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15.1Standing waves
When two waves of the same speed, wavelength and amplitude travelling in opposite directions meet, a standing wave is formed. According to the principle of superposition, the resulting wave has a displacement that is the sum of the displacements of the two travelling waves.
Figure 15.1 shows a red travelling wave moving to the left and an identical blue travelling wave moving to the right. Both waves travel on the same string. The graphs show the displacement due to each wave every one-twentieth of a period as a function of distance from the left end of the string. The wave shown in the thick purple line is the sum of the two and therefore shows the actual shape of the string. In the first diagram the waves are on top of each other.

Figure 15.1: A series of graphs showing the superposition of two identical travelling waves moving in opposite directions. The wave in blue travels to the right, and the wave in red travels to the left.

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In the top graph, at t = 0, the two travelling waves are on top of each other, and so the resultant wave has its maximum displacement at this time. At this instant the amplitude is twice the amplitude of one of the travelling waves.
In the following graphs the waves are moving apart and the amplitude of the purple wave decreases. In the last graph, a quarter of a period later, the two waves meet crest to trough and the resulting wave is zero: the entire string is flat at that particular instant of time.
Notice the following features that are characteristic of standing waves.
âThe position of crests and troughs of the purple wave in each graph are always at the same position x = 0.25 m, 0.75 m, 1.25 m and 1.75 m. They do not move right or left as they do in the case of travelling waves.
âThere are some points on the string where, as a result of destructive interference between the two waves, the displacement is always zero. We call these points nodes. The distance between two consecutive nodes is half a wavelength. In Figure 15.1 the nodes are at x = 0, 0.50 m, 1.0 m, 1.5 m and 2.0 m.
âHalf-way between nodes are points where, as a result of constructive interference, the displacement gets as large as possible. These points are called antinodes. Note that the nodes always have zero displacement whereas the antinodes are at maximum displacement for an instant of time only. In Figure 15.1 the antinodes are at x = 0.25 m, 0.75 m, 1.25 m and 1.75 m.
âThe amplitude of oscillation is different at different points on the string.
âA standing wave does not transfer energy: it consists of two travelling waves that transfer energy in opposite directions so the standing wave itself transfers no energy.
âThe ends of a standing wave are either nodes or antinodes. These âboundaryâ or âend conditionsâ determine the possible wavelength of the wave as we will see.
CHECK YOURSELF 1
All points on a standing wave have the same:
aamplitude
baverage speed
cfrequency
ddisplacement
EXAM TIP
You must be able to explain the formation of a standing wave in terms of the superposition of two oppositely moving travelling waves.
How do we create standing waves in practice? We will examine two cases: standing waves on strings and in pipes.
TEST YOUR UNDERSTANDING
1Describe what is meant by a standing wave.
2Outline how a standing wave is formed.
3List ways in which a standing wave differs from a travelling wave.
4In the context of standing waves describe what is meant by:
anode
bantinode
cwave speed.