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14.2The principle of superposition

Suppose that two pulses are produced in the same rope and are travelling towards each other from opposite ends. Something amazing happens when the two pulses meet. Figure 14.17 shows what happens in a sequence of pictures. For simplicity we have drawn idealised square pulses.

Figure 14.17: The superposition of two positive pulses.

The disturbance gets bigger when the two pulses meet, but subsequently the two pulses simply ‘go through each other’ with no ‘memory’ of what happened. You should contrast this with what happens in the motion of material particles: when two balls collide they bounce off each other.

What happens when two (or more) pulses meet at some point in space is described by the principle of superposition when two or more waves of the same type arrive at a given point in space at the same time, the displacement of the medium at that point is the algebraic sum of the individual displacements.

The principle of superposition states that when two or more waves of the same type arrive at a given point in space at the same time, the displacement of the medium at that point is the algebraic sum of the individual displacements.

So, if y 1 and y 2 are individual displacements, then the total displacement has the value:

y=y 1+y 2

Note the word ‘algebraic’. This means that if one pulse is ‘up’ and the other is ‘down’, then the resulting displacement is the difference of the individual pulses.

Let us look at Figure 14.17b in detail. The two pulses are partially overlapping—Figure 14.18 shows both of them separately (the pulse moving toward the right is drawn in dark blue and the one moving to the left in pale blue). Adding the two pulses according to the principle of superposition gives the shape in Figure 14.18.

Figure 14.18: The situation in Figure 14.17b analysed.

Figure 14.19 shows the superposition of a positive and a negative pulse on the same rope. In Figure 14.19a the positive and negative pulses are approaching each other. In Figure 14.19b the positive and negative pulses momentarily cancel each other out when they totally overlap. The pulses move through each other, and Figure 14.19c shows the positive and negative pulses continuing along the rope.

Figure 14.19: The superposition of a positive and a negative pulse.

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CHECK YOURSELF 2

Two identical pulses move towards each other with speed 7.5 cm s−1. Figure 14.20 shows the pulses at t = 0.

Figure 14.20: For check yourself question 2.

aAfter what time will the pulses completely overlap?

bCopy Figure 14.20 and add the shape of the pulses at the instant of complete overlap.

Links

Let us consider a simple problem in mechanics. A particle of mass m moving with speed v collides head on with another heavier particle of mass M that is at rest. Applying the laws of conservation of energy and momentum, we can find the velocities of both particles after the collision. Calling the velocity of m, u and that of M, w we find u=−M−m M+m v and w=2 m M+m v. The light particle bounces back and the heavy particle starts moving to the right. But there is another solution with u = v and w = 0. In mechanics we reject this solution because it implies that m goes through M leaving M at rest. But the equations are clever. This is what happens with pulses and waves; they go through each other. The equations of mechanics are general enough to allow for this possibility as well.

Reflection of pulses

What happens when a pulse created in a rope with one end fixed approaches that fixed end? Consider the pulse of Figure 14.21a. The instant the pulse hits the fixed end, the rope attempts to move the fixed end upward: that is, it exerts an upward force on the fixed end. By Newton’s third law, the wall will then exert an equal but opposite force on the rope. This means that a displacement will be created in the rope that will be negative and will start moving towards the left.

The pulse has been reflected by the wall and has been inverted. This is the same as saying that the wave experiences a phase change of 180° when reflected.

If the end of the rope is not fixed but free to move (imagine that the end of the rope is now tied to a ring that can slide up and down a vertical pole), the situation is different (Figure 14.21b). As the pulse arrives at the ring it pulls it upwards. Eventually the ring falls back down and in so doing creates a pulse moving to the left that is not inverted; that is, there is no phase change.

Figure 14.21: Reflection of a pulse from a a fixed end and b a free end. Notice the inversion in the case of the fixed end.

CHECK YOURSELF 3

Look at Figure 14.22. Draw the pulse after it has been reflected from the fixed end.

Figure 14.22: For check yourself question 3.

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Worked Example 14.3

Use the results about pulses reflecting from fixed and free ends to predict what happens when:

aa pulse in a heavy rope encounters a light rope (Figure 14.23a)

ba pulse in a light rope encounters a heavy rope (Figure 14.23b).

Figure 14.23: For worked example 14.3.

Answer

aWith the light rope to the right, the situation is similar to a pulse in a rope approaching a free end. So the reflected pulse will not be inverted.

bWith the heavy rope to the right, the situation is similar to a pulse in a rope approaching a fixed end. So the reflected pulse will be inverted.

Notice that in both cases, there will be a pulse transmitted into the rope to the right.

nature of science

Particles or a wave?

The early nineteenth century saw a revival of the wave approach to light, mainly due to the work of Young and Fresnel. In the eighteenth century, the Newtonian view of a particle nature of light prevailed, making research in directions that Newton would not ‘approve’ almost impossible. So the early researchers of the nineteenth century had to be brave as well as ingenious! It is interesting to note, however, that light waves were first thought to be longitudinal waves, like sound. The discovery of polarisation created enormous difficulties for the supporters of the wave theory because it could not be understood in terms of a longitudinal wave theory of light. It was Young who finally suggested that light was a transverse wave (an idea which Maxwell took much further when he proposed that light was an electromagnetic wave). So the phenomenon of polarisation that helps Monarch butterflies, bees and ants navigate (it may even have helped the Vikings reach Vinland, i.e. North America) is the phenomenon that for the first time introduced light as a transverse wave in physics.

TEST YOUR UNDERSTANDING

13In Figure 14.19b the two pulses completely cancelled each other out. What happened to the energy carried by each pulse?

14Figure 14.24 shows two pulses of equal width and height travelling in opposite directions on the same string. Draw the shape of the string when the pulses completely overlap.

Figure 14.24: For question 14.

15Figure 14.25 shows two pulses of equal width and height travelling in opposite directions on the same string. Draw the shape of the string when the pulses completely overlap.

Figure 14.25: For question 15.

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16The wave pulses shown in Figure 14.26 travel at 1 cm s−1, and both have width 2 cm. The heights are indicated on the diagram. In each case, draw the shape of the resulting pulse according to the principle of superposition at times t = 0.5 s, t = 1.0 s and t = 1.5 s. Take t = 0 s to be the time when the pulses are about to meet each other.

Figure 14.26: For question 16.

17Two pulses are simultaneously generated on a string. The graph (Figure 14.27) shows the variation of displacement y with distance x. Draw the actual shape of the string.

Figure 14.27: For question 17.