Standing waves occur when two waves of equal wavelength, frequency, and amplitude traveling in opposite directions interfere. The points of no displacement are called nodes, and the points of maximum displacement are called antinodes. A standing wave can be described by the equation D(x,t) = A(x)cos(ωt), where A(x) is the position-dependent amplitude. The distance between nodes and antinodes is λ/2, where λ is the wavelength. Standing waves are confined to a given space and do not transport energy like traveling waves.

1. LO 6
Standing Waves
Adria Lwin

2. What are standing waves?
✤ The interference of two waves with equal wavelength,
frequency, and amplitude that are moving in OPPOSITE
directions.

3. Demonstration of a Standing Wave
✤ Notice that the travelling blue and green waves are moving in opposite directions.
✤ The red wave is the sum of the blue and green waves (the standing wave).
✤ When the two travelling waves are aligned, the resultant wave achieves maximum amplitude (at antinodes).
✤ When the two travelling waves are directly opposite each other, the resultant wave is flat.
Click on the picture above to play the GIF
http://upload.wikimedia.org/wikipedia/commons/6/62/Harmonic_Standing_Wave.gif

4. Equation of a Standing Wave
D(x,t) = 2A sin(kx) cos(⍵t)
Let us define a position-dependent amplitude: A(x) = 2A sin(kx)
This lets us rewrite the previous equation:
D(x,t) = A(x) cos(⍵t)

5. Parts of a Standing Wave
✤ Node: Points on a standing wave with zero amplitude that remain at rest at all times.
✤ Each segment between two nodes oscillates with simple harmonic motion.
✤ Its energy continuously transforms between kinetic and potential energy.
✤ Antinode: Points on a standing wave which achieve maximum amplitude of 2A.
http://www.revisesmart.co.uk/images/resources/physics/nodes-antinodes.png

6. Locations of Nodes and Antinodes
✤ At a node, A(x) = 0, therefore,
x = 0, ±λ/2, ±3λ/2, ±2λ,…
✤ At an antinode, A(x) = ±2A,
therefore ,
x = ±λ/4, 3λ/4, 5λ/4,…
(Check page 224 for full derivations.)
✤ The distance between two consecutive nodes and antinodes is half a wavelength (λ/2).
✤ The distance between an adjacent node and antinode is a quarter of a wavelength apart.

7. Travelling Waves vs. Standing
Waves
✤ Carries energy ✤ Does not carry energy
✤ Has a fixed amplitude, A
✤ Amplitude varies as a function of
position, between 0 and 2A
✤ Not confined to a given space
✤ Is confined to a given space (ie.
between two walls)
✤ Can have any frequency ✤ Frequency is quantized
✤ Wavelength is the shortest distance
over which the waveform repeats itself
✤ Wavelength is two times the distance
between two nodes or two antinodes
✤ D(x,t) = A sin(kx ± ⍵t + Φ) ✤ D(x,t) = 2A sin(kx) cos(⍵t)
http://media.tumblr.com/tumblr_m8keuyIx7l1qf3sji.gif http://media.tumblr.com/tumblr_m8kevdFuZn1qf3sji.gif

8. Practice Questions
1. A 20 m long standing wave has three nodes (not including the first
and last nodes). What is the wavelength of this standing wave?
2. A standing wave is given by D(x, t) = 10 sin(4x) cos(4πt). What is:
A. The amplitude, frequency and equations of the constituent
waves.
B. The distance between two nodes or antinodes.
C. The amplitude of a particle whose position along the x-axis is 36
cm.

9. Answer 1
✤ The wavelength is 10 m.
✤ We know that:
✤ The length is 20 m
✤ There are 3 nodes
✤ Recall: The distance between two consecutive
nodes is λ/2

10. Solution 1

11. Answer 2
A. Amplitude = 5m, frequency = 2 Hz, wavelength = π/2 m
B. π/4 m
C. 9.91 m

12. Solution 2