1. Standing Waves & Resonance
A standing wave is created from two traveling waves, having the same frequency and the same amplitude and traveling in opposite directions in the same medium.
Using superposition, the net displacement of the medium is the sum of the two waves.
When 180° out-of-phase with each other, they cancel (destructive interference).
When in-phase with each other, they add together (constructive interference).
Diagram
Please watch jw

2. Simple example Standing Waves
If we tie a rope to a wall and shake the free end up and down, we produce a train of waves in the rope.
The wall is too rigid to shake, so the waves are reflected back along the rope.
By shaking the rope just right, we can cause the incident and reflected waves to form a standing wave.

3. Standing Waves
Nodes are the regions of minimal or zero displacement, with minimal or zero energy.
Antinodes are the regions of maximum displacement and maximum energy.
Antinodes and nodes occur equally apart from each other.

4. Standing Waves
If you shake the rope with twice the frequency, a standing wave of half the wavelength, having two loops results.
If you shake the tube with three times the frequency, a standing wave of one-third the wavelength, having three loops results.

5. Department of Physics & Astronomy
We use the term superimposed to mean two waves that are overlapping. Below, these two waves are travelling in opposite directions.
Moving to right
Moving to left
The sum of the
two waves
(“superposition”)
theory
Anti-Nodes
Nodes
Department of Physics & Astronomy
Page 5

7. Department of Physics & Astronomy
If the length remains unchanged, standing waves only occur at specific frequencies.
In our case, we have strings with nodes at both ends, which produces the following:
l/2 l
3l/2
theory
Department of Physics & Astronomy
Page 6

8. Topic 11: Wave phenomena 11.1 Standing (stationary) waves
Explain the formation of standing waves. Know what nodes and antinodes are, and how they form.
The standing wave has two important properties.
It does not travel to the left or the right as the blue and the green wave do.
Its “lobes” grow and shrink and reverse, but do not go to the left or the right.
Any points where the standing wave has no displacement is called a node (N).
The lobes that grow and shrink and reverse are called antinodes (A). N N N N N N N A A A A A A
© 2006 By Timothy K. Lund

9. Topic 11: Wave phenomena 11.1 Standing (stationary) waves
Discuss the modes of vibration of strings and air in open and in closed pipes.
You may be wondering how a situation could ever develop in which two identical waves come from opposite directions. Well, wonder no more.
When you pluck a stringed instrument, waves travel to the ends of the string and reflect at each end, and return to interfere under precisely the conditions needed for a standing wave.
Note that there are two nodes and one antinode.
Why must there be a node at each end of the string?
© 2006 By Timothy K. Lund L N N A
Because it is fixed at each end.

10. Topic 11: Wave phenomena 11.1 Standing (stationary) waves
Discuss the modes of vibration of strings and air in open and in closed pipes.
Observe that precisely half a wavelength fits along the length of the string.
Thus we see that  = 2L.
Since v = f we see that f = v/(2L) for a string.
This is the lowest frequency you can possibly get from this string configuration, so we call it the fundamental frequency f1.
The fundamental frequency of any system is called the first harmonic. L N A
1st harmonic
fundamental frequency f1 = v/(2L)
© 2006 By Timothy K. Lund

11. Topic 11: Wave phenomena 11.1 Standing (stationary) waves
Discuss the modes of vibration of strings and air in open and in closed pipes.
The next higher frequency has another node and another antinode.
We now see that  = L.
Since v = f we see that f = v/L.
This is the second lowest frequency you can possibly get and since we called the fundamental frequency f1, we’ll name this one f2.
This frequency is also called the second harmonic. L N N N
2nd harmonic A A
f2 = v/L
© 2006 By Timothy K. Lund

12. Nodes and Antinodes
Points oscillating with the biggest amplitude in a stationary wave are called antinodes.
Points undergoing zero oscillation are called nodes.

13. λ = 2L
Frequency = f
λ = L
Frequency = 2f
λ = 2/3 L
Frequency = 3f

14. Harmonic frequency = no. of loops x natural frequency
The lowest resonant frequency of the system is called the fundamental frequency (or 1st harmonic frequency). The next are called the 2nd harmonic, 3rd harmonic etc.
For a stationary wave in a string:
Harmonic frequency = no. of loops x natural frequency

15. Topic 11: Wave phenomena 11.1 Standing (stationary) waves
Because a standing wave consists of two traveling waves carrying energy in opposite directions, the net energy flow through the wave is zero.
Standing wave
Traveling wave
Energy
Not transferred but the wave has it in each antinode
Transferred from source to receiver at the wave speed
Amplitude
Maximum amplitude different for all points in medium between nodes
Maximum amplitude same for all points in medium
Frequency
All vibrations are SHM and same frequency
Wavelength
Same as component waves
Distance between crests
Phase
Same for each point in a lobe, but adjacent lobes are phase shifted by 180º
Different for each point along a single wavelength
Wave pattern
Does not move
Moves
© 2006 By Timothy K. Lund

16. Topic 11: Wave phenomena 11.1 Standing (stationary) waves
Solve problems involving standing waves.
PRACTICE: Complete the table below with both sketch and formula.
Remember that there are always nodes on each end of a string.
Add a new well-spaced node each time.
Decide the relationship between  and L.
We see that  = (2/3)L.
Since v = f we see that f = v/(2/3)L = 3v/2L.
f2 = v/L
f1 = v/2L
f3 =
3v/2L
© 2006 By Timothy K. Lund

17. Topic 11: Wave phenomena 11.1 Standing (stationary) waves
Discuss the modes of vibration of strings and air in open and in closed pipes.
We can also set up standing waves in pipes.
In the case of pipes, longitudinal waves are created (instead of translational waves), and these waves are reflected from the ends of the pipe.
Consider a closed pipe of length L which gets its wave energy from a mouthpiece on the left side.
Why must the mouthpiece end be an antinode?
Why must the closed end be a node?
© 2006 By Timothy K. Lund
(1/4)1 = L
(3/4)2 = L
(5/4)2 = L
f1 = v/4L
f2 = 3v/4L
f3 = 5v/4L
Source.
Air can’t move.

18. Topic 11: Wave phenomena 11.1 Standing (stationary) waves
Discuss the modes of vibration of strings and air in open and in closed pipes.
In an open-ended pipe you have an antinode at the open end because the medium can vibrate there (and, of course, at the mouthpiece).
(1/2)1 = L
2 = L
(3/2)2 = L
© 2006 By Timothy K. Lund
f1 = v/2L
f2 = 2v/2L
f3 = 3v/2L
FYI
The IBO requires you to be able to make sketches of string and pipe harmonics (both open and closed) and find wavelengths and frequencies.

19. Topic 11: Wave phenomena 11.1 Standing (stationary) waves
Solve problems involving standing waves.
PRACTICE: A tube is filled with water and a vibrating tuning fork is held above the open end. As the water runs out of the tap at the bottom sound is loudest when the water level is a distance x from the top. The next loudest sound comes when the water level is at a distance y from the top Which expression for  is correct, if v is the speed of sound in air?
A.  = x B.  = 2x C.  = y-x D.  = 2(y-x)
v = f and since v and f are constant, so is .
The first possible standing wave is sketched.
The sketch shows that  = 4x, not a choice.
© 2006 By Timothy K. Lund

20. Topic 11: Wave phenomena 11.1 Standing (stationary) waves
Solve problems involving standing waves.
PRACTICE: A tube is filled with water and a vibrating tuning fork is held above the open end. As the water runs out of the tap at the bottom sound is loudest when the water level is a distance x from the top. The next loudest sound comes when the water level is at a distance y from the top Which expression for  is correct, if v is the speed of sound in air?
A.  = x B.  = 2x C.  = y-x D.  = 2(y-x)
The second possible standing wave is sketched.
Notice that y – x is half a wavelength.
Thus the answer is  = 2(y - x).
y-x
© 2006 By Timothy K. Lund

21. Topic 11: Wave phenomena 11.1 Standing (stationary) waves
Solve problems involving standing waves.
PRACTICE: This drum head set to vibrating at different resonant frequencies has black sand on it, which reveals D standing waves.
Does the sand reveal nodes, or does it reveal antinodes?
Why does the edge have to be a node?
Nodes, because there is no displacement to throw the sand off.
© 2006 By Timothy K. Lund
The drumhead cannot vibrate at the edge.

22. Topic 11: Wave phenomena 11.1 Standing (stationary) waves
Solve problems involving standing waves.
Alternate lobes have a 180º phase difference.
See Slides 3 and 10.
© 2006 By Timothy K. Lund

23. Topic 11: Wave phenomena 11.1 Standing (stationary) waves
Solve problems involving standing waves.
Make a sketch. Then use v = f.
antinode
antinode
v = f
© 2006 By Timothy K. Lund L
f = v/
 / 2 = L
f = v/(2L)
 = 2L

24. Topic 11: Wave phenomena 11.1 Standing (stationary) waves
Solve problems involving standing waves.
© 2006 By Timothy K. Lund
Reflection provides for two coherent waves traveling in opposite directions.
Superposition is just the adding of the two waves to produce the single stationary wave.

25. Topic 11: Wave phenomena 11.1 Standing (stationary) waves
Solve problems involving standing waves.
© 2006 By Timothy K. Lund
A snapshot of Slide 3 shows the points between successive nodes.
For every point between the two nodes f is the same.
But the amplitudes are all different.
Therefore the energies are also different.

26. Topic 11: Wave phenomena 11.1 Standing (stationary) waves
Solve problems involving standing waves.
Energy transfer via a vibrating medium without interruption.
The medium itself does not travel with the wave disturbance.
© 2006 By Timothy K. Lund
Speed at which the wave disturbance propagates.
Speed at which the wave front travels.
Speed at which the energy is transferred.

27. Topic 11: Wave phenomena 11.1 Standing (stationary) waves
Solve problems involving standing waves.
Frequency is number of vibrations per unit time.
FYI: IB frowns on you using particular units as in
“Frequency is number of vibrations per second.”
© 2006 By Timothy K. Lund
FYI: There will be lost points, people!
Distance between successive crests (or troughs).
Distance traveled by the wave in one oscillation of the source.

28. Topic 11: Wave phenomena 11.1 Standing (stationary) waves
Solve problems involving standing waves.
© 2006 By Timothy K. Lund
Each of the waves traveling in opposite directions carry energy at same rate in both directions. Thus there is NO energy transfer.
The amplitude is always changing and reversing.

29. Topic 11: Wave phenomena 11.1 Standing (stationary) waves
Solve problems involving standing waves.
v = f L L
© 2006 By Timothy K. Lund
f = v/
P / 4 = L
Q / 2 = L
 = 4L
 = 2L
fP = v/(4L)
fQ = v/(2L)
v = 4LfP
fQ = 4LfP /(2L)
fQ = 2fP

30. Topic 11: Wave phenomena 11.1 Standing (stationary) waves
Solve problems involving standing waves.
The tuning fork is the driving oscillator (and is at the top).
© 2006 By Timothy K. Lund
The top is thus an antinode.
The bottom “wall” of water allows NO oscillation.
The bottom is thus a node.

31. Topic 11: Wave phenomena 11.1 Standing (stationary) waves
Solve problems involving standing waves.
© 2006 By Timothy K. Lund
Sound is longitudinal in nature.
Small displacement at P, big at Q.

32. Topic 11: Wave phenomena 11.1 Standing (stationary) waves
Solve problems involving standing waves.
© 2006 By Timothy K. Lund
If the lobe at T is going down, so is the node at U.

33. Topic 11: Wave phenomena 11.1 Standing (stationary) waves
Solve problems involving standing waves.
© 2006 By Timothy K. Lund
Pattern 1 is a 1/2 wavelength.
Pattern 2 is a 3/2 wavelength.
Thus f2 = 3f1 so that f1 / f2 = 1/3.