Page 1
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,)!.
Page 2
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.
Page 3
Page 1 of 1
Stickies 3/30/14, 3:53 PM
Page 1 of 1
Stickies 3/30/14, 3:54 PM
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¬&inverted
Page 4
Page 1 of 1
Stickies 3/29/14, 11:56 PM
Law$of$reflection$of$waves$at$a$boundary$or$interface:
θi =θr
Page 5
Chapters)11)and)12
Sound&and&Standing&Waves
Page 6
The$Nature$of$Sound$Waves
LONGITUDINAL*SOUND*WAVES
Speaker'makingsound'waves'ina'tube
Page 7
The$Nature$of$Sound$Waves
The$distance$between$adjacent$condensations$is$equal$to$the$wavelength$of$the$sound$wave.
Page 8
The$Nature$of$Sound$Waves
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.
Page 9
The$Nature$of$Sound$Waves
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$pitch.
Page 10
The$Nature$of$Sound$Waves
THE$PRESSURE$AMPLITUDE$OF$A$SOUND$WAVE
Loudness$is$an$attribute$ofa$sound$that$depends$primarily$on$the$pressure$amplitudeof$the$wave.
Page 11
The$Speed$of$Sound
Sound&travels&through&gases,&liquids,&and&solids&at&considerablydifferent&speeds.
Page 12
The$Speed$of$Sound
Conceptual$Example:$$Lightning,(Thunder,(and(a(Rule(of(Thumb
There%is%a%rule%of%thumb%for%estimating%how%far%away%a%thunderstorm%is.After%you%see%a%flash%of%lighting,%count%off%the%seconds%until%the%thunder%is%heard.%%Divide%the%number%of%seconds%by%five.%%The%result%gives%theapproximate%distance%(in%miles)%to%the%thunderstorm.%%Why%does%thisrule%work?
Page 13
The$Speed$of$Soundvsound 20o C( ) = 343 m/s× 1 mile
1600 m= 0.214 miles/s
vlight = c = 3.00×108 m/s× 1 mile1600 m
=188, 000 miles/s
1) Time for a light flash to travel distance x in miles
tlight =xc=
x188, 000 miles/s
= (5.32×10−6 s/mile)x
light travels 1 mile in 5.32 µs ⇒ ≈ instantaneous
2) Distance thunder travels x in miles in t seconds
x = vsoundt = 0.214 miles/s( ) t = t4.67 s
miles ≈ t5 s
miles
Page 14
Sound&Intensity
For$a$1000$Hz$tone,$the$smallest$sound$intensity$that$the$human$earcan$detect$is$about$1x10912$W/m2.$$This$intensity$is$called$the$thresholdof&hearing.$$
On$the$other$extreme,$continuous$exposure$to$intensities$greater$than$1$W/m2 can$be$painful.
As$we$saw$before,$if$the$source$emits$sound$uniformly*in*all*directions,*the$intensity$depends$on$the$distance$from$the$source$in$a$simple$way:
I = P4πr2 , P = power emitted from source
r = distance from source
Page 15
Decibels
The$decibel (dB)$is$a$measurement$unit$used$when$comparing$two$soundintensities.$$
Because$of$the$way$in$which$the$human$hearing$mechanism$responds$tointensity,$it$is$appropriate$to$use$a$logarithmic$scale$called$the$intensitylevel:
( ) !!"
#$$%
&=
oII
logdB 10'
212 mW1000.1 !"=oI
Note$that$log(1)=0,$so$when$the$intensity$of$the$sound$is$equal$to$the$threshold$of$hearing,$the$intensity$level$is$zero.$
Page 16
Decibels
Quick review of logarithms:
log x ≡ log10 x = y ⇒ 10y = x
For example, log1= 0 since 100 =1
logA− logB = log AB$
%&
'
() logA+ logB = log AB( )
log AN( ) = N logA
Page 17
Decibels
( ) !!"
#$$%
&=
oII
logdB 10' 212 mW1000.1 !"=oI
Page 18
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.
( ) !!"
#$$%
&=
oII
logdB 10'
Page 19
Decibels
( ) !!"
#$$%
&=
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
Page 20
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.
Page 21
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
Page 22
Transverse(Standing(Waves
!,4,3,2,1 2
=!"#
$%&= nLvnfnString(fixed(at(both(ends
Page 23
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
Page 24
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
Page 25
Transverse(Standing(Waves
!,4,3,2,1 2
=!"#
$%&= nLvnfn
Changing'the'pitch'of'a'guitar'string'by'fingering'it:'the'smaller'you'makeL,'the'higher'the'pitch.
Page 26
Longitudinal+Standing+Waves
A"longitudinal"standing"wave"pattern"on"a"slinky.
Page 27
Longitudinal+Standing+Waves
!,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$!
Page 28
Longitudinal+Standing+Waves
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.
Page 29
Longitudinal+Standing+Waves
!,4,3,2,1 2
=!"#
$%&= nLvnfn
( )( ) m 656.0
Hz 261.62sm3431
2 ===
nfnvL
Page 30
Longitudinal+Standing+Waves
!,5,3,1 4
=!"#
$%&= nLvnfnTube+open+at+one+end
One-quarter of the longest wavelength, λ1, can fit in the tube of length L
⇒ L = λ1
4⇒ f1 =
vλ1
=v
4L
Standing(sound(waves(in(a(tube(open(at(one(end
L
1st harmonic 3rd harmonic,/1st overtone
Page 31
Example:)The$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
Page 32
Longitudinal+Standing+Waves
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
Page 33
Complex(Sound(Waves
Page 34
Complex(Sound(Waves
Fourier'spectrum
Page 35
Complex(Sound(Waves