4.1 Kinematics of Simple Harmonic MotionEdit
Describe examples of oscillations.
Examples of oscillations include:
- The motion of a mass at the end of a spring after the mass has been displaced away from its equilibrium position;
- The motion of an aeroplane wing;
- The motion of a tight guitar string that has been set in motion by plucking it.
Define the terms displacement, amplitude, frequency, period, and phase difference.
Displacement: The distance of the oscillating object from equilibrium.
Amplitude: The maximum distance from equilibrium an oscillating particle reaches.
Frequency: The number of complete cycles of an oscillating particle per unit time.
Period: The amount of time it takes an oscillating particle to complete an oscillation.
Phase Difference: The measure of how "in step" different particles are. If they are moving together they are said to be in phase. If not they are said to be out of phase.
Students must understand the equation: , where f is frequency and T is period.
Medium : The substance through which the wave moves. The particles making up the medium are those which are moved (or rather displaced) as the wave moves through it.
Displacement : In the context of waves, this refers to the movement of particles above and below (or whatever) the mean position. Over a period of time, the average position of a particle in the medium will be the same as if there were no wave traveling through it.
Amplitude : The difference between the maximum displacement and the mean position, i.e. how 'big' the waves are.
Period : The amount of time for one complete cycle (ie from one peak to the next, or one compression to the next. Note, this is not the same thing as to the next point where a particle is in the same position. The symbol for period is T.
Frequency : The number of complete cycles passing a given point in 1 second (measured in Hertz, Hz).
where f is frequency and T is period.
Wavelength : The distance covered in by complete wave cycle (i.e. from one crest to the next).
where λ is wavelength, v is the wave speed, and f is the frequency. The equation is the equivalent of that given in the data packet, which is:
Wave speed : The speed at which a given point on the wave is traveling through the medium (i.e. how far a particular crest travels in a second).
Crest : Relevant only for transverse waves, this is the point of highest positive displacement (i.e. upwards) from the mean position.
Trough : The point of largest negative displacement (i.e. downwards) from the mean position (in a transverse wave).
Compression : If a compression (longitudinal) wave is drawn like this || | | | | || | | | | || | | | | || , the compressions are where the bars are close together. Specifically, it is where the particles are most compressed in the wave.
Rarefaction : The opposite of a compression, i.e. where the bars (or particles) are most spread out.
Different graphs of waves.
Displacement vs time : This graph tracks the movement of a particle as a wave moves through it. With displacement on the vertical axis, and time on the horizontal. The particle will move up and down in a sine curve type pattern. This graph allows us to find both frequency (which will be the number of crests in 1 sec) and period (which will be the time between crests), but tells us nothing about the wave speed or wavelength.
Displacement vs position : This is basically a 'snapshot' of the displacement of all the particles going through the medium at a given time. Displacement is on the vertical axis, and position (or ie distance from an arbitrary origin in the material) is on the x. The distance between peaks represents the wavelength. The wave speed can not be calculated directly from this graph, but can be found by combining the information from this and the displacement vs time graph (as described in the next section).
v = f × λ (wave speed = frequency × wavelength)
This equation can be used to find the speed of a wave given it's wavelength and frequency. Deriving this is really rather obvious, but described below.
If the unit of frequency is cycles/second and wavelength is meters/cycle, then when the two are multiplied, cycles cancel out, and we're left with meters/second, which is the unit of wave speed, and so the equation follows from the definitions of frequency and wavelength.
Note, the frequency for a given wave is constant (defined by the source) thus, if the wave speed changes (due to changing mediums) then the wavelength also changes, but frequency remains constant.
Electromagnetic waves are transverse waves, travelling at a wave speed of c (the speed of light = 3 × 108 m/s) when in a vacuum (they can travel without a medium unlike all other waves).
There are a number of sections of the spectrum which are commonly given the following names (in order of decreasing frequency and increasing wavelength).
- ultraviolet rays
- visible light
- infrared rays
- radio waves
The easy way to remember this that our physics teacher taught us was "Red Monkeys In Vegas Usually X-ray Girls." This goes from lowest frequency to highest, opposite of the list above (sorry).
Going down that list, frequency decreases and wavelength increases (because c is constant). The amount of 'energy' in the waves decreases down the list, which is why X-rays are dangerous, and radio waves aren't.
Visible light is split into colours from violet to red, violet having the highest frequency and red having the lowest. Visible light ranges from 400nm (1 nano meter is 1 × 10-9 meters) for violet to 700nm for red.
Electromagnetic waves are usually defined by their wavelength in a vacuum, which seems rather silly, since frequency never changes, and is what defines the characteristics (i.e. color), but who am I to argue.
However weird it may sound, a Microwave oven at Earth does not emit the same wavelength of wave as one in space. Albeit a small one, there's still a difference there. This is because light travels more slowly when it travels through a medium such as air; all electromagnetic radiation is slowed to some extent by the medium it is passing through. That small difference may correspond to millions of light years in determining the distance of stars, so it is, in fact, very important to refer to vacuum values all the time just for setting a common ground for experiments.
Vacuum is chosen as the common reference point because all electromagnetic radiation, no matter what the frequency, travels at the same speed in vacuum. As mentioned, the speed of light is slower when it's traveling through something, and higher-energy radiation is slowed down less. It is only in vacuum that it all travels at the same speed no matter what the frequency or energy is.
4.2 The behavior of wavesEdit
During SHM, the pendulum swings from left to right and back the same way. When the pendulum reaches it's highest point or highest amplitude, it is momentarily at rest and has zero kinetic energy. The kinetic energy it previously possessed when it was moving has been converted to potential energy at this high point. When he swings back down, the potential energy is changed back to kinetic energy and this is highest at the equilibrium point. At this point, though, there is zero potential energy. This model also follows the law of conservation of energy.
Longitudinal waves travel in one dimension, and so when they strike a boundary, they will be reflected back in the same direction, though the will experience a phase change (i.e. when a compression hits the boundary, a rarefaction is emitted back from it, and vice versa.
This also applies to standing waves travelling in a stretched string. If both ends are connected to a boundary, then nodes (points where the string doesn't move up and down) will occur at both ends, and a number of antinodes will occur through the string, separated by nodes.
In an air column, it is possible to have both open and closed boundaries. At an open boundary, and antinode will occur, while at a closed one and node will occur.
Whenever a wave is reflected from a boundary, the angle of reflection will equal the angle of incidence. Thus, if the wave strikes the boundary at 90°, then it will be reflected straight back, but other angles will reflect the waves away from the source.
Note that it is common for waves to travel in a full, or semicircle out from the source rather than in one line, which complicates reflection, because each wave is entering at a different angle.
Also, curved boundary's must be handled. The basic technique here is to draw in a few important lines representing different waves, see where they would reflect to, and then fill in the rest.
When waves in water strike a boundary, the crests will be reflected as troughs, the same goes for sound. Phase changes in light are a little more complex, but we'll come to those later.
Waves can be refracted when they move from one medium to another, and when they have different wave speeds in these two media. It is easiest to consider this as a series of wavefront lines entering a boundary at an angle. The frequency (the time between lines) must remain constant, but the speed slows (so they must become closer together). As they enter at an angle, the wavefront on one side slows down first, which effectively pulls the entire wave around towards that corner. In ray diagrams, light simply enters at one angle to the normal, and leaves the boundary at another (we get to how to find these angles next).
This phenomena can have some weird effects. When looking into a swimming pool, light from objects at the bottom is diverged (refracted away from the normal). this means that when the virtual rays are traced back, a virtual image is formed much closer to the surface than the actual object (apparent vs real depth).
The angles described above can be found with Snell's law.
n1 × sin i = n2 × sin R
n1 is the refractive index of the initial medium (vacuum = 1, air = 1 (or close enough to 1 that it doesn't mater)), and n2 is the index of the medium it's entering. i is the angle of incidence, and R the angle of Refraction, both of which are measured from the normal.
When light goes from a more to a less dense medium, then there comes a point where the angle of refraction will be 90°. The angle of incidence where this occurs is called the critical angle. If the angle of incidence is above the critical angle, then the light is totally internally reflected. Angle of incidence = angle of reflection applies, and the light is reflected from the boundary.
Unusual examples of this include water ripples travelling slower in shallow water, and sound travelling at different speeds through hot and cold air. Most of the time, though, the problems relate to light entering / leaving water or a glass prism.
When two waves are moving in the same medium, the displacements of the particles add together. It is therefore possible for two waves to produce one wave of larger amplitude, or to produce two waves where the total amplitude is zero. Note, the waves and energies are still there, it's just that the two waves are adding to zero.
Questions about this generally involve two waves traveling in opposite directions down a string (they're usually rather easy).
Constructive interference is what occurs when two waves add together, while destructive interference is what happens when two waves add to zero.
If, for example, we have two point sources producing waves in a circle, they will interfere differently at different points. The easiest way to see this is to draw circles out from the source representing the crests. When two of these crests coincide, constructive interference produces a bigger crest. When two gaps coincide, we get a bigger trough, and when one crest and one trough coincide, there is destructive interference, and they add to zero. The same thing can be applied to waves in strings as above.
Young's double slit experiment is basically where two slits act as point sources, and form a diffraction pattern, thus demonstrating the wave nature of light. When light is shone onto the backs of the slits, they act as point sources which are initially exactly in phase. Light from each of these travels to a screen, striking it and producing light on it.
The issue is, however, that the light from each slit has to travel a different distance to reach the screen. When the difference between these distance is exactly N × wavelength + 1/2 wavelength of the light, the two waves will destructively interfere and produce a dark spot on the screen. When the difference is a multiple of of the wavelength, the two waves arrive in phase, and produce a bright spot.
The resulting pattern is a series of bright and dark bands when monochromatic light is used. When white light is used, different colors will construct and destruct at different points, producing a series of spectra which will eventually overlap as we move away from the centre. In both cases, the center point will have a bright spot, of the original light colour.
The equation for this experiment is given in the optics section of the data book. m × λ = a sin Θ, or the order of the band × wavelength = the distance between the centres of the slits × sin of the angle of the bright band.
Since there are a series of bright bands, different values of m can be substituted. m=0 gives the central bright band, then m=1,2,3,4...give the subsequent band angles.
When two waves which have different frequencies interfere, beats will be heard. The beats are points where the amplitude (volume in the usual case of sound) reaches a peak. The frequency of the beats can be calculated by fbeats = |f1-f2|. Meaning the beat frequency will be the difference between the two frequencies.
This can be seen by drawing two sine curves, say sin x and sin 2x, then adding them. Both high and low points will be found, showing the beats (any graphing calculator will show you).
Diffraction of waves.
Water : When there is something blocking waves in the water, say, a log floating in it, immediately behind the log will be calm water, but eventually the waves wrap around it. This is due to diffraction, meaning as the waves go through, the motion of the particles affect those, not just in the direction of propagation, but also to the side, allowing the wave to spread to the side as well as forward. If waves were passed through a thin slit, they would form a semi circle point source, just like light in the double slit thing.
Sound : Just like water, sound can trace around obstacles, and join back up on the other side, or pass through a thin slit and form a sort of point source. This also shows that both longitudinal and transverse waves act in the same way with respect to diffraction.
Light : As was seen in the double slit experiment, light can be diffracted through a thin slit. Because it moves so fast, we tend not to notice light waves bending around corners, but it can.
Polarisation : Light naturally travels as a transverse wave in all planes, i.e. the 'particles' move left and right, up and down, and at every angle between. It is possible to cut out all but, say, the the up and down motion. Light to which this has been done is called polarised.
(More info is in the Optics section if you think you need it.)
4.3 Standing wavesEdit
Standing waves occur when a source sets up a continual wave, which interacts with waves being reflected back at the source from a boundary to form a stable pattern of nodes and antinodes (nodes being points where the two waves add to zero and anti-nodes being where the waves always add to maximum amplitude.)
At a closed boundary, a node will always be formed, and a antinode will occur at an open boundary. This can occur in a stretched string, or in an air column. In each case, the fundamental frequency is the wave with the longest wavelength which satisfies this (i.e. the lowest frequency). After this, harmonic frequencies can be found by adding half a cycle into the string/air column. With two closed boundaries, the fundamental wavelength will be 2 × the length, with one open, one closed, 4 × wavelength, and with 2 open, 2 × length.
All bodies have a certain frequency at which they will most naturally resonate. When a source produces such waves into this body (or medium) then it will vibrate 'in sympathy' with it. When this occurs, the amplitude of these vibrations will be at a maximum.
As stated above, with two closed boundaries, the fundamental wavelength will be 2 × the length. with one open boundary and one closed, 4 × length, and with 2 open boundaries, 2 × length.
The first, second and so on harmonics can be found by adding 1/2 of a cycle in to the diagram, so with one open, one closed, we have 3/4 of a cycle in the pipe, so the wavelength is 4/3 × length. This applies to all other types (2 open and 2 closed), and allows the wavelength to be found. We can then calculate the frequency given speed or vice versa, etc.