AG26_Superposition

AG26_Superposition - Chapter 26: The Principle of...

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1 Chapter 26: The Principle of Superposition for Waves © 2007 Alan Giambattista The flute and the clarinet are two popular wind instruments that are roughly the same length. In both, the musician blows air in a precisely controlled way into one end of the instrument, setting the column of air inside into vibration. However, the mouthpieces of the two instruments are quite different. A flautist blows across an open hole, while a clarinet has a thin wooden reed that vibrates, covering and uncovering the hole. How do the different styles of mouthpiece explain these differences between the two instruments? The sound quality is strikingly different—a listener can easily distinguish the two even when they are playing tones of the same pitch and loudness. The clarinet can play much lower pitches than the flute. 26.1 THE PRINCIPLE OF SUPERPOSITION Suppose two mechanical waves of the same type pass through the same region of space. Do the waves affect each other? If the amplitudes of the waves are large enough, then particles in the medium are displaced far enough from their equilibrium positions that Hooke’s law (restoring force displacement) no longer holds; in that case the waves do affect each other. However, for small amplitudes, the waves can pass through each other and emerge unchanged . A more general statement is that, when the amplitudes are not too large, the principle of superposition applies to mechanical waves: Principle of superposition When two or more waves overlap, the net disturbance at any point is the sum of the individual disturbances due to each wave. Whenever waves are described by the wave equation, Eq. 25-9, the principle of superposition is valid. Suppose that y 1 ( x , t ) and y 2 ( x , t ) are two solutions of the wave equation. Then 22 22 11 11 2222 and y yyy txtx ∂∂∂∂ == ∂∂ 22 vv Adding the two wave equations and combining terms, ( ) ( ) 12 y yy y tx ∂+ = 2 v Then y ( x , t ) = y 1 ( x , t ) + y 2 ( x , t ) is also be a solution of the wave equation. The principle of superposition holds because the wave equation is a linear partial differential equation: each term contains only y or one of its derivatives to the first power.

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2 Checkpoint For which of the following wave equations does the principle of superposition hold? (1) 24 y y A tx ∂∂ = (2) 22 y yy AB t =+ (3) 3 y t ⎛⎞ ⎜⎟ ⎝⎠ * Suppose two wave pulses are traveling toward each other on a string (Fig. 26.1a). If one of the pulses (acting alone) would produce a displacement y 1 at a certain point and the other would produce a displacement y 2 at the same point, the result when the two overlap is a displacement of 12 y y + . Figures 26.1b, c show in detail how y 1 and y 2 add together to produce the net displacement when the pulses overlap. The dashed curves represent the individual pulses; the solid line represents the superposition of the pulses. In Fig. 26.1b the pulses are starting to overlap and in Fig. 26.1c they are just about to coincide.
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AG26_Superposition - Chapter 26: The Principle of...

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