Download Oscillations and Waves notes 2016-2017

Oscillations
Periodic Motion
•
•
Repeats itself over a fixed
and reproducible period of
time.
Mechanical devices that do
this are known as
oscillators.
Simple Harmonic Motion
(SHM)
•
•
Periodic motion which can be
described by a sine or cosine
function.
Springs and pendulums are
common examples of Simple
Harmonic Oscillators (SHOs).
 Springs;
as the spring compresses or
stretches, the spring force accelerates it
back toward its equilibrium position.
 Pendulums:
as the pendulum swings,
gravity accelerates it back toward
its equilibrium position.
 Uniform
circular motion: as an object
moves around a circle, its vertical
position (y-position) is continuously
oscillating between +r and –r.

Waves: waves passing through some
medium (such as water or air) cause
the medium to oscillate up and down,
like a duck sitting on the water as
waves pass by.
Equilibrium
•
•
The midpoint of the
oscillation of a simple
harmonic oscillator.
Position of minimum
potential energy and
maximum kinetic energy.
All oscillators obey…
Law of
Conservation of
Energy
Amplitude (A)
•
•
How far the wave is from
equilibrium at its maximum
displacement.
Waves with high amplitude
have more energy than waves
with low amplitude.
Period (T)
•
The length of time it takes for
one cycle of periodic motion to
complete itself.
Frequency (f):
•
•
•
•
How fast the oscillation is
occurring.
Frequency is inversely related to
period.
f = 1/T
The units of frequency is the
Hertz (Hz) where 1 Hz = 1 s-1.
Restoring force
 The
restoring force is the secret
behind simple harmonic motion.
 The force is always directed so
as to push or pull the system
back to its equilibrium (normal
rest) position.


What is the restoring force for an
oscillating spring?
What about a pendulum?
Hooke’s Law
A restoring force directly
proportional to displacement is
responsible for the motion of a
spring.
F = -kx
where
F: restoring force
k: force constant
x: displacement from equilibrium
Hooke’s Law
Fs = -kx
m
Fs
mg
The force constant of a
spring can be determined
by attaching a weight and
seeing how far it
stretches.
Hooke’s Law
Equilibrium position
F
m
x
F = -kx
Spring compressed,
restoring force out
Spring at equilibrium,
restoring force zero
m
F
x
m
Spring stretched,
restoring force in
Hooke’s Law
Equilibrium position
m
ν
a
F
Us
0 amax Fmax


vmax 0
0
x=A
m
K
0
0
x=0
m
x=A
m
x=0
0
amax Fmax


vmax 0
0
0
0
Mechanical Energy is
CONSERVED!!!
(in a closed frictionless system)
(i is any point in the motion)
Period of a spring
T
= 2m/k
– T:
period (s)
– m: mass (kg)
– k: force constant (N/m)
Period of a pendulum
T
= 2l/g
– T:
period (s)
– l: length of string (m)
– g: gravitational acceleration
(m/s2)
Waves
Mechanical Wave
A disturbance which
propagates through a
medium with little or no net
displacement of the
particles of the medium.
Wave types: transverse
A transverse wave is a wave in
which particles of the medium
move in a direction
perpendicular to the direction
which the wave moves.
Example: waves on a string
Parts of a Transverse Wave
T
3
A
2
Equilibrium point
-3
x(m)
4
6
t(s)
Parts of a Transverse Wave
: wavelength
3
equilibrium
2
-3
y(m)
crest
A: amplitude
4
trough
6
x(m)
Wave types: longitudinal
A longitudinal wave is a wave in which
particles of the medium move in a
direction parallel to the direction which
the wave moves.
Longitudinal waves are also called
compression waves.
Example: sound
Longitudinal Wave Anatomy
Longitudinal vs Transverse
Transverse vs Longitudinal
Speed of a wave
 The
speed of a wave is the
distance traveled by a given
point on the wave (such as a
crest) in a given interval of time.
v
= d/t
d: distance
– t: time
–
Speed of a wave
v
= ƒ
–v
: speed (m/s)
–  : wavelength (m)
–1
– ƒ : frequency (s , Hz)
And the answer
is?
J 3300 Hz
At 0°C sound travels through air at a speed
of 330 m/s. If a sound wave is produced
with a wavelength of 0.10 m, what is the
wave’s frequency?
F 0.0033 Hz
Use the formula chart!!!
G 33 Hz
Velocity
=
f
λ
OR
H 330 Hz
J 3300 Hz
330 m/s = f x 0.10 m
Period of a wave
T
= 1/ƒ
–T
: period (s)
– ƒ : frequency (s-1,
Hz)
Putting it all together to find position on
the wave
3X
T
Period = time to
oscillate once
A
2
-3
x(m) Position
Frequency = cycles
per second
4
6
t(s)
X = A cos (2πft)
MUST be in RADIANS (mode)
Reflection of waves
• Occurs when a wave strikes a
medium boundary and
“bounces back” into original
medium.
• Completely reflected waves
have the same energy and
speed as original wave.
Wave Interactions
Reflection- bounce off barriers in regular
ways
The angle the wave hits the barrier is the
angle it will leave the barrier
Reflection of waves

Fixed-end reflection: wave reflects
with inverted phase. This occurs
when reflecting medium has greater
density.
Reflection of waves

Open-end reflection: wave reflects
with same phase. This occurs when
reflecting medium has lesser density.
Reflection of pulses
Principle of Superposition
When two or more waves pass a
particular point in a medium
simultaneously, the resulting
displacement at that point in the medium
is the sum of the displacements due to
each individual wave.
 The waves interfere with each other.

Types of interference.


If the waves are “in phase”, that is crests
and troughs are aligned, the amplitude is
increased. This is called constructive
interference.
If the waves are “out of phase”, that is
crests and troughs are completely
misaligned, the amplitude is decreased and
can even be zero. This is called destructive
interference.
When two pulses
Destructive
Constructive
Interference travel towards one Interference
another, the waves
pass through each
other, exhibiting
constructive and/or
destructive
interference as they
pass. Then they
continue on, like
SHIPS PASSING IN
THE NIGHT.
Constructive
interference
– wave pulses
“in phase”
Destructive
interference
– wave pulses
“out of
phase”
Waves Traveling Together
(Road Trip!!!)


If the waves are “in phase”, that is crests
and troughs are aligned, the amplitude is
increased. This is called constructive
interference.
If the waves are “out of phase”, that is
crests and troughs are completely
misaligned, the amplitude is decreased and
can even be zero. This is called destructive
interference.
Constructive Interference
crests aligned with crest
waves are
“in phase”
Destructive Interference
crests aligned with troughs
waves are
“out of
phase”
Interference Patterns
When two progressive waves propagate into each other’s
space, the waves produce interference patterns. This
diagram shows how interference patterns form:
The resulting interference pattern looks like the following picture:
In this picture, the bright regions are wave peaks, and
the dark regions are troughs. The brightest
intersections are regions where the peaks interfere
constructively, and the darkest intersections are
regions where the troughs interfere constructively.
Sound is a longitudinal wave
Standing Wave
A standing wave is a wave which is
reflected back and forth between
fixed ends (of a string or pipe, for
example).
 Reflection may be fixed or openended.
 Superposition of the wave upon itself
results in constructive interference
and an enhanced wave.


In a standing wave pattern there are
points along the medium which appear
as if they are always standing still.
These points are known as nodes and
are easily remembered as the points
of no displacement. There is always an
antinode positioned between two
adjacent nodes. Antinodes are points
of maximum positive and negative
displacement.
Harmonics

In standing wave patterns, there is a
unique half-number relationship
between the length of the medium and
the wavelength of the waves
Open vs. Closed End Pipes
Open vs. Closed End Pipes
Doppler Effect
 The
Doppler Effect is the raising
or lowering of the perceived pitch
of a sound based on the relative
motion of the observer and the
source of the sound.
Doppler Effect
When an ambulance is racing toward
you, the sound of its siren appears to
be higher in pitch.
 When the ambulance is racing away
from you, the sound of its siren appears
to be lower in pitch.

Beats
The characteristic loud-soft pattern
that characterizes two nearly (but
not exactly) matched frequencies.
 Musicians call this “being out of tune”

Resonance
Occurs when a vibration
from one oscillator occurs
at a natural frequency for
another oscillator.
The first oscillator will
cause the second to
vibrate.
Resonance

Is the vibration of an object caused
by being struck by a wave at a certain
frequency
A final word about “pitch”
 Pitch
rises when frequency
rises and the sound is
“higher”.
 Pitch is lowered when the
frequency is lowered and the
sound is “lower”.
Wave Interactions
Refraction- waves can
change direction when
speed changes
Refraction of waves
• Transmission of wave from
one medium to another.
• Refracted waves may change
speed and wavelength.
• Refracted waves do not
change frequency.

When a wave is refracted through an
object, it will ALWAYS bend
TOWARDS the thicker part of the
object.
Diffraction
 The
bending of a wave around
a barrier.
 Diffraction combined with
interference of diffracted
waves causes “diffraction
patterns”.
Wave Interactions
 Diffraction
– waves bend
around obstacles
Double-slit or multi-slit
diffraction
n=2
n=1

n=0
n=1
n = dsin
Formula for strings or openended pipes
v
fn  n
2L
Fixed-end standing waves
(violin string)
L
Fundamental
First harmonic
 = 2L
First Overtone
Second harmonic
=L
Second Overtone
Third harmonic
 = 2L/3
Open-end standing waves
(organ pipes)
L
Fundamental
First harmonic
 = 2L
Second harmonic
=L
Third harmonic
 = 2L/3
Closed-ended pipes
(some organ pipes)
L
First harmonic
 = 4L
Second harmonic
 = (4/3)L
Third harmonic
 = (4/5)L
Formula for closed-ended pipes
v
fn  n
4L
Only can have ODD harmonics (1, 3, 5…)