3 Superposition

3.1 Introductory Concepts

Consider Alan and Bill taking batting practice (figure 4232):

Figure 42: Interaction of particles in space and time.

• Suppose that the pitching machines are arranged such that:

– Machine A throws balls towards Bill.

32Knight, Figure 21.1(a), page 647

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– Machine B throws balls towards Alan.

• Suppose that each machine launches a baseball at the same time, trav-

eling with the same speed. What happens?

– The two balls collide at the crossing point, and move away.

Two particles cannot occupy the same point in space and time.

Consider Alan and Bill listening to a stereo through two loudspeakers (figure

43 33):

Figure 43: Interaction of sound waves in space and time.

• Both Alan and Bill hear the emitted sound waves quite well – there is

no distortion or missing sound.

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– Loudspeakers A and B emits a sound wave towards Alan.

– Loudspeakers A and B emits a sound wave towards Bill.

• What happens to the medium at a point where two waves are present

simultaneously?

– Wave 1 displaces the particle in the medium by u1.

– Wave 2 simultaneously displaces the particle in the medium by

u2.

– The net displacement of the particle is simply u1 + u2.

This method of combining waves is called the principle of superposition:

when two or more waves are simultaneously present at a single point

in space, the displacement of the medium at that point is the sum

of the displacements due to each individual waves.

Mathematically, if n waves are simultaneously present, where the displace-

ment caused by wave i is ui, then the net displacement unet of a particle in the

medium is

unet = u1 + u2 + · · · + un =n∑

i=1

ui . (3.1)

Note: although in general this is not the case, we will make the simplifying

assumption that the displacements of the individual waves are along the same

line – i.e. so that the displacements add as scalars rather than vectors.

Figure 4434 shows snapshots of the superposition of two traveling waves (both

of speed 1 m s−1, moving in opposite directions), taken 1 s apart.

We see that

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Figure 44: Superposition of two waves on a string.

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• t = 0 s: waves approach each other.

• t = 1 s: waves make first contact.

• t = 2 s, 3 s: waves interact – the net displacement is a point–by–

point summation of each wave, in accordance with (3.1) for n = 2.

• t = 4 s: waves emerge unchanged – the original shapes are maintained.

3.2 Standing Waves

We consider a so-called standing wave, i.e. a wave in which the crests and

troughs stand in place as the wave oscillates.

For example, figure 4535 is a time-lapse photograph of a standing wave in a

vibrating string (e.g. plucked string in a musical instrument).

A standing wave is actually a superposition of two traveling waves. To see

this, consider two sinusoidal waves traveling through a medium with:

• Equal amplitudes.

• Equal frequencies.

• Equal wavelengths.

• Equal speeds.

• Traveling in opposite directions.

Figure 4636 shows a snapshot of nine graphs, taken at an interval T/8 apart.

• Let us identify two crests on the two waves:

– Red wave is moving to the right.

– Green wave is moving to the left.

35see http://www1.union.edu/∼newmanj/lasers/Light as a Wave/light as a wave.htm36Knight, 21.4(a), page 649

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Figure 45: Standing wave.

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Figure 46: Superposition of sinusoidal waves: construction.

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• At each point, the net displacement of the medium is found by adding

the red and green displacements.

• The superposed wave is the blue wave, shown more clearly in figure 4737.

– So the superposed wave is the actual wave observed in the medium.

Note: the blue dot shown in figure 47 is moving neither to the left nor

right:

• The motion of the blue dot shows that the superposition does give a

wave, but not a traveling wave.

• In fact, the superposed wave is a standing wave, because the crests

and troughs stand in place as it oscillates.

More interesting is figure 4838, showing the nine separate graphs of figure 47

on the same graph.

• There are points on the standing wave that never actually move – these

are called nodes.

– These are points of zero displacement.

– They occur a distance of λ/2 apart.

– At every instant in time, the displacements of the two traveling

waves have equal magnitudes but opposite signs.

– Consequently, the superposition is always zero – there is destruc-

tive interference (see later notes).

– At a point where u1 is always equal to −u2, the two waves are said

to be out of phase with each other.

• There are points on the standing wave where the particles of the medium

oscillate with maximum displacement – these are called antinodes.

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Figure 47: Superposition of sinusoidal waves: observed wave.

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