Topic 4: Oscillations and waves Flashcards
Describe where KE and PE are of a mass moving between two horizontal springs
Kinetic energy: Moving mass
Potential energy: Elastic potential energy in the springs
Describe where KE and PE are of a mass moving on a vertical spring
Kinetic energy: Moving mass
Potential energy: Elastic potential energy in the springs and gravitational potential energy
Describe where KE and PE are of a simple pendulum
Kinetic energy: Moving pendulum bob
Potential energy: Gravitational potential energy of bob
Describe where KE and PE are of a buoy bouncing up and down in water
Kinetic energy: Moving buoy
Potential energy: Gravitational potential energy of buoy and water
Define: displacement (in terms of SHM).
The instantaneous distance of the moving object from its mean position in a specified direction.
Define: amplitude (in terms of SHM).
The maximum displacement from the mean position.
Define: frequency (in terms of SHM).
The number of oscillations completed per unit time.
Define: period (in terms of SHM)
The time taken for one complete oscillation.
Define: phase difference
A measure of how in phase different particles are.
When are particles said to be in phase?
If they are moving together; at 0; 360, …
What is the symbol for phase difference?
phi, ϕ
When are particles said to be completely out of phase?
at 180º or π rad
Define: simple harmonic motion
The motion that takes place when the acceleration, a, of an object is always directed towards, and is proportional to, its displacement from a fixed point. This acceleration is caused by a restoring force that must always be pointed towards the mean position and also proportional to the displacement from the mean position.
What is the defining equation for SHM?
a = -ω2 x
- ω2 is the gradient of the line
Describe the interchange between kinetic energy and potential energy during SHM
In SHM, the total energy is interchanged between kinetic energy and potential energy. If no resistive forces that dissipate energy act on the motion, the total energy is constant and the oscillation is said to be undamped.
Potential energy increases as we move away from the equilibrium position and kinetic energy decreases. As we come closer to the equilibrium position its vice versa. Potential energy can be expressed as a sine curve, kinetic energy as a cosine curve.
Define: damping
Involves a frictional force that is always in the opposite direction to the direction of motion of the oscillating particle. As the particle oscillates, it does work against this resistive (or dissipative) force and so the particle loses energy.
total energy of the particle ∝ (amplitude)<span>2</span>
This means that the amplitude decreases exponentially with time.
Define: light damping
The resistive force is small, so a small fraction of the total energy is removed with each cycle and hence the amplitude decreases. The time period of the oscillations is not affected and the oscillations continue for a significant number of cycles. The time taken for the oscillation to die out can be long.
Define: critical damping
Involves an intermediate value for a resistive force such that the time taken for the particle to return to zero displacement is a minimum. There is no overshoot.
Define: heavy damping
Involves large resistive forces and can completely prevent oscillations from taking place. The time taken for the particle to return to zero displacement can be long.
Define: natural frequency of vibration
When the system is temporarily displaced from its equilibrium position and the system oscillates as a result.
Define: forced oscillations.
It is possible to force a system to oscillate at any frequency by subjecting it to a changing force that varies with the chosen frequency. This periodic driving frequency must be provided from outside the system. When the driving frequency is first applied, a combination of natural and forced oscillations take place, producing complex transient oscillations. Once the amplitude of the transient oscillations dies down, a steady condition is achieved.
What are the conditions of forced oscillations?
- system oscillates at the driving frequency
- amplitude of the forced oscillations is fixed (each cycle, energy is dissipated as a result of damping and the driving force does not work on the system, the overall result is that the energy of the system remains constant)
- amplitude of forced oscillations depends on:
- comparative values of the natural frequency and the driving frequency
- amount of damping present in the system.
Define: resonance
Occurs when a system is subject to an oscillating force at exactly the same frequency as the natural frequency of oscillation of the system.
Describe resonance in vibrations in machinery
When in operation, the moving parts of machinery provide regular driving forces on the other sections of the machinery. If the driving frequency is equal to the natural frequency, the amplitude of a particular vibration may get dangerously high.
Define: wave pulse.
Involves one oscillation.
Define: continuous progressive (travelling wave)
Involves a succession of individual oscillations.
Define: transverse waves
Have a crest and a trough. The oscillations are at right angles to the direction of energy transfer. Cannot be propagated through fluids
Define: longitudinal waves
Oscillations are parallel to the direction of energy transfer
Describe four examples of transverse waves.
- Water ripples
- Light waves
- Waves along a stretched rope
Describe three examples of longitudinal waves.
- Soundwaves
- Compression waves down a spring
Define: wave front
Highlight parts of the waves that are moving together.
Define: ray
Highlight the direction of energy transfer.
Define: crest
The top of the wave.
Define: trough
The bottom of the wave
Define: compression
A high pressure point on the wave
Define: rarefaction
A low pressure point on the wave.
Define: displacement (waves)
Measures the change that has taken place as a result of a wave passing a particular point. Zero displacement refers to the mean position. For mechanical waves, the displacement is the distance that the particle moves from its undisturbed position.
Define: wavelength
The shortest distance along the wave between two points that are in phase.
Define: wave speed
Speed at which the wave fronts pass a stationary observer.
Define: intensity (waves)
Power per unit area that is received by the observer.
intensity ∝ amplitude2
Wavelength of microwaves
10-1 – 10-3
Wavelength of radio waves
105 – 10-1
What speed do electromagnetic waves travel at in a vacuum?
Same speed
What happens when a wave meets a boundary?
It is partially reflected and partially transmitted.
State Snell’s Law.
The ratio (sin i) / (sin r) is a constant for a given frequency.
Equation for refractive index.
Define: refractive index
A measure of the change in speed of the wave as it passes between two mediums.
Diagram for refraction.
What happens to a light ray when it moves from a less dense medium to a more dense one?
Wave speed is greatest in the less dense medium
Wavelength is greatest in the less dense medium
Frequency is the same
The reflected pulse becomes inverted when a wave in a less dense medium is heading towards a boundary with a more dense medium
The amplitude of the incident pulse is always greater than the amplitude of the reflected pulse.
Less dense -> more dense: light bends towards the normal
Diffraction diagram where slit is bigger than wavelength
Diffraction diagram where slit is about the same size as the wavelength
Define: superposition.
The overall disturbance at any point and at any time where the waves meet is the vector sum of the disturbances that would have been produced by each of the individual waves.
Explain constructive interference.
Explain destructive interference.
What is the path difference in constructive interference?
n λ
What is the phase difference in constructive interference?
Zero
What is the path difference in destructive interference?
(n + ½ ) λ
What is the phase difference in destructive interference?
180° / π radians
Apply the principle of superposition to determine the resultant of two waves
Add two waves.
In which direction is the restoring force acting in?
Always to the mean position/equilibrium point.
For a mass moving between two horizontal springs, what is the where is the kinetic energy and what is the potential energy store?
KE: the moving mass
PE: Elastic potential in the springs
For a mass moving on a vertical spring, what is the where is the kinetic energy and what is the potential energy store?
KE: the moving mass
PE: Elastic potential energy in the spring; gravitational potential energy
For a simple pendulum, what is the where is the kinetic energy and what is the potential energy store?
KE: the moving pendulum bob
PE: the gravitational potential energy of the bob
For a buoy bouncing up and down in water, what is the where is the kinetic energy and what is the potential energy store?
KE: the moving buoy
PE: the gravitational potential energy of the buoy and water
How do displacement, velocity and acceleration curves relate?
The velocity curve will be the derivative of the displacement curve i.e. the rate of change
The acceleration curve will be the derivative of the velocity curve - which happens to be the same function as the displacement curve just with the sign flipped.
What is the phase change between acceleration and velocity?
Acceleration is ahead of velocity by 90º, so therefore, should have a phase change of 90º or -270º
What is the phase change between velocity and displacement?
Velocity is ahead of displacement by 90º, so therefore, should have a phase change of 90º or -270º
What is the phase change between acceleration and displacement?
Acceleration and displacement have a phase difference of 180º so are completely out of phase.
If the total energy is constant in SHM, what does this indicate?
There is zero dampening i.e. no energy is lost.
What is energy in SHM proportional to?
- the mass, m
- the (amplitude)2
- the (frequency)2
What are mechanical and electromagnetic waves?
Mechanical waves are waves which require a material medium through which to travel.
Electromagnetic waves can travel without a medium, i.e. through a vacuum.
What are wavefronts?
A highlighted line of the part of a wave that are moving together (usually drawn at peaks).
What are rays?
Lines that highlight the direction of energy transfer.
What is the angle between rays and wavefronts?
It is always a right angle (90º)
Can transverse waves go through fluids?
No, they cannot propagate through fluids (liquids and gases).
At which frequencies and wavelengths might you find gamma rays?
Frequency: >1018 Hz
Wavelength: <10-10 m
At which frequencies and wavelengths might you find x-rays?
Frequency: >1017Hz
Wavelength: <10-9m
At which frequencies and wavelengths might you find UV waves?
Frequency: 1015< f < 1017Hz
Wavelength: 10-9-7m
At which frequencies and wavelengths might you find visible light?
Frequency: ≈ 1015Hz
Wavelength: ≈ 10-6m
At which frequencies and wavelengths might you find IR waves?
Frequencies: 1012< f <1015Hz
Wavelength: 10-6-3m
At which frequencies and wavelengths might you find microwaves?
Frequencies: 109< f <1012Hz
Wavelength: 10-3
At which frequencies and wavelengths might you find radio waves?
Frequencies: <109Hz
Wavelength: >1 m
What are electromagnetic waves constructed of?
Oscillating electric and magnetic fields at right angles to each other.
What is intensity proportional to?
Intensity ∝ (amplitude)2
What is the inverse square law for intensity?
Intensity ∝ 1/(distance from source)2
How do you find the intensity of a sound wave in a spherical space?
By using the surface area of a sphere, we can derive the equation:
Intensity = Power/(4πr2)
How can rays be used to show the intensity inverse square law?
As rays should the path taken by the wave energy, for a given distance of x, the ray lines will be closer together, therefore more power per area, so a higher intensity.
As you move out, the rays are further and further apart, representing a fall in intensity.
Why do unpolarized waves exist?
There is an infinite number of ways for the fields to be orientated, therefore are usually unpolarized.
Define: Unpolarized light
Light with a plane of vibration that varies randomly
Define: plane-polarized light
Light that has a fixed place of vibration
Define: partially-polarized light
Light that is a mixture of both polarized and unpolarized light.
How can light be polarized?
- Reflection
- Refraction
- Using a polaroid
Define: circularly polarized light
Light which has a plane of vibration that rotates uniformly.
What does a polarizer do?
A polarizer is any device that produces plan-polarized light.
What does an analyser do?
A polarizer used to detect polarized light.
What is a polaroid?
A material which preferentially absorbs any light in one particular plane of polarization allowing transmission only in the plane at 90º to this.
i.e. allows all light that travels in parallel to the lines, block all light which is perpendicular to it.
When a wave reflects off a surface what is it always?
The reflected wave is always partially-polarized.
What does Brewster’s law state?
That for light incident on a boundary of two media, the angle of incidence for when the reflected ray and refracted ray are perpendicular (at 90º), the angle of incidence is the polarizing angle and the reflected ray is totally plane-polarized.
How is the refractive index related to the incident angle?
- n* = sinθi / sinθr
- n* = sinθi / cosθ<span>i</span>
- n* = tanθi
What does Malus’ law state?
When plane-polarized light is incident on an analyser, the component of the electric field transmitted to the analyser is the wave electric field multiplied by the cosine between the wave’s plane of vibration and the lines on the analyser.
As intensity is amplitude2, the intensity received is the incident intensity multiplied by cos2θ.
I = I0 cos2θ
What is an optically active substance?
A substance that rotates the plane of polarization of light that passes through it. An example is certain sugar solutions.
How can polarization be used?
- Polaroid sunglasses and photography
- Calculating the concentrations of solutions
- Stress analysis
- LCDs
What waves can be polarized?
Transverse only.
Longitudinal waves, i.e. sound, cannot.
What is the law of reflection?
The incident angle = the reflected angle
θi = θr
What is diffuse reflection and when does it occur?
Diffuse reflection is when the light is reflected in all different directions. It happens when the reflective surface is disturbed or uneven.
What is the absolute refractive index?
It is the refractive index defined in terms of electromagnetic wave speeds.
n = (EM wave speed in vacuum)/(EM wave speed in medium)
How can you find the ratio between refractive indices and wave speed?
sinθ2/sinθ1 = n1/n2 = V2/V1
When can total internal reflection occur?
When the initial medium is more optically dense than another.
What the critical angle?
It is the incident angle on a boundary when the refracted angle is 90º.
When does total internal reflection occur?
When the incident angle is greater than the critical angle and the rays reflect back inside.
How can you calculate the critical angle?
n12. The light starts in n2
n1 sinθr = n2 sinθi
As θr is 90º, sinθr=1:
n1 = n2 sinθi
therefore sinθi = n1/n2
so the critical angle is θi = sin-1(n1/n2)
What uses are there to knowing critical angles or to using total internal reflection?
- What fish see underwater (when trying to catch flies)
- periscopes and binoculars
- optical fibre cable
What can diffraction be used for?
- CDs and DVDs
- Electron microscopes
- Radio telescopes
For a full standing wave, what is the wavelength with respect of string length and harmonic number?
λ = 2L/n
What is a standing wave?
A wave where energy is not transferred, it is stored.
What is a node?
A point on the standing wave where there is 0 displacement as a result of destructive interference. A point of 180º phase difference of the first and reflected wave.
What is an antinode?
A point of maximum displacement caused by constructive displacement. The first and reflected wave are in phase.
What is another name for the first harmonic?
Fundamental frequency or natural frequency.
For full standing waves, how might you find out what harmonic it is?
The harmonic number is the number of “humps” of the wave i.e. the number of anti-nodes.
For a closed-end pipe, what does the first harmonic look like?
What is the length of the pipe as a fraction of the wavelength?
1 node at the close end, 1 antinode at the open end.
The length of the pipe would be 1/4λ.
For close-ended pipes with standing waves, what is the length of the pipe as a fraction of the wavelength for all harmonics?
L = nλ/4
where n is the harmonic number (n cannot be even for close end pipes)
For open-ended pipes with standing waves, what is the length of the pipe as a fraction of the wavelength for all harmonics?
L = nλ/2
where n is the harmonic number (n can be any integer for open-end pipes)
For a closed-end pipe, what does the second harmonic look like?
What is the length of the pipe as a fraction of the wavelength?
Trick question, you cannot have even harmonics for a close-ended pipe.
For a closed-end pipe, what does the third harmonic look like?
What is the length of the pipe as a fraction of the wavelength?
2 nodes, 2 anti-nodes
For close-ended pipes, how are the number of nodes and the number of anti-nodes related?
They are equal.
For open-ended pipes, how are the number of nodes and the number of anti-nodes related?
The number of nodes is the harmonic number, the number of anti-nodes is 1 more.
How can you decide what to draw for standing waves in close-ended pipes?
For a given number, the number of nodes+antinodes is one more than the harmonic number. The number of nodes and the number of antinodes is the same.
So for the fifth harmonic, there will be 6 points. 6/2 = 3, so there must be 3 nodes and 3 antinodes.
How can you decide what to draw for standing waves in open-ended pipes?
The number of nodes is the harmonic number and there is always two anti-nodes on either side of a node.
For an open-end pipe, what does the first harmonic look like?
What is the length of the pipe as a fraction of the wavelength?
One node in the centre, two anti-nodes on either side (at the openings)
L = λ/2
For an open-end pipe, what does the second harmonic look like?
What is the length of the pipe as a fraction of the wavelength?
Two nodes in the pipe, with three anti-nodes equally spaced out.
L = 1λ
For an open-end pipe, what does the third harmonic look like?
What is the length of the pipe as a fraction of the wavelength?
3 nodes in the pipe, with 4 anti-nodes spaced equally.
L = 3λ/2
How does the frequency of n harmonics relate to that of the first harmonic?
It the fundamental frequency multiplied by the harmonic number.
