PHYA22 Chapter 21.doc

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Physics and Astrophysics
Brian Wilson

Chapter 21 SuperpositionThe Principle of SuperpositionPrinciple of Superposition when two or more waves are simultaneously present at a single point in space the displacement of the medium at that point is the sum of the displacements due to each individual waveStanding WavesStanding Wave the superposition of two waves where the crests and troughs stand in place as the wave oscillatesWe assume that two waves have the same frequency the same wavelength and the same amplitudeNodes points that never move nodes are spaced l2 apartAntinodes points of maximum amplitude and are also spaced l2 apartConstructive Interference when two waves are in phase such that the superposition at that point yields a wave whose amplitude is twice that of the individual wavesDestructive Interference when two waves are out o phase such that the superposition gives a wave with zero amplitudeIntensity is maximum at points of constructive interference and zero at points of destructive interference if the waves have equal amplitudesSinusoidal Wave Traveling to the right DasinkxwtSinusoidal Wave Traveling to the left DasinkxwtNet Displacement of the Medium when both waves are present DxtAxcoswt where Ax2asinkxAmax2a at points where sinkx1Frequency fw2pWave Number k2plAngular Frequency w2pfTransverse Standing WavesWhenever a wave encounters a discontinuity some of the waves energy is transmitted forward and some is reflectedWhen a wave travels from a string with a large linear density thick connected to a string with a small linear density thin wave speed increases faster tension is the sameWhen a wave reflects from a boundary there is no transmitted wave and all the waves energy is reflected the amplitude of a wave reflected from a boundary is unchangedWhen you wiggle a string in the middle the sinusoidal waves travel outward in both directions and soon reach the boundariesoBecause the speed of a reflected wave does not change the wavelength and frequency of a reflected sinusoidal wave are unchangedoReflections at the ends of the string cause two waves of equal amplitude and wavelength to travel in opposite directions along the stringBoundary Condition a mathematical statement of any constraint that must be obeyed at the boundary or edge of a mediumoDisplacements at the boundary must be zero at all timesoStanding Wave Boundary Conditions Dx0 t0 and DxL t0oNodes are required at both ends of the string closedclosed
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