Wa7 interference of waves phaseshifter mixer two

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Wa7: Interference of waves – phase/shifter mixer 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. 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
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28 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) Sinusoidal Waves with Destructive Interference ± When ± = ³ , then cos ( ± /2) = 0 ± Also any odd multiple of ³ ± 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 - ² t + ± /2)
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29 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) 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 + ± )
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30 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) 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)
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