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Chapter 18

PHYSICS 1E03 Chapter 18: Super Position and Standing waves

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McMaster University
David Chettle

Chapter 18: Superposition and Standing Waves  Analysis Model: Waves in interference o 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 functions of the individual waves  Linear waves obey this. o Amplitudes of these waves are much smaller than their wavelengths  Non-linear waves violate this o Characterized by large amplitudes o Interference  The combination of separate waves in the same region of space to produce a resultant wave  Constructive and destructive: o The superposition principle is the centerpiece of the analysis model called waves in interference. o In many cases, waves combine according to this principle and exhibit interesting phenomena with practical applications  Superposition of Sinusoidal Waves o If we have two waves, traveling in the same direction with the same frequency, wavelength, and amplitude, differing only in phase, we can express their wave function as y = y1 + y2 o The result is this:  The resultant wave function y also is sinusoidal and has the same frequency and wavelength as the individual waves  The amplitude of the resultant wave is 2Acos(phi/2)  In general, constructive interference occurs when cos(phi/2) = +/- 1  True when Phi = 0, 2π, 4π …  Destructive interference occurs when cos(phi/2) = 0  When the crest of one wave occurs at the same time the trough of another wave occurs  Interference of Sound waves o Sound from loudspeaker S is sent into a tube at point P where there is a T-shaped junction. o Half the sound energy travels in one direction, half the other way o The sound waves that reach the receiver R can travel along either path o Distance along any path from the speaker to the receiver is called path length r. Can create constructive or destructive interference by sliding the metal covering up or down on the tube  Standing Waves: o Suppose we had two speakers, and they faced each other: o Two identical waves travel in opposite directions in the same medium. o The waves combine according to the interference model. o The standing wave function is as follows:  o Standing waves have no sense of motion in the direction of propagation of either original wave. o Describes a special type of Simple harmonic motion. o Points of zero amplitude are called nodes  Located when x = 0, o Points of greatest amplitude are called antinodes.  Located where x = o Features:  The distance between adjacent antinodes is equal to λ/2  The distance between adjacent nodes is the same as above  The distance between a node an
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