UMD Physics - Lecture 10 · 2008. 2. 26. · Musical Instruments • fundamental mode: 1/4 in...
Transcript of UMD Physics - Lecture 10 · 2008. 2. 26. · Musical Instruments • fundamental mode: 1/4 in...
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Lecture 10• this week:
superposition (combination of 2 or more waves) applications to lasers, musical instruments...
• today: basic principle standing waves (2 waves traveling in opposite direction)
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Principle of Superposition• Two particles can’t occupy same point of
at same time, waves can... (pass thru’ each other)
• displacement of medium due to 2 or more waves present at same time and same point is given by sum of displacements due to each wave
Dnet =∑n
i=1 Di
observe solid blue line
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Standing Waves: graphical (I)• Two sine waves (same f, A, ) traveling in opposite
directions: superposition is wave, but not movingλ
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Standing Waves: graphical (II)• Nodes (point never move, apart):
displacements of 2 waves have same magnitude, but opposite sign at all times (destructive interference)
• Antinodes (maximum amplitude): two waves in phase, net displacement twice... (constructive interference)
• intensity : maximum/zero at antinodes/nodes
λ/2
∝ A2
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Standing Waves: mathematical
• not a function of , not a traveling wave: each point has SHM with amplitude A(x)
• nodes: (spacing of nodes )
DR = a sin(kx− ωt), DL = a sin(kx + ωt)D(x, t) = DL + DR = A(x) cos ωt
amplitude function
λ/2A = 0 for kxm = 2πλ xm = mπ ⇒ xm = m
λ2
(x± vt)
several times
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Transverse standing waves• generate standing waves on string fixed at
both ends: traveling wave encounters a boundary...
(phase change of on reflection)
π
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Standing waves on string• Reflection does not change f, • boundary condition (constraint obeyed at edge):
D = 0 at ends (nodes)
• allowed standing waves:
• normal modes: fundamental frequency:
only envelope shown
A(x) = sin kx = 0 at x = 0 and at x = Lif sin kL = 0 ⇒ kL = mπ
f1 = v2L , λ = 2L (only half λ and no node in-between)
harmonics: fm = mf1 (m = 2, 3, ...) (m− 1 nodes and m antinodes in-between)
λ
master formulae
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Standing EM waves
• light wave has node at each mirror...similar to string....
• microwave oven: turntable to avoid part of food being always a node
λ ∼ cm ⇒
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Standing Sound Waves (I)
• closed end: node• open end (sound wave partly reflected back into
tube): antinode not literally a snapshot, size of tube unrelated to A
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Musical Instruments
• fundamental mode: 1/4 in length L (cf.1/2 for closed-closed or open-open) frequency is half of open-open/closed-closed...
• stringed: fundamental frequency: change density or tension
• wind: change effective length by covering holes/opening valves
Standing Sound Waves (II)λ
speed along string
f1 = v2L =1
2L
√Tsµ
sound speed in air