From Last Time… Wave Properties - Department of...

Fri. Sep 17 Phy107 Lecture 6

From Last Time… Wave Properties

• Amplitude is the maximum displacement ofstring above the equilibrium position

• Wavelength, λ, is the distance between twosuccessive points that behave identically

• Period: required to complete one cycle• Frequency = 1/Period = rate at which cycles

are completed

• Velocity = Wavelength/Period,v = λ / T, or v = λf


Hour Exam #1 Scores

0

5

10

15

20

25

30

15 30 45 60 75 90 105

HOUR EXAM 1 SCORES

# SC

ORES

SCORE ( % )

Mean = 65%Min = 27 %Max = 93%

65% =high BC


Quality of Sound – Tuning Fork

• Tuning fork producesonly thefundamentalfrequency


Quality of Sound – Flute

• The same note playedon a flute soundsdifferently

• The second harmonic isvery strong

• The fourth harmonic isclose in strength to thefirst


Quality of Sound – Clarinet

• The fifth harmonic isvery strong

• The first and fourthharmonics are verysimilar, with the thirdbeing close to them


Timbre

• In music, the characteristic sound of anyinstrument is referred to as the quality ofsound, or the timbre of the sound

• The quality depends on the mixture ofharmonics in the sound


Pitch

• Pitch is related mainly, although notcompletely, to the frequency of the sound

• Pitch is not a physical property of the sound• Frequency is the stimulus and pitch is the

response– It is a psychological reaction that allows humans

to place the sound on a scale


Frequency Response Curves

• Bottom curve is thethreshold of hearing– Threshold of hearing is

strongly dependent onfrequency

– Easiest frequency tohear is about 3000 Hz

• When the sound isloud (top curve,threshold of pain) allfrequencies can beheard equally well


Combining waves• Two traveling waves can meet and pass through

each other without being destroyed or evenaltered

• Waves obey the Superposition Principle– If two or more traveling waves are moving through a

medium, the resulting wave is found by addingtogether the displacements of the individual wavespoint by point

– Constructive interference: waves reinforce– Destructive interference: waves tend to cancel


Constructive Interference in a String

• Two pulses are traveling inopposite directions

• The net displacement whenthey overlap is the sum ofthe displacements of thepulses

• Note that the pulses areunchanged after theinterference


Constructive Interference

• Two waves, a and b, havethe same frequency,amplitude, and start point– Are in phase

• The combined wave, c, hasthe same frequency and agreater amplitude

Combined wave

+

=


Destructive Interference in aString

• Two pulses are traveling inopposite directions

• The net displacement whenthey overlap thedisplacements of the pulsessubtract

• Note that the pulses areunchanged after theinterference


Destructive Interference

• Two waves, a and b, havethe same amplitude andfrequency

• They are 1/2 wavelengthout of phase

• When they combine, thewaveforms cancel

+

=


Interference of sound waves

• Interference arises when waves change their‘phase relationship’.

• Can vary phase relationship of two waves bychanging physical location of speaker.

Constructive Destructive

‘in-phase’ ‘1/2 λ phase diff’


Example

• Speed of sound ~ 340 m/s• So f=340 Hz gives λ=v/f = 1 meter• Change of 1/2 wavelength is 1/2 meter.


Spherical Waves

• A spherical wavepropagates radiallyoutward from theoscillating sphere

• The energypropagates equallyin all directions


Representations of Waves

• Wave fronts are theconcentric arcs– The distance between

successive wave fronts isthe wavelength

• Rays are the radial linespointing out from thesource andperpendicular to thewave fronts


Shock Waves and Sonic Booms

• A shock waveresults when thesource velocityexceeds the speedof the wave itself

• The circlesrepresent thewave frontsemitted by thesource

Fig 14.11, p. 439

Slide 15


Sonic Boom

• What happens if the source of soundapproaching the listener is equal toor faster than the speed of sound

• Each successive wave is superimposedon the previous one

– If the speed is exceeded• No sound received till after the source

passes the listener - then a sonic boom- followed by normal sound from thesource

• Conical bow wake• Much like triangular

bow wake behind aduck swimming in apond


Superposition of Unequal FrequencyWaves

• Consider two harmonic waves A and B meeting atx=0.– Same amplitudes, but ω2 = 1.15 ω1.

• The displacement versus time for each is shownbelow:

What does C(t) = A(t) + B(t) look like??

A(ω1t)

B(ω2t)


Superposition & Interference• Consider two harmonic waves A and B meeting at x = 0.

– Same amplitudes, but ω2 = 1.15 x ω1.• The displacement versus time for each is shown below:

A(ω1t)

C(t) = A(t) + B(t)CONSTRUCTIVEINTERFERENCE

DESTRUCTIVEINTERFERENCE

B(ω2t)


Beats

( ) ( )tcostcosA2)tcos(A)tcos(A HL21 ωω=ω+ω

( )ω ω ωL = −12 1 2 ( )21H 2

1ω+ω=ω

The ‘beat’ freqency is exactly half the difference of the twosource freqeuncies.

The ‘pitch’ is the average (half the sum) of the two sourcefrequencies

where and

cos(ωLt)


Doppler Effect

• A Doppler effect is experienced wheneverthere is relative motion between a source ofwaves and an observer.– When the source and the observer are moving

toward each other, the observer hears a higherfrequency

– When the source and the observer are movingaway from each other, the observer hears alower frequency

• Although the Doppler Effect is commonlyexperienced with sound waves, it is aphenomena common to all waves


Doppler Effect, Case 1• An observer is moving

toward a stationarysource

• Due to his movement,the observer detectsan additional numberof wave fronts

• The frequency heard isincreased

Fig 14.8, p. 435

Slide 12


Doppler Effect, Case 2

• An observer ismoving away from astationary source

• The observerdetects fewer wavefronts per second

• The frequencyappears lower

Fig 14.9, p. 436

Slide 13


Doppler Effect, Source inMotion

• As the source movestoward the observer (A),the wavelength appearsshorter and the frequencyincreases

• As the source moves awayfrom the observer (B),the wavelength appearslonger and the frequencyappears to be lower

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