Wave Pheomea Physics 5c Lecture Fourier Aalysis (H&L Sectios 3. 4) (Georgi Chapter )
Admiistravia! Midterm average 68! You did well i geeral! May got the easy parts wrog, e.g. Problem (a) ad 3(a)! erm Paper! 5 pages o a topic relevat to the course! Discuss your topic with me before Sprig Break! ype it usig ay software you like! I do t wat to read your hadwritig! Plots ad pictures may be glued o! Deadlie is 4/5, 5 PM
What We Did Last ime! Studied reflectio of mechaical waves! Similar to reflectio of electromagetic waves! Mechaical impedace is defied by F = ±Zv! For trasverse/logitudial waves:! Useful i aalyzig reflectio Z = [ or K] ρ l! Studied stadig waves! Created by reflectig siusoidal waves! Oscillatio patter has odes ad atiodes! Musical istrumets use stadig waves to produce their distict soud
Goals For oday! Defie Fourier itegral! Fourier series is defied for repetitive fuctios! Discreet values of frequecies cotribute = + + =! Exted the defiitio to iclude o-repetitive fuctios! Sum becomes a itegral! Discuss pulses ad wave packets! Sedig iformatio usig waves! Sigal speed ad badwidth ( ω ω ) f() t a a cos t b si t! Coectio with Quatum Mechaics
Lookig Back! I Lecture #5, we solved the wave equatio ξ( xt, ) = c w ξ( xt, ) t x ω i( kx± ωt) c! Normal-mode solutios # ξ( xt, ) = ξe w = k! Usig Fourier series, we ca make ay arbitrary waveform with liear combiatio of the ormal modes! Example: forward-goig repetitive waves π ω = ξ( xt, ) = f( x ct w ) = ( acos( kx ωt) + bsi( kx ωt) ) = k = cwω! No-repetitive waves also OK if we make #! his makes ω cotiuous A little math work eeded
Fourier Series! For repetitive fuctio f(t) a = f ( t) dt = + + = ( ω ω ) f() t a a cos t b si t a f t tdt = ()cosω = ω = π b f()si t ωtdt! Express cosω t ad siω t with complex expoetials a ib iωt a + ib iωt acosωt+ bsi ωt = e + e am + ibm iωmt a + ib iωt = e + e m= = ( ) = = m = a m = a b m = b ω m = ω
Fourier Series Sum icludes = a + ib! Defie F ad! How do we calculate F? iω t F = ()cos ()si () f t ω tdt + i f t ω tdt = f t e dt F = f() t dt =! It s useful later if I shift the itegratio rage here iω t F = f() t e dt! Now we take it to the cotiuous limit F = a same i f() t = Fe ω = OK because f(t) is repetitive t
Fourier Itegral i π t f() t Fe ω = iω t F = f() t e dt ω = =! Make # iω t F i ωt π f() t = lim F e = lim e ω ω = = ω i t lim Fe ω = d ω π iωt = iωt F( ω) e dω F( ω) lim F ( ) = f t e dt π π! F(ω) is the Fourier itegral of f(t) iωt f() t = F( ω) e dω iωt F( ω) = f( t) e dt π
Fourier Itegral iωt f() t = F( ω) e dω! Fourier itegral F(ω) is! A decompositio of f(t) ito differet frequecies! A alterative, complete represetatio of f(t)! Oe ca covert f(t) ito F(ω) ad vice versa! f(t) is i the time domai! F(ω) is i the frequecy domai iωt F( ω) = f( t) e dt π F(ω) ad f(t) are two equally-good represetatios of a same fuctio
Warig! Differet covetios exist i Fourier itegrals!!! iωt f() t = F( ω) e dω ad iωt f() t = F( ω) e dω π iωt F( ω) = f( t) e dt π ad! Watch out whe you read other textbooks F( ω) = f( t) e iωt dt iωt f() t = iωt F( ω) e dω ad F( ω) = f( t) e dt π π
Square Pulse t < f() t = t > iωt iωt ω F( ω) = f( t) e dt e dt si π = π = πω F() = π! Cosider a short pulse with uit area Fourier! F(ω) is a buch of little ripples aroud ω =! Height is /π! Area is / π ω
Pulse Width ω F( ω) = si π width πω! he shorter the pulse, the wider the F(ω)! Pulse of duratio #! (width i t) (width i ω) = π = cost! his is a geeral feature of Fourier trasformatio! Example: Gaussia fuctio f() t = π e t F( ω) = e π ω
Sedig Iformatio! Cosider sedig iformatio usig waves! Voice i the air! Voice coverted ito EM sigals o a phoe cable! Video sigals through a V cable! You ca t do it with pure sie waves cos(kx ωt)! It just goes o # Completely predictable # No iformatio! You eed waves that chage patters with time! What you really eed are pulses! Pulse width determies the speed! Pulses must be separated by at least
Amplitude Modulatio! Audio sigals rage from to khz! oo low for efficiet radio trasmissio! Use a better frequecy ad modulate amplitude Carrier wave Audio sigal! Modulated waves are o loger pure sie waves! What is the frequecy compositio? Amplitude-modulated waves
Wave Packet! Cosider carrier waves modulated by a pulse! his makes a short trai of waves # A wave packet f() t = ω e t <! = /( khz) for audio sigals! Fourier itegral is i t t > f() t iω ( t iωt ω ω) F( ω) = e e dt = si π π( ω ω)
Wave Packet ( ω ω) F( ω) = si πω ( ω )! Similar to the square pulse! Width is π/! Cetered at ω = ω o sed pulses every secod, your sigal must have a miimum spread of π/ i ω, which correspods to / i frequecy! his is called the badwidth of your radio statio! his limits how close the frequecies of radio statios ca be! You eed khz for HiFi audio ω! It s more like 5 khz i commercial AM statios π ω
Badwidth! Speed of iformatio trasfer = # of pulses / secod! Determied by the pulse width i the time domai! raslated ito badwidth i the frequecy domai! We say badwidth to mea speed of commuicatio! Broadbad meas fast commuicatio! Each medium has its maximum badwidth! You ca split it ito smaller badwidth chaels! Radio wave frequecies # Regulated by the govermet! Cable V # 75 MHz / 6 MHz = 5 chaels! You wat to miimize the badwidth of each chael! elephoes carry oly betwee 4 ad 34 Hz
Delta Fuctio! ake the square pulse agai! Make it arrower by! he height grows /! We get a ifiitely arrow pulse with uit area! Dirac s delta fuctio δ(t) δ () t t = = t ad! For ay fuctio f(t) δ () tdt= f() t δ () t dt = f() f t δ t t dt = f t () ( ) ( )
Delta Fuctio! What is the Fourier itegral of δ(t)? iωt F( ω) = δ( t) e dt = π π! δ(t) cotais all frequecies equally iωt δ () t = e dω Aother way of defiig δ(t) π You ca get this also by makig # i ω F( ω) = si πω
Pure Sie Waves! Cosider pure sie waves with agular frequecy ω f() t = i t e ω F ω = e e dt e dt δω ω π = π = iωt i( ) t ( ) i ω t ω ω ( ) f() t F( ω) t ω ω
How higs Fit ogether Waveform t domai t width ω domai ω width Siusoidal uiform ifiite δ(ω ω) δ pulse δ(t) uiform ifiite Fiite pulse ad everythig else f(t) F(ω) /! Pure sie waves ad δ pulses are the two extreme cases of all waves! Everythig falls i betwee! Widths i t ad ω are iversely proportioal to each other! Wait Did I prove it?
Arbitrary Sigal Width! Now we cosider a sigal with a arbitrary shape f () t F( ω) Fourier! Let s defie the average time ad the average frequecy t = t f() t f() t dt dt ω = ω F( ω) dω F( ω) dω! Because (eergy desity) (amplitude)! Now we defie the r.m.s. widths i t ad ω ( t) = ( t t ) ( ω) = ( ω ω ) r.m.s. = root mea square
Arbitrary Sigal Width ( ) ( ) ( ) () t = t t = f() t dt t t f t dt ( ) ( ) ( ) F( ) ω = ω ω = F( ω) dω ω ω ω dω What ca we do with this mess??! We ca express F(ω) with f(t) as F ω dω f t f se dωdtds 4π * = f () t f ( s) δ ( t s) dtds π = f() t dt π * iω ( t s) ( ) = ( ) ( ) = π ( ) ( ) i ω F ω f t e t dt
Arbitrary Sigal Width! Next we take iωt f() t = F( ω) e dω d [ f () t ] i F ( ) e iωt d with t dt = ω ω ω i t d ω ωf( ω) e dω = i f ( t) dt! We ca use this to costruct Differetiate * ( ) ( ) = ( ) ( )( ) ( ) ( ) [ ] ω ω F ω dω ω ω F ω ω ω F ω δ ω ω dωdω * i ω ω = ( ω ω ) ( ω) ( ω ω ) ( ω ) ω ω π d = i ω f() t dt π dt ( ) t F F e d d dt
Arbitrary Sigal Width! Now we have ( t) = ( ) t t f() t dt f() t! Here comes the trick: we calculate the itegral I ( κ )! It s a positive umber divided by a positive umber! κ is a real umber dt ( ω ) = d t t iκ i ω f() t dt dt = > f() t dt d i ω f() t dt dt f() t dt
Arbitrary Sigal Width I ( κ) ( t) κ ( ω) = + + ( ) κ ω + κ ω ( ) * * t t f() t i i f () t i i f() t t t f () t dt! he itegral i the deomiator becomes κ! Itegrate the first term i parts d d dt dt f() t d d dt dt ( ) + ( ) * * t t f () t f () t f () t t t f () t dt d * d * κ ( t t ) f() t + κ ( t t ) f() t f () t + f() t ( t t ) f () t dt dt dt = because the pulse has a fiite extet κ d * d * ( tf () t ) f () t + f () t tf () t dt = κ f () t dt dt dt dt
Arbitrary Sigal Width! We ve come a log way! Now we got I ( ) ( t) ( ) κ = + κ ω κ >! If a quadratic fuctio of κ is always positive, ( ) ( ) D= 4 t ω < fially! t ω > For ay sigal, the product of the r.m.s. widths t ad ω i the time ad frequecy domai is greater tha /
Space ad Waveumber! We have studied Fourier trasformatio i time t ad frequecy ω! We ca also do it i space x ad waveumber k! Everythig works the same way ikx f( x) = ikx F( k) e dk Fk ( ) = f ( xe ) dx π! I particular, x k > for ay sigal travelig i space! Why is it importat?
Ucertaity Priciple! I Quatum Mechaics, particles are wave packets! Ulike a classical particle, wave packet has a legth! he positio caot be determied more accurately tha x! Mometum is related to the waveumber by p =! k! his meas h! = Plack s costat = 6.63 34 J s π x p =! x k >! Heiseberg s Ucertaity Priciple
Summary! Defied Fourier itegral iωt f() t = iωt F( ω) e dω F( ω) = f( t) e dt π! f(t) ad F(ω) represet a fuctio i time/frequecy domais! Aalyzed pulses ad wave packets! ime resolutio t ad badwidth ω related by! Proved for arbitrary waveform! Rate of iformatio trasmissio badwidth! Dirac s δ(t) a limitig case of ifiitely fast pulse t ω >! Coectio with Heiseberg s Ucertaity Priciple i QM