6B_Chapter_14_GG

6B_Chapter_14_GG - Chapter 14 Superposition and Standing...

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Chapter 14 Superposition and Standing Waves
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Superposition Principle • If two or more traveling waves are moving through a medium and combine at a given point, the resultant position of the element of the medium at that point is the sum of the positions due to the individual waves • Waves that obey the superposition principle are linear waves – In general, linear waves have amplitudes much smaller than their wavelengths
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Superposition Example 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 to the left is y 2 The pulses have the same speed but different shapes The displacement of the elements is positive for both When the waves start to overlap (b), the resultant wave function is y 1 + y 2 The shapes of the pulses remain unchanged
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Superposition in a Stretch Spring • Two equal, symmetric pulses are traveling in opposite directions on a stretched spring • They obey the superposition principle
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Superposition and Interference • Two traveling waves can pass through each other without being destroyed or altered – A consequence of the superposition principle • The combination of separate waves in the same region of space to produce a resultant wave is called interference
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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 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
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Destructive Interference Example • Two pulses traveling in opposite directions • Their displacements are inverted with respect to each other • When they overlap, their displacements partially cancel each other
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Superposition of Sinusoidal Waves • Assume two waves are traveling 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 - t + φ ) y = y 1 + y 2 = 2 A cos ( /2) sin ( kx - t + /2)
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Superposition of Sinusoidal Waves, cont • The resultant wave function, y , is also sinusoidal • The resultant wave has the same frequency and wavelength as the original waves • The amplitude of the resultant wave is 2 A cos ( φ / 2) • The phase of the resultant wave is /2
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Sinusoidal Waves with Constructive Interference • When φ = 0, then cos ( /2) = 1 • The amplitude of the resultant wave is 2 A – The crests of one wave coincide with the crests of the other wave • The waves are everywhere in phase • The waves interfere constructively
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Sinusoidal Waves with Destructive Interference • When φ = π , then cos ( /2) = 0 – Also any even multiple of • The amplitude of the resultant wave is 0 – Crests of one wave coincide with troughs of the other wave • The waves interfere destructively
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6B_Chapter_14_GG - Chapter 14 Superposition and Standing...

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