Consider a string, such as a guitar string, that is stretched between two clamps. Suppose we send a continuous sinusoidal wave of a certain frequency along the string, say, toward the right. When the wave reaches the right end, it reflects and begins to travel back to the left. That left-going wave then overlaps the wave that is still traveling to the right. When the left-going wave reaches the left end, it reflects again and the newly reflected wave begins to travel to the right, overlapping the left-going and right-going waves. In short, we very soon have many overlapping traveling waves, which interfere with one another.
For certain frequencies, the interference produces a standing wave pattern (or oscillation mode) with nodes and large antinodes like those in Fig. 16-22. Such a standing wave is said to be produced at resonance, and the string is said to resonate at these certain frequencies, called resonant frequencies. If the string is oscillated at some frequency other than a resonant frequency, a standing wave is not set up. Then the interference of the right-going and left-going traveling waves results in only small (perhaps imperceptible) oscillations of the string.
Fig. 16-22 Stroboscopic photographs reveal (imperfect) standing wave patterns on a string being made to oscillate by an oscillator at the left end. The patterns occur at certain frequencies of oscillation.
Let a string be stretched between two clamps separated by a fixed distance L. To find expressions for the resonant frequencies of the string, we note that a node must exist at each of its ends, because each end is fixed and cannot oscillate. The simplest pattern that meets this key requirement is that in Fig. 16-23a, which shows the string at both its extreme displacements (one solid and one dashed, together forming a single “loop”). There is only one antinode, which is at the center of the string. Note that half a wavelength spans the length L, which we take to be the string’s length. Thus, for this pattern, λ/2 = L. This condition tells us that if the left-going and right-going traveling waves are to set up this pattern by their interference, they must have the wavelength λ = 2L.
A second simple pattern meeting the requirement of nodes at the fixed ends is shown in Fig. 16-23b. This pattern has three nodes and two antinodes and is said to be a two-loop pattern. For the left-going and right-going waves to set it up, they must have a wavelength λ = L. A third pattern is shown in Fig. 16-23c. It has four nodes, three antinodes, and three loops, and the wavelength is λ = 
L. We could continue this progression by drawing increasingly more complicated patterns. In each step of the progression, the pattern would have one more node and one more antinode than the preceding step, and an additional λ/2 would be fitted into the distance L.
Fig. 16-23 A string, stretched between two clamps, is made to oscillate in standing wave patterns. (a) The simplest possible pattern consists of one loop, which refers to the composite shape formed by the string in its extreme displacements (the solid and dashed lines). (b) The next simplest pattern has two loops. (c) The next has three loops.
Thus, a standing wave can be set up on a string of length L by a wave with a wavelength equal to one of the values
The resonant frequencies that correspond to these wavelengths follow from Eq. 16-13:
Here v is the speed of traveling waves on the string.
Equation 16-66 tells us that the resonant frequencies are integer multiples of the lowest resonant frequency, f = v/2L, which corresponds to n = 1. The oscillation mode with that lowest frequency is called the fundamental mode or the first harmonic. The second harmonic is the oscillation mode with n = 2, the third harmonic is that with n = 3, and so on. The frequencies associated with these modes are often labeled f1, f2, f3, and so on. The collection of all possible oscillation modes is called the harmonic series, and n is called the harmonic number of the nth harmonic.
The phenomenon of resonance is common to all oscillating systems and can occur in two and three dimensions. For example, Fig. 16-24 shows a two-dimensional standing wave pattern on the oscillating head of a kettledrum.
Fig. 16-24 One of many possible standing wave patterns for a kettle-drum head, made visible by dark powder sprinkled on the drumhead. As the head is set into oscillation at a single frequency by a mechanical oscillator at the upper left of the photograph, the powder collects at the nodes, which are circles and straight lines in this two-dimensional example.
CHECK POINT 7 In the following series of resonant frequencies, one frequency (lower than 400 Hz) is missing: 150, 225, 300, 375 Hz. (a) What is the missing frequency? (b) What is the frequency of the seventh harmonic?
In Fig. 16-25, a string, tied to a sinusoidal oscillator at P and running over a support at Q, is stretched by a block of mass m. The separation L between P and Q is 1.2 m, the linear density of the string is 1.6 g/m, and the frequency f of the oscillator is fixed at 120 Hz. The amplitude of the motion at P is small enough for that point to be considered a node. A node also exists at Q.
Fig. 16-25 A string under tension connected to an oscillator. For a fixed oscillator frequency, standing wave patterns occur for certain values of the string tension.
(a) What mass m allows the oscillator to set up the fourth harmonic on the string?
Solution: One Key Idea here is that the string will resonate at only certain frequencies, determined by the wave speed v on the string and the length L of the string. From Eq. 16-66, these resonant frequencies are
To set up the fourth harmonic (for which n = 4), we need to adjust the right side of this equation, with n = 4, so that the left side equals the frequency of the oscillator (120 Hz).
We cannot adjust L in Eq. 16-67; it is set. However, a second Key Idea is that we can adjust v, because it depends on how much mass m we hang on the string. According to Eq. 16-26, wave speed 
. Here the tension τ in the string is equal to the weight mg of the block. Thus,
Substituting v from Eq. 16-68 into Eq. 16-67, setting n = 4 for the fourth harmonic, and solving for m give us
(b) What standing wave mode is set up if m = 1.00 kg?
Solution: If we insert this value of m into Eq. 16-69 and solve for n, we find that n = 3.7. A Key Idea here is that n must be an integer, so n = 3.7 is impossible. Thus, with m = 1.00 kg, the oscillator cannot set up a standing wave on the string, and any oscillation of the string will be small, perhaps even imperceptible.
PROBLEM-SOLVING TACTICS
TACTIC 2:<<Harmonics on a String
When you need to obtain information about a certain harmonic on a stretched string of given length L, first draw that harmonic (as in Fig. 16-23). If you are asked about, say, the fifth harmonic, you need to draw five loops between the fixed support points. That would mean that five loops, each of length λ/2, occupy the length L of the string. Thus, 5(λ/2) = L, and λ = 2L/5. You can then use Eq. 16-13 (f = v/λ) to find the frequency of the harmonic.
Keep in mind that the wavelength of a harmonic is set only by the length L of the string, but the frequency depends also on the wave speed v, which is set by the tension and the linear density of the string via Eq. 16-26.
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