17.1

The Principle of Linear Superposition

Often, two or more sound waves are present at the same place at the same time, as is the case with sound waves when everyone is talking at a party or when music plays from the speakers of a stereo system. To illustrate what happens when several waves pass simultaneously through the same region, let’s consider Figures 17-1 and 17-2, which show two transverse pulses of equal heights moving toward each other along a Slinky. In Figure 17-1 both pulses are “up,” while in Figure 17-2 one is “up” and the other is “down.” Part a of each drawing shows the two pulses beginning to overlap. The pulses merge, and the Slinky assumes a shape that is the sum of the shapes of the individual pulses. Thus, when the two “up” pulses overlap completely, as in Figure 17-1b, the Slinky has a pulse height that is twice the height of an individual pulse. Likewise, when the “up” pulse and the “down” pulse overlap exactly, as in Figure 17-2b, they momentarily cancel, and the Slinky becomes straight. In either case, the two pulses move apart after overlapping, and the Slinky once again conforms to the shapes of the individual pulses.

Figure 17-1 Two transverse “up” pulses passing through each other.

Figure 17-2 Two transverse pulses, one “up” and one “down,” passing through each other.

The adding together of individual pulses to form a resultant pulse is an example of a more general concept called the principle of linear superposition.

| The Principle of Linear Superposition
| When two or more waves are present simultaneously at the same place, the resultant disturbance is the sum of the disturbances from the individual waves.

This principle can be applied to all types of waves, including sound waves, water waves, and electromagnetic waves such as light. It embodies one of the most important concepts in physics, and the remainder of this chapter deals with examples related to it.

Check Your Understanding 1

The drawing shows two pulses traveling toward each other at t = 0 s. Each pulse has a constant speed of 1 cm/s. When t = 2 s, what is the height of the resultant pulse at (a) x = 2 cm, (b) x = 4 cm, and (c) x = 6 cm? (The answers are given at the end of the book.)

Background: Understanding the principle of linear superposition is the key to answering this question.
For similar questions (including conceptual counterparts), consult Self-Assessment Test 17.1, which is described at the end of Section 17.4.

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