Oscillations and Waves
What is a wave?
How do the particles move?
Some definitions…
1) Amplitude – this is
“how high” the wave is:
Define the terms
displacement, amplitude,
frequency and period.
2) Wavelength () – this is the
distance between two
corresponding points on the
wave and is measured in metres:
3) Frequency – this is how many waves pass by
every second and is measured in Hertz (Hz)
Describing waves
Displacement
1 cycle is described by 2π
radians of phase
T
Hyperlink
T
Time
2π
ω = 2π/T
f = 1/T
ω = Angular frequency
Hyperlink and
scroll down
ω = 2πf
What are radians?
f = Frequency
T = Time period
Phase and angle
Examples of phase difference
Oscillations
http://www.acoustics.salford.ac.uk/feschools/waves/shm.htm#motion
Simple harmonic motion
Any motion that repeats itself after a certain period is known as a
periodic motion, and since such a motion can be represented in terms
of sines and cosines it is called a harmonic motion.
The following are examples of simple harmonic motion:
a test-tube bobbing up and down in water (Figure 1)
a simple pendulum
a compound pendulum
a vibrating spring
atoms vibrating in a crystal lattice
a vibrating cantilever
a trolley fixed between two springs
a marble on a concave surface
a torsional pendulum
liquid oscillating in a U-tube
a small magnet suspended over a horseshoe magnet
an inertia balance
Data loggers
• Use the data loggers to find the
variation of displacement with
time for an oscillating mass on a
spring.
• Process this data to find velocity
and acceleration with time.
• Now use the data to obtain a
graph of Acceleration versus
displacement.
Analysing your graphs
•
•
•
•
•
From the graph, find the
Time period
Angular frequency
Amplitude
The value of the velocity at maximum
displacement
• The value of the acceleration at zero
displacement
• The relationship between displacement and
acceleration.
SHM definition
• Find the gradient of your graph of
acceleration and displacement.
• Use this to calculate ω
• Calculate T from the graph.
• How the graph fit in with the definition of
SHM?
SHM
Free body diagram for SHM
• Draw the free body diagram for the mass
when it is
• in the centre of the motion
• At the top of the motion
• Between the bottom and the middle, down
• Between the bottom and the middle,
heading upwards.
Hyperlink
http://www.acoustics.salford.ac.uk/feschools/waves/shm2.htm
Spring pendulum
Spring Pendulum
Hyperlink
Oscillations to wave motion
Restoring forces
Simple
Harmonic
Motion
Consider a pendulum bob:
Let’s draw a graph of displacement against time:
Equilibrium position
Displacement
“Sinusoidal”
Time
Pendulum
Simple Pendulum
Hyperlink
Displacement
SHM Graphs
Time
Velocity
T
Time
Acceleration
Time
Definition of SHM
Acceleration
Displacement
Now write your OWN definition of SHM
The Maths of SHM
Displacement
Time
Therefore we can describe the motion mathematically as:
x = x0cosωt
v = -x0ωsinωt
a = -x0ω2cosωt
a = -ω2x
Students are expected to understand the
significance of the negative sign in the equation
and to recall the connection between ω and T.
ω = 2π/T
a
5
SHM 1)questions
Calculate the gradient of this
graph
x
2
2) Use it to work out the value
of ω
3) Use this to work out the time
period for the oscillations
4) Ewan sets up a pendulum and lets it swing 10
times. He records a time of 20 seconds for
the 10 oscillations. Calculate the period and
the angular speed ω.
5) The maximum displacement of the pendulum
is 3cm. Sketch a graph of a against x and
indicate the maximum acceleration.
a
x
Questions
•
•
•
•
•
•
•
Q’s 1 – 9 from the worksheet
Using the equations
V = V0cosωt
V = V0sinωt
x = x0cosωt
x = x0sinωt
V = ± ω√(x02-x2)
When do you use cos or sin?
4.2 Energy changes during
simple harmonic motion (SHM)
At which points are
-max displacement?
-max velocity?
-max acceleration?
- max Ek
Total energy
-max Ep
Energy
-max total energy?
-x0
x0
Displacement (x)
SHM: Energy change
Equilibrium position
Energy
GPE
K.E.
Time
Energy formulae
Ek = ½ mω2(x02 – x2)
Ep = ½ mω2x2
Etotal = ½ mω2x02
Total energy
Energy
-x0
x0
Displacement (x)
Questions
Answers
4.3 Forced oscillations and
resonance
4.3.1 State what is meant by damping.
“It is sufficient for students to know that damping
involves a force that is always in the opposite
direction to the direction of motion of the
oscillating particle and that the force is a
dissipative force.”
Free and Forced oscillations
Forcing frequency too slow
Forcing frequency too fast
Forcing frequency equals natural
frequency
Resonance
Resonance and frequency
Hyperlink
Physics Applets
Resonance and frequency
The width of the curve (Q value) is determined by the damping in the
system. The value of the resonant frequency depends factors such as
the size of the object…..
Tacoma Narrows
Useful resonance
• Musical instruments
• Microwave ovens
• Electrical resonance when tuning a radio
Damping
Damped oscillations
Amplitude of
driven system
Damping
Low damping
High damping
Driver frequency
Damping
How much damping is best?
Critical damping
Wave characteristics
The wave
pulse
transfers
energy
If the source continues to oscillate, then a
continuous progressive wave is produced.
Students should be
able to distinguish
between oscillations
and wave motion, and
appreciate that in
many examples, the
oscillations of the
particles are simple
harmonic.
Travelling Waves
Definition: A travelling wave (or “progressive wave”)
is one which travels out from the source that made
it and transfers energy from one point to another.
Energy dissipation
Clearly, a wave will get weaker the further it travels.
Assuming the wave comes from a point source and travels out
equally in all directions we can say:
Energy flux = Power (in W)
(in Wm-2)
Area (in m2)
φ=
P
4πr2
An “inverse square law”
Example questions
1) Darryl likes doing his homework. His work is 2m from a
100W light bulb. Calculate the energy flux arriving at his
book.
2) If his book has a surface area of 0.1m2 calculate the total
amount of energy on it per second (what assumption did you
make?).
3) Matti doesn’t like the dark. He switches on a light and
stands 3m away from it. If he is receiving a flux of
2.2Wm-2 what was the power of the bulb?
4) Matti walks 3m further away. What affect does this have
on the amount of flux on him?
State that progressive (travelling)
waves transfer energy.
Students should understand that there is no net motion
of the medium through which the wave travels.
Transverse
waves are when
the displacement
is at right angles
to the direction
of the wave…
Displacement
Transverse vs. longitudinal waves
Displacement
Direction
Direction
Longitudinal
waves are when
the displacement
is parallel to the
direction of the
wave…
Transverse wave
Transverse waves
Students should describe the waves
in terms of the direction of
oscillation of particles in the wave
relative to the direction of transfer
of energy by the wave. Students
should know that light waves and
water waves are transverse and that
water waves cannot be propagated
in gases or liquids.
Longitudinal waves
Sound waves and earthquake P-waves are
longitudinal
Longitudinal slinky
Loudspeaker
Describe waves in two dimensions,
including the concepts of wavefronts
and of rays.
Watch the wavefront(s) propagate
Energy is transferred in 2 dimensions
Wavefronts and rays
.
Wavefronts and rays
rays
Wavefronts
Rays show the direction of travel of the
energy. The wavefronts are where the
crests of the waves are. The rays are
always at 90 deg to the wavefronts.
Ray point of view
Wave point of view
Light travels out in
straight lines from a
small source
Light spreads out in
spherical wavefronts
from a small source
Light in a parallel beam or
from a very distant source
has rays (approximately)
parallel to one another
Wavefronts in a parallel
beam or from a very distant
source are straight (not
curved) and parallel
Ray and wave points of view show the same thing but
in different ways
Longitudinal waves
Compressions and rarefactions
Transverse waves
Crests
Troughs
Displacement graphs
Define the terms displacement,
amplitude, frequency, period,
wavelength, wave speed and intensity
WAVELENGTH
- the distance from one crest to another or one trough to
another. (In fact generally from any point on the wave to
the next exactly similar point i.e. 2 consecutive points in
phase)
FREQUENCY
- the number of vibrations of any part of the wave per
second. The bigger the frequency the higher the pitch of
the note or the bluer the light
AMPLITUDE
- the maximum distance that any point on the wave moves
from its mean position. The bigger the amplitude the
louder the sound, the rougher the sea, or the brighter
the light
Period (T)
The time it takes for one complete cycle of the wave.
Displacement (x)
How far the “particle” has travelled from its mean position.
Wave speed (v)
The speed at which the wavefronts pass a stationary observer
Intensity (I)
The power per unit area that is received by an observer.
Students should know that intensity α amplitude2
Derive and apply the relationship between
wave speed, wavelength and frequency.
Speed = Dist/time
For 1 cycle of the wave, dist = λ and time =T
Speed = λ/T
Therefore
f = 1/T
V=f λ
x
The Wave Equation
The wave equation relates the speed of the wave to its
frequency and wavelength:
Wave speed (v) = frequency (f) x wavelength ()
in m/s
in Hz
in m
V
f

Some example wave equation
questions
1) A water wave has a frequency of 2Hz and a wavelength
of 0.3m. How fast is it moving?
0.6m/s
2) A water wave travels through a pond with a speed of
1m/s and a frequency of 5Hz. What is the wavelength
of the waves?
0.2m
3) The speed of sound is 330m/s (in air). When Dave
hears this sound his ear vibrates 660 times a second.
What was the wavelength of the sound?
0.5m
4) Purple light has a wavelength of around 6x10-7m and a
frequency of 5x1014Hz. What is the speed of purple
light?
3x108m/s
Electromagnetic waves
Click to play
4.5 Wave properties
Wave diagrams
1) Reflection
2) Refraction
3) Refraction
4) Diffraction
• Describe the reflection and transmission of
waves at a boundary between two media.
• This should include the sketching of
incident, reflected and transmitted waves.
The amount of
transmission and
reflection
depends upon the
difference in the
“density” of the
2 media. i.e the
bigger the
difference, the
greater the
amount of
reflection.
Refraction through a glass block:
Wave slows down and bends
towards the normal due to
entering a more dense medium
Wave slows down but is
not bent, due to entering
along the normal
Wave speeds up and bends
away from the normal due to
entering a less dense medium
Refraction of Light applet
Hyperlink
Finding the Critical Angle…
1) Ray gets refracted
3) Ray still gets refracted (just!)
THE CRITICAL
ANGLE
2) Ray still gets refracted
4) Ray gets
internally reflected
Optical fibres
Uses of Total Internal Reflection
Optical fibres:
An optical fibre is a long, thin, _______ rod made of
glass or plastic. Light is _______ reflected from one
end to the other, making it possible to send ____
chunks of information
Optical fibres can be used for _________ by sending
electrical signals through the cable. The main advantage
of this is a reduced ______ loss.
Words – communications, internally, large, transparent, signal
Other uses of total internal reflection
1) Endoscopes (a medical device
used to see inside the body):
2) Binoculars and periscopes (using “reflecting prisms”)
Huygen’s principle
1. Velocity decreases
2. Wavelength
decreases
3. Frequency same
Snell’s law
Questions
• 10,11,12
• Practice Q 2
•
•
•
•
nair = 1.00
nwater = 1.33
ndiamond = 2.42
nglass= 1.50
Diffraction
More diffraction if the size of the gap is similar to the wavelength
More diffraction if wavelength is increased (or frequency decreased)
Diffraction
Diffraction at a single aperture
Single slit
distant screen
Hyperlink
intensity
across
screen
Sound can also be diffracted…
The explosion can’t be seen over the hill, but it can be
heard. We know sound travels as waves because sound
can be refracted, reflected (echo) and diffracted.
Diffraction depends on frequency…
A high frequency (short wavelength)
wave doesn’t get diffracted much – the
house won’t be able to receive it…
Diffraction depends on frequency…
A low frequency (long wavelength) wave
will get diffracted more, so the house
can receive it…
i) Diffraction by a
"large" object
ii) Diffraction at a
"large" aperture
iii) Diffraction by a
"small" object
iv) Diffraction by a
"narrow" aperture
Superposition
Superposition is seen when two waves of the same type cross.
It is defined as “the vector sum of the two displacements of
each wave”:
Superposition
Interference of 2 pulses
Click to play
Constructive interference i.e. Loud
or bright. Waves are in phase
Destructive interference i.e.
dark or quiet. Waves are π
rads out of phase.
Interference of sound waves
Where are the positions of constructive and destructive
interference?
Interference of 2 point sources
Click to play
Hyperlink
Superposition patterns
Consider two point sources (e.g. two dippers or a barrier with
two holes):
Superposition of Sound Waves
Path Difference
Destructive
Constructive
interference
interference
1st Max
Min
Max
Min
1st Max
2nd Max