Optional Practical: Phase and reflection (GeoGebra)
In this exercise you will use GeoGebra to model the reflection of waves leading to standing waves and thin film interference. A wave is made up of a series of out of phase oscillation so first let's model SHM.
It can be shown that if the acceleration of a body is proportional to the distance from a fixed point then it's displacement will vary sinusoidally with time. In other words y = a sin ωt or y = a sin 2πft
a = amplitude
f = frequency
t = time
- Make sliders for a, f and t all between 0 and 5 with increment 0.01
- Input equation y = a*sin(2*pi*f*t)
- Animate t by double clicking the t slider and selecting animate. Observe what happens when you vary a and f (set animation as increasing (object properties) or you will start to go back in time).
what you should see is a line moving up and down with SHM. This is not a wave since all parts of the line move at the same time, they are in phase.
- Add a slider w from -5 to 5 with increment 0.01
- Input equation y = a*sin(2*pi*f*t + pi*w)
- set d to 1 and run the animation.
wπ is the phase angle, when this is π the oscillations are out of phase when 2π they are in phase
- Vary d and observe the phase change
- Before going onto the next stage delete both of the lines.
The equation of a wave
To model wave motion we need to make the phase depend on the position along the x axis. The wavelength is defined as the distance between two points that are in phase. So if a point is distance λ from the origin it will be 2π out of phase with the origin. Point x will therefore be 2πx/λ out of phase with the origin (check it out: if x = λ the phase difference is 2π).
Note that if the wave travels from left to right, points to the right of the origin will lag behind the origin. So the wave equation is
y = a sin (2πft - 2πx/λ)
- Make a slider for λ (Greek letters can be selected by clicking the little alpha on the right).
- Input g(x) = a sin(2*pi*f*t - 2*pi*x/λ )
- Observe the changes when you vary a, f and λ.
Phase and waves
To change the phase of all parts of the wave we can add a phase angle.
- Input h(x) = a sin(2*pi*f*t - 2*pi*x/λ + w*pi )
- Vary the phase of the wave by changing w (it is simplest to see this without the animation but try both).
When waves superpose the displacements add, this can be modeled by simply adding the two equations together.
- Input h+g
- Observe how the sum of the waves changes as you change w.
Phase and distance
The phase of two identical waves is related to the distance they have travelled. To show this we first need to define the starting point of one wave as the origin.
- Type function in the input line and select Function[
, , ]
- replace the part in [ ] with [g,0,10]. This will display function g from 0 to 10
- hide all lines apart from this new one.
A wave starting to the left of the origin will be lagging behind the origin by the time it gets there. If it is distance d from the origin the phase difference will be 2πd/λ. First we will make a marker for the start of wave 2.
- Create a slider for d from -5 to +5 increment 0.1
- change the equation of your second wave (h) to a sin(2π f t - 2π x / λ + 2π d / λ)
- You can delete the w slider now.
- To make the wave start from position d input Function[h,d,10] (where h is the second wave function)
- Hide everything except the two wave sections.
- Vary d and see how the phase changes.
Notice how the waves start in phase but when the meet they are out of phase unless the path difference is a whole number of wavelengths.
Before continuing it is best to tidy up a bit:
- Delete all the functions apart from the original wave g(x)
- Delete the slider for d
First we will make a reflection from the y axis. A reflected wave simply goes in the other direction so the points to the right of the origin are ahead of the origin. This is achieved by adding the phase angle instead of subtracting it.
- Input h(x) = a sin(2*pi*f*t + 2*pi*x/λ )
- Change the colour of the new line in object properties.
- Run the animation and see how the new line is a mirror reflection of the original line in the y axis.
- Hide the lines to the lines to the right of the y axis using Function[g,-10,0] and Function[h,-10,0]
Note that the on the y axis both lines are always in phase
Mirror not at the origin
If we have a mirrors at some distance d1 from the origin we must the reflected wave will be such that at the mirror it is in phase with the oncoming wave (actually when mechanical waves reflect they have a phase change of π but we can easily add that later). To make this happen we must fix the phase difference. Now when the reflected wave gets back to the origin it will have travelled a distance 2d so will be 2d2π/λ out of phase. So adding a phase angle of 4πd/λ to the reflected wave should do the trick.
- delete the last two line segments.
- Add a slider for d1 (d_1 will give you a subscript)
- Input x= d_1 to add a line for the reflector.
- Change the reflected wave equation to y = a sin(2*pi*f*t + 2*pi*x/λ+4*pi*d_1/λ)
- Use "Function[ ]" to display on the part of the wave to the left of the reflector.
- Move the reflector and see how the phase of the reflected wave changes.
- To make the phase of the wave change by π on reflection add π to the equation of the reflected wave.
A standing wave is formed when an incident wave superposes with its reflection.
- Add the incident and reflected waves.
- Hide the incident and reflected waves.
You now have standing wave. Notice that there are points that have zero amplitude (nodes) and points with 2a amplitude (antinodes). Let's fix the length of the string so it equals d1
- Use "Function[ ] to display the resultant wave from 0 to d1 .
The two ends of a string are fixed so there must be a node at each end. Fix the length of the string so that the x = 0 end doesn't move (you can add a point to the end so you can see it more easily)
For a wave in a closed pipe one end is a node the other an antinode.
What do you have to do to model the wave in an open pipe?
Thin film interference
This film interference occurs when light is reflected off two parallel reflectors.
- Add a new slider for d2
- Position a second reflector
- Add a wave that represents reflection off d2
- Vary the separation of the reflectors and observe the changing phase of the reflected waves.
- Add the two reflections to get the resultant.