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)

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

Standing Waves: graphical (I)• Two sine waves (same f, A, ) traveling in opposite

directions: superposition is wave, but not movingλ

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

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

Transverse standing waves• generate standing waves on string fixed at

both ends: traveling wave encounters a boundary...

(phase change of on reflection)

π

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

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 ⇒

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

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