1. Speed of a wave on a string?
Which of the following determines the wave speed of a wave on a string?
1. the frequency at which the end of the string is shaken up and down
2. the coupling between neighboring parts of the string, as measured by the tension in the string
3. the mass of each little piece of string, as characterized by the mass per unit length of the string.
4. Both 1 and 2
5. Both 1 and 3
6. Both 2 and 3
7. All three.

2. A wave on a string
What parameters determine the speed of a wave on a string?
Properties of the medium: the tension in the string, and how heavy the string is.
where μ is the mass per unit length of the string.

3. Adding waves: the principle of superposition
When more than one wave is traveling in a medium, the waves simply add.
The principle of superposition: the net displacement of any point in the medium is the sum of the displacements at that point due to each individual wave.

4. Constructive interference
When the displacements of individual waves go in the same direction at a point, the result is a large amplitude there, because the displacements add. This is known as constructive interference. Simulation.
A neat feature of waves is that, after passing through one another, waves (or pulses) travel as if they had never met.

5. Destructive interference
When the displacements of individual waves are in opposite directions at a point, the waves cancel (at least partly). This is known as destructive interference. Simulation.
How is it possible for the two pulses to re-emerge from the flat string? Where is the energy to do this?

6. Reflections (fixed end)
How waves reflect at the ends of a medium, or at the interface between two media, is critical to understanding things like musical instruments.
When a wave encounters a fixed end, for instance, it comes back upside down Simulation

7. Reflections (free end)
When a wave encounters a free end, it comes back upright.

8. Standing waves
When two waves of the same frequency and amplitude travel in opposite directions in a medium, the result is a standing wave - a wave that does not travel one way or the other.
If the waves are identical except from their direction of propagation, they can be described by the equations:
y1 = A sin(kx - ωt) and y2 = A sin(kx + ωt)
The resultant wave is their
sum, and can be written as:
y = 2A sin(kx) cos(ωt)

9. Standing waves The resultant wave is their sum, and can be written as:
y = 2A sin(kx) cos(ωt)
This is quite different from the equation for a traveling wave, because the spatial part is separated from the time part. It tells us that the string is totally flat at certain points in time, and that there are certain positions where the amplitude is always zero - these points are called nodes. There are other points halfway between the nodes where the amplitude is maximum - these are the anti-nodes.

10. Standing waves: a string fixed at both ends
A wave traveling in one direction on the string reflects off the end, and returns inverted because the end is fixed. This gives two identical waves traveling in opposite directions on the string, just what is needed for a standing wave. Simulation
The waves reflect from both ends of the string. Completely constructive interference takes place only when the wavelength is related to the length L of the string by:
Using , the corresponding frequencies are:
, where n = 1, 2, 3, ...
where n = 1, 2, 3, ...

11. Standing waves: a string fixed at both ends
The lowest resonance frequency (n = 1) is known as the fundamental frequency for the string. All the higher frequencies are known as harmonics - these are integer multiples of the fundamental frequency.

12. Standing waves: a string fixed at both ends
fundamental (n = 1)
second harmonic (n = 2)
third harmonic (n = 3)
fourth harmonic (n = 4)

13. Standing waves: a string fixed at both ends
All stringed musical instruments have strings fixed at both ends. When they are played, the sound you hear is some combination of the fundamental frequency and the different harmonics - it's because the harmonics are included that the sound sounds musical. A pure sine wave does not sound nearly so nice.

14. Standing waves: a tube open at both ends
For a tube open at both ends, reflections of the sound at both ends produce a large-amplitude wave for particular resonance frequencies. For the standing waves, an open end is an anti-node (maximum amplitude point) for displacement.

15. Standing waves: a tube open at both ends
Simulation: transverse representation
Simulation: longitudinal representation
The resonance frequencies are given by the same equation we used for the string:

16. Standing waves: a tube closed at one end only
For a tube closed at one end, the closed end is a node (zero displacement) while the open end is an anti-node (maximum displacement). This leads to a different equation for the resonance frequencies.
, where n can only be odd integers
Simulation: transverse Simulation: longitudinal

17. Standing waves: a tube closed at one end only
For a tube closed at one end, the closed end is a node (zero displacement) while the open end is an anti-node (maximum displacement). This leads to a different equation for the resonance frequencies.
, where n can only be odd integers
Simulation: transverse representation
Simulation: longitudinal representation

18. Whiteboard