 .03 .04
.05 .06 .07 .08
Wave Boundaries
• What happens when a wave hits a
boundary between two mediums?
– Part of the wave is transmitted
– Part is reflected
• The amount that gets transmitted versus
reflected depends on the difference
between two mediums.
Waves at Boundaries
Low Density Medium
High Density Medium
Note: Both amplitudes get smaller
Reflected pulse
Transmitted pulse
Waves at Boundaries
High Density Medium
Low Density Medium
Transmitted pulse
Reflected pulse
Wave Boundaries
• The frequency of a wave being transmitted
from one medium to another does NOT
change.
• e.g. If I’m moving a string up and down, I
don’t change the velocity that I vibrate it.
Wave Types
Mechanical Waves: require a medium
(material) to propagate.
Water
Springs
Rope
Sound
3 types of Mechanical Waves
– Transverse
– Longitudinal
Longitudinal Waves
• A Longitudinal: A wave in which the vibration is in the
same direction that the wave is traveling.
• Notice how the atom in the box below never leaves
the box even though the wave is obviously traveling to
the right.
Animation courtesy of Dr. Dan Russell, Kettering University
Transverse Mechanical Waves
• A transverse wave is one in which the individual atoms or
particles vibrate in a direction perpendicular to the direction of
motion of the wave.
• Notice how the atoms in the box below never leave the box
even though the wave is obviously traveling to the right.
Animation courtesy of Dr. Dan Russell, Kettering University
As the Wheel Turns
• Watch how the sine function traces out as a wheel turns.
• The vertical axis represents horizontal position and the
horizontal axis represents time.
• Note that one revolution (2π radians) is one sine wave cycle
Rotations
1.5
1
0.5
0
0
0.2
0.4
0.6
-0.5
-1
-1.5
t
0.8
1
1.2
Simple Harmonic Motion
Ts  2
m
k
Period of a swing pendulum
l
T  2
g
Simple Harmonic Motion
Simple Harmonic Motion: Motion caused by
a linear restoring force that has a period
independent of amplitude.
Period: The time required to repeat
one complete cycle
Amplitude: Maximum displacement
from equilibrium.
Position versus time
x  A sin(2 ft )
Derive v, in terms of x and A
KE  EPEx  EPEA
Simple Harmonic Motion
x  A sin(2 ft )
vmax  A
K
m
Derive v in terms of x and A
v  vmax
vf
x2
1 2
A
Doppler Effect
• Sound waves propagate out from the source in
all direction.
• If the source isn’t moving, the wavelengths are
constant
Police
Sonic boom from real player library
Doppler Effect
• Source moving towards you:
• Source moving away from you:
Police
fo  f s (
1
1
vs
fo  f s (
1
1
vs
)
v
)
v
Doppler Effect Example
A train is approaching you at
31m/s and blows its whistle of
305hz.
1
f o  f s ( vs )
1 s
a)What frequency do you
hear?
1
f o  305Hz ( 31m / s )
1  343m / s
fo  335Hz
Doppler Effect Example
A train is approaching you
at 31m/s and blows its
whistle of 305hz.
b) What frequency does
your friend hear if the train
has past him and continues
moving away?
fo  f s (
1
1
f o  305 Hz (
1  31m / s
)
343m / s
fo  279.7 Hz
1
vs
)
v
Doppler Effect Example
Towards
Away
1
f o  f s ( vs )
1 v
1
f o  f s ( vs )
1 v
fo  f s (1  )
fo  f s (1  )
vo
v
f s (v  vo )
fo 
(v  vs )
vo
v
Superposition of Waves
• Principle of superposition: The displacement of
a medium caused by two or more waves is the
algebraic sum of each wave.
• Waves pass each other so the original wave
continues unaltered.
• Interference is the result of the superposition of
two or more waves.
Superposition of Waves
• Constructive Interference: Occurs when
the displacements are in the same
direction
• Destructive Interference: Occurs when the
displacements are on opposite sides of
equilibrium.
• Show excel demo
Wave Superposition
Antinodes
Wave Sum
Graph A
Nodes
Constructive
Destructive Interference
Interference
Antinodes
•
•
•
•
Think First then ACT!
Lets think about what is happening in terms
of position, speed, and acceleration as the
particle moves in simple harmonic motion.
Which way does acceleration act at all times
during the motion of an object moving in
either circular or simple harmonic motion?
Center seeking acceleration.
At what point(s) on the graph below is the
blue dot moving at the fastest speed?
• Slowest speed?
A
B
• What is the point physically doing as it
approaches these two points?
C
Simple Harmonic and Circular Motion
• Simple harmonic motion may be thought of as the
projection of circular motion into a linear scale.
• Notice how the laser strikes the mass at all times
while the mass oscillates and while the disc
rotates
• This demonstrates that both the rotating disk and
the oscillating spring have the same period and
angular frequency.
Phasors – the Displacement Vector
• Any vector that rotates is known as a
Phasor.
• The displacement vector “A” that
always points to the yellow point in the
animation below is an example of a
phasor.
• The horizontal and vertical
components of the phasor A may be
determined using the following
equation.
x  A cos 
y  A sin 
A
Mathematical Description of a Wave
y  A sin 
• The equation for a sinusoidal
wave is as follows.
• Recall that the angular distance is
and can be expressed in terms of
frequency
• After substitution we get an
expression for the vertical position
of a particle y in terms of the
horizontal position x and time t.
  t
  2 f
y  Asin t 
y  Asin  2 ft 
t
t1
t2
The Phase Angle
• In order to better understand the physical meaning of the phase angle, we will look
at the graphs generated by the black and blue points on the edge of the circle below.
• Recall that the phase angle of the black point was f = 0 radians and the phase angle
of the blue point was f = /2.
• This phase angle can be thought of as the lag or lead angular distance of one point
when compared to another.
• The blue graph leads the black graph by /2 radians.
• Conversely, the black graph lags the blue graph by /2 radians.
• This fact is true at every point
Simple Harmonic Motion and Phase Shifts
on the graph.
f 0
f  /2
1.50
x  A cos t  f 
x  cos t 


x  cos  t     sin t 
2

Amplitude
1.00
0.50
0.00
-0.50
0
0.2
-1.00
-1.50
f
0.4

2
0.6
Time
0.8
Amplitude Variations
• Variations in amplitude only change the height of the waveform.
• In the graph below, the three waves are in phase with each other (f = 0),
and they have the same angular frequency.
• The amplitude of the black line is 1.
• What is the amplitude of the green line?
• What is the amplitude of the red line?
• The equations of the three lines are as follows.
Variations in Amplitude
2.50
2.00
x  cos t 
1
x  cos t 
2
1.00
Amplitude
x  2cos t 
1.50
0.50
0.00
-0.50 0
0.2
0.4
0.6
-1.00
-1.50
-2.00
-2.50
Time
0.8
Standing Waves
• Standing Wave: has stationary nodes and
antinodes. It is the results of identical
waves traveling in opposite direction.
• Node: The medium is not displaced as the
waves pass through
• Antinode: The displacement caused by
interfering waves is largest.