Standing Waves

Halfway between adjacent notes are antinodes where

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Unformatted text preview: s. Halfway between adjacent notes are antinodes, where the amplitude of the resultant wave is a maximum. See Figure 2 for a diagram of nodes and antinodes. Wave patterns that do not move left or right are called standing waves. Also the locations of the maxima and minima do not change. 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. Figure 2 Diagram of nodes and antinodes of a standing wave. www.wfu.edu/physics/pira/PhysicsDrawings.htm 4 To analyze a standing wave, we represent the two combining waves with the equations , , sin sin (12) (13) The principle of superposition (equation (11)) yields , sin sin (14) cos (15) cos (16) After applying the trigonometric identity sin sin 2 sin equation (14) becomes , 2 sin The quantity 2 ym sin(kx) in the brackets of equation (16) 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 2 ym 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. This is not true for a standing wave, in which the amplitude varies with position. In the standing wave of equation (16), the amplitude is zero for values of kx that give sin(kx)=0. Those are: for 0,1,2, … (17) By using substituting equation (10) into equation (18) we get (nodes) 2 (18) as the positions of zero amplitude—the nodes—for the standing wave of equation (16). Note that the distance between nodes is λ/2, half a wavelength. The amplitude of the standing wave of equation (16) has a maximum value of 2y, which occurs for values of kx that give | sin(kx)=1 |. Those values are , 1 2 (19) ,… for (20) 0, 1, 2, … By substituting equation (10) into equation (20) we get 1 22 (antinodes) (21) as the positions of maximum amplitude—the antinodes—of the standing wave of equation (16). The antinodes are separated by λ/2 and are located halfway between pair of nodes. 5 2.7 Standing Waves and Resonance 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 waves reach 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. In short, we very soon have many overlapping traveling waves, which interfere with one another. For certain frequ...
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This document was uploaded on 03/20/2014 for the course PHYS 215 at Lafayette.

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