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Unformatted text preview: Superposition Principle Multiple waves simultaneously can be present in the same medium 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 Notes y resultant (x,t) = y 1 (x,t)+y 2 (x,t)+ Waves that obey the superposition principle are linear waves Waves of small amplitudes are typically linear The waves do not interact with each other they can pass through each other without changing shape or speed Superposition Example: Constructive Interference Two pulses are traveling in opposite directions The wave function of the pulse moving to the right is y 1 and for the one moving the left is The combination of separate waves in the same region of space to produce a resultant wave is called interference Notes to the left is y 2 The pulses have the same speed but different shapes The displacement of the elements is positive for both Superposition Example: Constructive Interference The resultant wave function is y 1 (x,t)+ y 2 (x,t) at any moment of time When crest meets crest (c ) the resultant wave has a larger Notes amplitude than either of the original waves this is called constructive interference Superposition Example: Constructive Interference The two pulses separate They continue moving in their original directions Notes The shapes of the pulses remain unchanged Types of Interference Constructive interference occurs when the displacements caused by the two pulses are in the same direction The amplitude of the resultant pulse is greater than either individual pulse estructive interference ccurs when the Notes Destructive interference occurs when the displacements caused by the two pulses are in opposite directions The amplitude of the resultant pulse is less than either individual pulse Destructive Interference Example Two pulses traveling in opposite directions Their displacements are inverted with respect to each other Notes When they overlap, their displacements partially cancel each other Superposition of Sinusoidal Waves Assume two sinusoidal waves are traveling in the same direction , with the same frequency, wavelength and amplitude The waves differ in phase: y 1 = A sin (k x t) y 2 = A sin ( k x t + ) The resultant wave is given by: Notes y = y 1 +y 2 = 2A cos ( /2) sin (k x t + /2) where we have used the trigonometric identity: ( ) ( ) +  = + 2 sin 2 cos 2 sin sin b a b a b a Superposition of Sinusoidal Waves, cont The resultant wave function, y , is also sinusoidal y = y 1 +y 2 = 2A cos ( /2) sin (k x t + /2) The resultant wave has the same frequency and wavelength as the original waves Notes The amplitude...
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 Winter '10
 IlyaKrivorotov
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