04 Wave Motion, Superposition, Standing Waves

Two waves of equal amplitude and frequency are

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Two waves of equal amplitude and frequency are displayed on a CRO along with their resultant. By shifting the phase between the two both constructive and destructive interference can be shown. Because our hearing sensitivity is logarithmic, it is very difficult to get complete cancellation: if the waves cancel to 99% you don't see much of a wave, but you can usually hear a sound that is only 40dB down.

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Superposition of Sinusoidal Waves Assume two waves are travelling in the same direction, with the same frequency, wavelength and amplitude The waves differ in phase y 1 = A sin ( kx - ω t ) y 2 = A sin ( kx - ω t + φ ) y = y 1 + y 2 = 2 A cos ( φ /2) sin ( kx - ω t + φ /2) Sinusoidal Same frequency and wavelength as original waves Amplitude 2 A cos ( φ /2) Phase φ /2 See Active Figure 18.03 Remember sin A + sin B = 2sin A + B 2 cos A B 2
Sinusoidal Waves with Constructive Interference When φ = 0, then cos ( φ /2) = 1 Amplitude of resultant wave = 2 A The crests of one wave coincide with the crests of the other wave Waves are everywhere in phase Waves interfere constructively y =2 A cos ( φ /2) sin ( kx - ω t + φ /2)

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Sinusoidal Waves with Destructive Interference When φ = π , then cos ( φ /2) = 0 Also any odd multiple of π The amplitude of the resultant wave is 0 Crests of one wave coincide with troughs of the other wave The waves interfere destructively y =2 A cos ( φ /2) sin ( kx - ω t + φ /2)
Sinusoidal Waves, General Interference When φ is other than 0 or an even multiple of π , the amplitude of the resultant is between 0 and 2 A The wave functions still add y =2 A cos ( φ /2) sin ( kx - ω t + φ /2)

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Summary of Interference Constructive interference occurs when: φ = 0 Amplitude of the resultant is 2 A Destructive interference occurs when: φ = n π where n is an odd integer Amplitude is 0 General interference occurs when: 0 < φ < 2 π Amplitude is 0 < A resultant < 2 A y =2 A cos ( φ /2) sin ( kx - ω t + φ /2) y 1 = A sin ( kx - ω t ), y 2 = A sin ( kx - ω t + φ )
Standing Waves Active Figure 18.08 The diagrams above show standing-wave patterns produced at various times by two waves of equal amplitude travelling in opposite directions In a standing wave, the elements of the medium alternate between the extremes shown in (a) and (c)

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Standing Waves: I Assume two waves with the same amplitude, frequency and wavelength, travelling in opposite directions in a medium y 1 = A sin ( kx ω t ) y 2 = A sin ( kx + ω t ) They interfere according to the superposition principle y = y 1 +y 2 = 2 A sin ( kx ) cos( ω t)
Standing Waves: II The resultant wave will be y = (2 A sin kx ) cos ω t This is the wave function of a standing wave There is no ( kx ω t ) term, and therefore it is not a travelling wave In observing a standing wave, there is no sense of motion in the direction of propagation of either of the original waves 2 A sin kx

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Standing Waves: III A node occurs at a point of zero amplitude i.e. when sin( kx) =sin(2 π x / λ ) =0 These correspond to positions of x where An antinode

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