Lectures on standing waves

Lectures on standing waves - Superposition Principle •...

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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 – ω ω ω ω t) y 2 = A sin ( k x – ω ω ω ω t + φ ) • The resultant wave is given by: Notes y = y 1 +y 2 = 2A cos ( φ φ φ φ /2) sin (k x – ω ω ω ω t + φ φ φ φ /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 ( φ φ φ φ /2) sin (k x – ω ω ω ω t + φ φ φ φ /2) • The resultant wave has the same frequency and wavelength as the original waves Notes • The amplitude...
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Lectures on standing waves - Superposition Principle •...

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