Lecture 7• this/next week:

superposition (combination of 2 or more waves) applications to lasers, musical instruments...

• today: basic principle standing waves (2 waves traveling in opposite direction)

Example of Doppler effect

• A friend of yours is loudly singing a single note at 400 Hz while racing toward you at 25.0 m/s on a day when the speed of sound is 340 m/s. What frequency (a) do you hear?(b) does your friend hear if you suddenly start singing at 400 Hz?

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 = 2π

λ xm = mπ ⇒ xm = mλ2

(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

Example• A 121-cm-long, 4.0 g string oscillates in its m=3 mode

with a frequency of 180 Hz and a maximum amplitude of 5.0 mm. What are (a) the wavelength and (b) the tension in the string?

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