Defining'wave'fronts'and'rays. - Physicshumanic/p1200_lecture24.pdf · Wave%Fronts%and%Rays...

Wave%Fronts%and%Rays

Defining'wave'fronts'and'rays.

Consider)a)sound)wave)since)itis)easier)to)visualize.

Shown)is)a)hemispherical)view)of)a)sound)wave)emitted)by)a)pulsating)sphere.

The)rays are)perpendicularto'the'wave'fronts (e.g.)crests))which)are)separated)from)each)other)by)the)wavelength)of)the)wave,)!.

Wave%Fronts%and%Rays

The$positions$of$two$spherical$wave$fronts$are$shown$in$(a)$with$theirdiverging$rays.$

At$large$distances$from$the$source,$the$wave$fronts$become$lessand$less$curved$and$approach$the$limiting$case$of$a$plane&waveshown$in$(b).$A$plane$wave$has$flat$wave$fronts$and$rays$parallel$toeach$other.

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Reflection*of*string*pulses*at*boundaries*and*interfaces

}}

From&less&dense&mediumto&more&dense&medium

From&more&dense&mediumto&less&dense&medium

Reflection&inverted&&&&&&&&&Reflection&not&inverted

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Law$of$reflection$of$waves$at$a$boundary$or$interface:

θi =θr

Chapters)11)and)12

Sound&and&Standing&Waves

The$Nature$of$Sound$Waves

LONGITUDINAL*SOUND*WAVES

Speaker'makingsound'waves'ina'tube


The$distance$between$adjacent$condensations$is$equal$to$the$wavelength$of$the$sound$wave.


Individual)air)molecules)are)not)carried)along)with)the)wave.

When)the)sound)hits)your)ear)it)causes)your)eardrum)to)vibrate)and)your)braininterprets)the)vibration)as)thepitch and)loudness of)thesound.


THE$FREQUENCY$OF$A$SOUND$WAVE

The$frequency is$the$number$of$cyclesper$second.

A$sound$with$a$single$frequency$is$calleda$pure$tone.

The$brain$interprets$the$frequency$in$termsof$the$subjective$quality$called$pitch.


THE$PRESSURE$AMPLITUDE$OF$A$SOUND$WAVE

Loudness$is$an$attribute$ofa$sound$that$depends$primarily$on$the$pressure$amplitudeof$the$wave.

The$Speed$of$Sound

Sound&travels&through&gases,&liquids,&and&solids&at&considerablydifferent&speeds.

The$Speed$of$Sound

Conceptual$Example:$$Lightning,(Thunder,(and(a(Rule(of(Thumb

There%is%a%rule%of%thumb%for%estimating%how%far%away%a%thunderstorm%is.After%you%see%a%flash%of%lighting,%count%off%the%seconds%until%the%thunder%is%heard.%%Divide%the%number%of%seconds%by%five.%%The%result%gives%theapproximate%distance%(in%miles)%to%the%thunderstorm.%%Why%does%thisrule%work?

The$Speed$of$Soundvsound 20o C( ) = 343 m/s× 1 mile

1600 m= 0.214 miles/s

vlight = c = 3.00×108 m/s× 1 mile1600 m

=188, 000 miles/s

1) Time for a light flash to travel distance x in miles

tlight =xc=

x188, 000 miles/s

= (5.32×10−6 s/mile)x

light travels 1 mile in 5.32 µs ⇒ ≈ instantaneous

2) Distance thunder travels x in miles in t seconds

x = vsoundt = 0.214 miles/s( ) t = t4.67 s

miles ≈ t5 s

miles

Sound&Intensity

For$a$1000$Hz$tone,$the$smallest$sound$intensity$that$the$human$earcan$detect$is$about$1x10912$W/m2.$$This$intensity$is$called$the$thresholdof&hearing.$$

On$the$other$extreme,$continuous$exposure$to$intensities$greater$than$1$W/m2 can$be$painful.

As$we$saw$before,$if$the$source$emits$sound$uniformly*in*all*directions,*the$intensity$depends$on$the$distance$from$the$source$in$a$simple$way:

I = P4πr2 , P = power emitted from source

r = distance from source

Decibels

The$decibel (dB)$is$a$measurement$unit$used$when$comparing$two$soundintensities.$$

Because$of$the$way$in$which$the$human$hearing$mechanism$responds$tointensity,$it$is$appropriate$to$use$a$logarithmic$scale$called$the$intensitylevel:

( ) !!"

#$$%

&=

oII

logdB 10'

212 mW1000.1 !"=oI

Note$that$log(1)=0,$so$when$the$intensity$of$the$sound$is$equal$to$the$threshold$of$hearing,$the$intensity$level$is$zero.$

Decibels

Quick review of logarithms:

log x ≡ log10 x = y ⇒ 10y = x

For example, log1= 0 since 100 =1

logA− logB = log AB$

%&

'

() logA+ logB = log AB( )

log AN( ) = N logA

Decibels

( ) !!"

#$$%

&=

oII

logdB 10' 212 mW1000.1 !"=oI

Decibels

Example:..Comparing*Sound*Intensities

Audio&system&1&produces&a&sound&intensity&level&of&90.0&dB,&and&system2&produces&an&intensity&level&of&93.0&dB. Determine&the&ratio&of&intensities.

( ) !!"

#$$%

&=

oII

logdB 10'

Decibels

( ) !!"

#$$%

&=

oII

logdB 10'

( ) !!"

#$$%

&=

oII1

1 logdB 10' ( ) !!"

#$$%

&=

oII2

2 logdB 10'

( ) ( ) ( ) ( ) !!"

#$$%

&=!!

"

#$$%

&=!!

"

#$$%

&'!!"

#$$%

&='

1

2

1

21212 logdB 10logdB 10logdB 10logdB 10

II

IIII

II

II

o

o

oo

((

( ) !!"

#$$%

&=

1

2logdB 10dB 0.3II

0.210 30.0

1

2 ==II

!!"

#$$%

&=

1

2log0.30II

Transverse(Standing(Waves

In#reflecting#from#the#wall,#aforward3traveling#half3cyclebecomes#a#backward3travelinghalf3cycle#that#is#inverted.

Unless#the#timing#is#right,#thenewly#formed#and#reflected#cyclestend#to#offset#one#another.

Repeated#reinforcement#betweennewly#created#and#reflected#cyclescauses#a#large#amplitude#standingwave#to#develop.

Transverse(Standing(WavesTransverse(standing(wave(patterns(on(a(string.

One-half of the longest wavelength, λ1, can fit on the string of length L

⇒ L = λ1

2⇒ f1 =

vλ1

=v

2L

node%%%%%%anti)node

L


!,4,3,2,1 2

=!"#

$%&= nLvnfnString(fixed(at(both(ends

4/5/2014 Note Frequencies

http://www.seventhstring.com/resources/notefrequencies.html 1/2

Note Frequencies Seventh String Contact

Transcribe! Overview Screen shots Reviews Download Buy Affiliates Video Pedals Lost Licenses History FAQ

Fake Book Index Search

Utilities Metronome Tuner Tuning Fork

Resources Transcription How to Transcribe How to Slow Down Note Frequencies Other

Here is a table giving the frequencies in Hz of musical pitches, covering the full range of allnormal musical instruments I know of and then some. It uses an even tempered scale with A= 440 Hz.

C C# D Eb E F F# G G# A Bb B0 16.35 17.32 18.35 19.45 20.60 21.83 23.12 24.50 25.96 27.50 29.14 30.871 32.70 34.65 36.71 38.89 41.20 43.65 46.25 49.00 51.91 55.00 58.27 61.742 65.41 69.30 73.42 77.78 82.41 87.31 92.50 98.00 103.8 110.0 116.5 123.53 130.8 138.6 146.8 155.6 164.8 174.6 185.0 196.0 207.7 220.0 233.1 246.94 261.6 277.2 293.7 311.1 329.6 349.2 370.0 392.0 415.3 440.0 466.2 493.95 523.3 554.4 587.3 622.3 659.3 698.5 740.0 784.0 830.6 880.0 932.3 987.86 1047 1109 1175 1245 1319 1397 1480 1568 1661 1760 1865 19767 2093 2217 2349 2489 2637 2794 2960 3136 3322 3520 3729 39518 4186 4435 4699 4978 5274 5588 5920 6272 6645 7040 7459 7902

The octave number is in the left column so to find the frequency of middle C which is C4,look down the "C" column til you get to the "4" row : so middle C is 261.6 Hz.

Some Specific Notes

Middle C is C4=261.6Hz

Standard tuning fork A is A4=440Hz

Piano range is A0=27.50Hz to C8=4186Hz

Guitar strings are E2=82.41Hz, A2=110Hz, D3=146.8Hz, G3=196Hz, B3=246.9Hz,E4=329.6Hz

Bass strings are (5th string) B0=30.87Hz, (4th string) E1=41.20Hz, A1=55Hz, D2=73.42Hz,G2=98Hz

Mandolin & violin strings are G3=196Hz, D4=293.7Hz, A4=440Hz, E5=659.3Hz

Viola & tenor banjo strings are C3=130.8Hz, G3=196Hz, D4=293.7Hz, A4=440Hz

Cello strings are C2=65.41Hz, G2=98Hz, D3=146.8Hz, A3=220Hz

Coda

Bear in mind that everything here is in relation to the even tempered (aka equal tempered)scale, where an octave is a frequency ratio of exactly two and a semitone is a frequency ratioof exactly the twelfth root of two. In the real world however many different temperamentsmay be used -‐‑ see en.wikipedia.org/wiki/Musical_temperament -‐‑ and octaves too can vary insize, see en.wikipedia.org/wiki/Stretched_octave.

Also we call middle C "C4" : this is the commonest octave numbering but some people callmiddle C "C3" or even "C5".

Recommend this page to others, on these social bookmarking sites:

Note%frequencies%(Hz)%of%the%chromatic%music%scale

Example:)The$A$string$on$a$violin$has$a$fundamental$frequency$of$440$Hz.$The$length$of$the$string$is$32.0$cm$and$it$has$a$mass$of$0.450$g.$Under$what$tension$must$the$string$be$placed?

v = FTm L

⇒ FT = v2 mL

f1 =v

2L⇒ v = 2Lf1

∴FT = 2Lf1( )2 mL= 4Lf1

2m = 4 0.320( ) 440( )2 0.450×10−3( ) =112 N


!,4,3,2,1 2

=!"#

$%&= nLvnfn

Changing'the'pitch'of'a'guitar'string'by'fingering'it:'the'smaller'you'makeL,'the'higher'the'pitch.

Longitudinal+Standing+Waves

A"longitudinal"standing"wave"pattern"on"a"slinky.


!,4,3,2,1 2

=!"#

$%&= nLvnfnTube+open+at+both+ends

Standing(sound(waves(in(a(tube(open(at(both(ends

The$anti)nodes$occur$at$the$open$ends$of$the$tube.

One-half of the longest wavelength, λ1, can fit in the tube of length L

⇒ L = λ1

2⇒ f1 =

vλ1

=v

2L

L

Note$that$the$string$fixed$at$both$ends$and$the$tube$open$at$both$endshave$the$same$equation$for$the$standing$wave$frequencies$!


Example:++Playing(a(Flute

When(all(the(holes(are(closed(on(one(type(offlute,(the(lowest(note(it(can(sound(is(middleC((261.6(Hz).((If(the(speed(of(sound(is(343(m/s,and(the(flute(is(assumed(to(be(a(cylinder(openat(both(ends,(determine(the(distance(L.


!,4,3,2,1 2

=!"#

$%&= nLvnfn

( )( ) m 656.0

Hz 261.62sm3431

2 ===

nfnvL


!,5,3,1 4

=!"#

$%&= nLvnfnTube+open+at+one+end

One-quarter of the longest wavelength, λ1, can fit in the tube of length L

⇒ L = λ1

4⇒ f1 =

vλ1

=v

4L

Standing(sound(waves(in(a(tube(open(at(one(end

L

1st harmonic 3rd harmonic,/1st overtone

Example:)The$fundamental$frequency$of$an$open$organ$pipe$corresponds$to$the$note$D2$(f1 =$73.42$Hz$on$the$chromatic$musical$scale).$The$third$harmonic$of$another$organ$pipe$that$is$closed$at$one$end$has$the$same$frequency.$Compare$the$lengths$of$these$two$pipes.$

!,4,3,2,1 2

=!"#

$%&= nLvnfn

!fn = !n v4 !L"

#$

%

&' !n =1,3, 5,…

Tube$open$at$both$ends

Tube$closed$at$one$end

f1 =v

2L⇒ L = v

2 f1=

3432 73.42( )

= 2.34 m

!f3 = 3 v4 !L"

#$

%

&'= f1 ⇒ !L =

3v4 f1

=3 343( )

4 73.42( )= 3.50 m


Tube+open+at+one+end

Conceptual+Example:+Why+does+inhaling+Helium+raise+the+pitch+of+a+voice?

!"Warning:"It"is"dangerous"to"inhale"Helium"!

!,5,3,1 4

=!"#

$%&= nLvnfn

Assume&that&the&mouth/larynx&system&acts&as&a&tube&open&at&one&end&oflength&0.25&m.&

In&air,&where&vsoundair&=&343&m/s,&the&fundamental&frequency&of&a&voice&is

In&Helium,&where&vsoundHelium =&965&m/s,&the&fundamental&frequency&of&a&voice&is

f1air =

vsoundair

4L=

3434 0.25( )

= 343 Hz ⇒ ≈ an octave 4 F note

f1Helium =

vsoundHelium

4L=

9654 0.25( )

= 965 Hz ⇒ ≈ an octave 5 B note

Complex(Sound(Waves

Complex(Sound(Waves

Fourier'spectrum

Complex(Sound(Waves

Defining'wave'fronts'and'rays. - Physicshumanic/p1200_lecture24.pdf · Wave%Fronts%and%Rays...

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