SJ16_waves_1toapage

# By shifting the phase between the two both

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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 - ± t ) ± y 2 = A sin ( kx - ± t + ² ) ± y = y 1 + y 2 = 2 A cos ( ² /2) sin ( kx - ± t + ² /2) ± Sinusoidal ± Same frequency and wavelength as original waves ± Amplitude 2 A cos ( ² /2) ± Phase ² /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 ± = 0, then cos ( ± /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 ( ± /2) sin ( kx - ² t + ± /2)

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Sinusoidal Waves with Destructive Interference ± When ± = ² , then cos ( ± /2) = 0 ± Also any odd multiple of ² ± 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 ( ± /2) sin ( kx - ³ t + ± /2)
Sinusoidal Waves, General Interference ± When ± is other than 0 or an even multiple of ² , the amplitude of the resultant is between 0 and 2 A ± The wave functions still add y =2 A cos ( ± /2) sin ( kx - ³ t + ± /2)

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Summary of Interference ± Constructive interference occurs when ± = 0 ± Amplitude of the resultant is 2 A ± Destructive interference occurs when ± = n ² where n is an odd integer ± Amplitude is 0 ± General interference occurs when 0 < ± < 2 ² ± Amplitude is 0 < A resultant < 2 A y =2 A cos ( ± /2) sin ( kx - ³ t + ± /2) y 1 = A sin ( kx - ³ t ), y 2 = A sin ( kx - ³ t + ± )
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 ± t ) ± y 2 = A sin ( kx + ± t ) ± They interfere according to the superposition principle ± y = y 1 +y 2 = 2 A sin ( kx ) cos( ± t)
Standing Waves: II ± The resultant wave will be y = (2 A sin kx ) cos ± t ± This is the wave function of a standing wave ± There is no ( kx ± 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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