In Section 16-10, we discussed two sinusoidal waves of the same wavelength and amplitude traveling in the same direction along a stretched string. What if they travel in opposite directions? We can again find the resultant wave by applying the superposition principle.
Figure 16-19 suggests the situation graphically. It shows the two combining waves, one traveling to the left in Fig. 16-19a, the other to the right in Fig. 16-19b. Figure 16-19c shows their sum, obtained by applying the superposition principle graphically. The outstanding feature of the resultant wave is that there are places along the string, called nodes, where the string never moves. Four such nodes are marked by dots in Fig. 16-19c. Halfway between adjacent nodes are antinodes, where the amplitude of the resultant wave is a maximum. Wave patterns such as that of Fig. 16-19c are called standing waves because the wave patterns do not move left or right; the locations of the maxima and minima do not change.
Fig. 16-19 (a) Five snapshots of a wave traveling to the left, at the times t indicated below part (c) (T is the period of oscillation). (b) Five snapshots of a wave identical to that in (a) but traveling to the right, at the same times t. (c) Corresponding snapshots for the superposition of the two waves on the same string. At t = 0, 
T, and T, fully constructive interference occurs because of the alignment of peaks with peaks and valleys with valleys. At t = 
T and 
T, fully destructive interference occurs because of the alignment of peaks with valleys. Some points (the nodes, marked with dots) never oscillate; some points (the antinodes) oscillate the most.
Fig. 16-20 The resultant wave of Eq. 16-60 is a standing wave and is due to the interference of two sinusoidal waves of the same amplitude and wavelength that travel in opposite directions.
If two sinusoidal waves of the same amplitude and wavelength travel in opposite directions along a stretched string, their interference with each other produces a standing wave.
To analyze a standing wave, we represent the two combining waves with the equations
The principle of superposition gives, for the combined wave,
Applying the trigonometric relation of Eq. 16-50 leads to
which is displayed in Fig. 16-20. This equation does not describe a traveling wave because it is not of the form of Eq. 16-17. Instead, it describes a standing wave.
The quantity 2ym sin kx in the brackets of Eq. 16-60 can be viewed as the amplitude of oscillation of the string element that is located at position x. However, since an amplitude is always positive and sin kx can be negative, we take the absolute value of the quantity 2ym sin kx to be the amplitude at x.
In a traveling sinusoidal wave, the amplitude of the wave is the same for all string elements. That is not true for a standing wave, in which the amplitude varies with position. In the standing wave of Eq. 16-60, for example, the amplitude is zero for values of kx that give sin kx = 0. Those values are
Substituting k = 2π/λ in this equation and rearranging, we get
as the positions of zero amplitude—the nodes—for the standing wave of Eq. 16-60. Note that adjacent nodes are separated by λ/2, half a wavelength.
The amplitude of the standing wave of Eq. 16-60 has a maximum value of 2y, which occurs for values of kx that give | sin kx | = 1. Those values are
Substituting k = 2π/λ in Eq. 16-63 and rearranging, we get
as the positions of maximum amplitude—the antinodes—of the standing wave of Eq. 16-60. The antinodes are separated by λ/2 and are located halfway between pairs of nodes.
Reflections at a Boundary
We can set up a standing wave in a stretched string by allowing a traveling wave to be reflected from the far end of the string so that the wave travels back through itself. The incident (original) wave and the reflected wave can then be described by Eqs. 16-58 and 16-59, respectively, and they can combine to form a pattern of standing waves.
Fig. 16-21 (a) A pulse incident from the right is reflected at the left end of the string, which is tied to a wall. Note that the reflected pulse is inverted from the incident pulse. (b) Here the left end of the string is tied to a ring that can slide without friction up and down the rod. Now the pulse is not inverted by the reflection.
In Fig. 16-21, we use a single pulse to show how such reflections take place. In Fig. 16-21a, the string is fixed at its left end. When the pulse arrives at that end, it exerts an upward force on the support (the wall). By Newton’s third law, the support exerts an opposite force of equal magnitude on the string. This second force generates a pulse at the support, which travels back along the string in the direction opposite that of the incident pulse. In a “hard” reflection of this kind, there must be a node at the support because the string is fixed there. The reflected and incident pulses must have opposite signs, so as to cancel each other at that point.
In Fig. 16-21b, the left end of the string is fastened to a light ring that is free to slide without friction along a rod. When the incident pulse arrives, the ring moves up the rod. As the ring moves, it pulls on the string, stretching the string and producing a reflected pulse with the same sign and amplitude as the incident pulse. Thus, in such a “soft” reflection, the incident and reflected pulses reinforce each other, creating an antinode at the end of the string; the maximum displacement of the ring is twice the amplitude of either of these pulses.
CHECK POINT 6 Two waves with the same amplitude and wavelength interfere in three different situations to produce resultant waves with the following equations:
(1) y′(x, t) = 4 sin(5x − 4t)
(2) y′(x, t) = 4 sin(5x) cos(4t)
(3) y′(x, t) = 4 sin(5x + 4t)
In which situation are the two combining waves traveling (a) toward positive x, (b) toward negative x, and (c) in opposite directions?
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