18 - Superposition and Standing Waves

# 18 - Superposition and Standing Waves - Chapter 18...

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543 543 543 Superposition and Standing Waves CHAPTER OUTLINE 18.1 Superposition and Interference 18.2 Standing Waves 18.3 Standing Waves in a String Fixed at Both Ends 18.4 Resonance 18.5 Standing Waves in Air Columns 18.6 Standing Waves in Rods and Membranes 18.7 Beats: Interference in Time 18.8 Nonsinusoidal Wave Patterns ± Guitarist Carlos Santana takes advantage of standing waves on strings. He changes to a higher note on the guitar by pushing the strings against the frets on the ﬁngerboard, shorten- ing the lengths of the portions of the strings that vibrate. (Bettmann/Corbis) Chapter 18

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544 I n the previous two chapters, we introduced the wave model. We have seen that waves are very different from particles. A particle is of zero size, while a wave has a characteristic size—the wavelength. Another important difference between waves and particles is that we can explore the possibility of two or more waves combining at one point in the same medium. We can combine particles to form extended objects, but the particles must be at different locations. In contrast, two waves can both be present at the same location, and the ramiﬁcations of this possibility are explored in this chapter. When waves are combined, only certain allowed frequencies can exist on systems with boundary conditions—the frequencies are quantized . Quantization is a notion that is at the heart of quantum mechanics, a subject that we introduce formally in Chapter 40. There we show that waves under boundary conditions explain many of the quan- tum phenomena. For our present purposes in this chapter, quantization enables us to understand the behavior of the wide array of musical instruments that are based on strings and air columns. We also consider the combination of waves having different frequencies and wave- lengths. When two sound waves having nearly the same frequency interfere, we hear variations in the loudness called beats. The beat frequency corresponds to the rate of alternation between constructive and destructive interference. Finally, we discuss how any nonsinusoidal periodic wave can be described as a sum of sine and cosine functions. 18.1 Superposition and Interference Many interesting wave phenomena in nature cannot be described by a single traveling wave. Instead, one must analyze complex waves in terms of a combination of traveling waves. To analyze such wave combinations, one can make use of the superposition principle: If two or more traveling waves are moving through a medium, the resultant value of the wave function at any point is the algebraic sum of the values of the wave func- tions of the individual waves. Superposition principle
when two pebbles are thrown into a pond and hit the surface at different places, the expanding circular surface waves do not destroy each other but rather pass through each other. The complex pattern that is observed can be viewed as two independent sets of expanding circles. Likewise, when sound waves from two sources move through air, they pass through each other.

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18 - Superposition and Standing Waves - Chapter 18...

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