Acoustical Interference Phenomena

Transcription

1 cousical Inerference Phenomena When wo (or more) periodic signals are linearly superposed (i.e. added ogeher), he resulan/overall waveform ha resuls depends on he ampliude, frequency and phase informaion associaed wih he individual signals. Mahemaically, his is mos easily and ransparenly described using complex noaion. Basics of/primer on Complex Variables Complex variables are used whenever phase informaion is imporan. complex funcion, Z = X+iY consiss of wo porions, a so-called real par of Z, denoed X = Re(Z) and a socalled imaginary par of Z, denoed Y = Im(Z). The number i (-). The magniude of he complex variable, Z is designaed as Z = (Z), = (ZZ*), wih Z ZZ* where Z* is he socalled complex conjugae of Z, i.e. Z* = (Z)* = (X+iY)* = XiY, wih i* = (i)* (-). Thus, Z = (ZZ*) = (X+iY)(X+iY)* = (X+iY)(XiY) = (X + ixy ixy + Y ) = (X +Y ). Thus he magniude of Z, Z is analogous o he hypoenuse, c of a righ riangle (c = a + b ) and/or he radius of a circle, r cenered a he origin (r = x + y ). Because complex variables Z = X+iY consis of wo componens, Z can be graphically depiced as a -componen vecor Z = (X,Y) lying in he so-called complex plane, as shown in he figure below. The real componen of Z, X = Re(Z) is by convenion drawn along he x, or horizonal axis (i.e. he abscissa). The imaginary componen of Z, Y = Im(Z) is by convenion drawn along he y, or verical axis (i.e. he ordinae), as shown in he figure below. Imaginary xis: Y = Im (Z) Z = (X + Y ) ½ Z = X + iy Y X Real xis: X=Re (Z) Professor Seven Errede, Deparmen of Physics, Universiy of Illinois a Urbana- Champaign, Illinois ll righs reserved.

2 I can be readily seen from he above diagram ha he endpoin of he complex vecor, Z = X+iY, lies a a poin on he circumference of a circle, cenered a (X,Y) = (0,0), wih radius (i.e. magniude) Z = (X + Y ) ½ and phase angle, (defined relaive o he X-axis), of an (Y / X) (or equivalenly: an (X / Y), defined relaive o he Y-axis). Insead of using Caresian coordinaes, we can alernaively and equivalenly express he complex variable, Z in polar coordinae form, Z = Z ( + isin), since X = Z and Y = Z Sin. Recall also he rigonomeric ideniy, + Sin =, which is used in obaining he magniude of Z, Z from Z iself. If we now redefine he variable such ha ( +), i can hen be seen ha Z( = Z {( + )+ isin( +)}, wih real componen X( = Re(Z() = Z ( +), and imaginary componen Y( = Im(Z() = Z Sin( +). ime = 0, hese relaions are idenical o he above. If (for simpliciy s sake) we ake he phase angle, = 0, hen Z( = Z {(+ isin(}. ime = 0, i can be seen ha he complex variable Z(=0) is purely real, Z(=0) = X(=0), i.e. lying enirely along he x-axis, Z(=0) = Z cos0 = Z. s ime, progresses, he complex variable Z( = Z {(+ isin(} roaes in a couner-clockwise manner wih (consan angular frequency, = f radians/second, where f is he frequency (in cycles/second {cps}, or Herz {=Hz}), compleing one revoluion in he complex plane every = /f = / seconds. The variable is also known as he period of oscillaion, or period of vibraion. Linear Superposiion (ddiion) of Two Periodic Signals I is illusraive o consider he siuaion of linear superposiion of wo periodic, equalampliude, idenical-frequency signals, bu in which one signal differs in phase from he oher by 90 degrees. Since he zero of ime is arbirary, we can hus chose one signal o be purely real a ime = 0, such ha Z( = and he oher signal, Z( = isin, wih purely real ampliude, and angular frequency, for each. Then, using he rigonomeric ideniy (-B) = B + SinSinB, we see ha Sin = (90 o ) = 90 o + SinSin90 o. Thus, for his example, here, he signal Z( = isin lags (i.e. is behind) he signal Z( = by 90 degrees in phase. The resulan/oal complex ampliude, Z( is he sum of he wo individual complex ampliudes, Z( = Z( + Z( = ( + isin. We can also wrie his relaion in exponenial form, since exp(i) = e i ( + isin), exp(i) = e i ( isin), and hus ½(e i + e i ) and Sin ½i(e i e i ). Then for our example, he oal complex ampliude, Z( = Z( + Z( = ( + isin = e i, wih magniude Z( = {Z(Z*(} = =. If we had insead chosen he second ampliude o be Z( = isin, hen he signal Z( would lead (i.e. be ahead of) he signal Z( = by 90 degrees in phase. Then for his siuaion, he oal complex ampliude, Z( = Z( + Z( = ( isin = e i, wih magniude Z( = (i.e. he same as before). Thus, a change in he sign of a complex quaniy, Z( Z( physically corresponds o a phase change/shif in phase/phase advance of +80 degrees (n.b. which is also mahemaically equivalen o a phase reardaion of 80 degrees). Professor Seven Errede, Deparmen of Physics, Universiy of Illinois a Urbana- Champaign, Illinois ll righs reserved.

3 The same mahemaical formalism can be used for adding ogeher wo arbirary (bu sill periodic) signals, Z( = (exp{i(( + ()} and Z( = (exp{i(( + ()}. The individual ampliudes, frequencies and phases may be ime-dependen. The resulan overall complex ampliude is Zo( = Z( + Z( = (exp{i(( + ()} + (exp{i(( + ()}. Because he zero of ime is (always) arbirary, we are free o choose/redefine = 0 in such a way as o roae away one of he wo phases absorbing i as an overall/absolue phase (which is physically unobservable). Since e (x+y) = e x e y, he above formula can be rewrien as: Zo( = Z( + Z( = (exp{i(}exp{i(} + (exp{i(}exp{i(}. Muliplying boh sides of his equaion by exp{i(}: Zo( exp{i( = (exp{i(} + (exp{i(}exp{i((()}. This shif in overall phase, by an amoun exp{i(}is formally equivalen o a redefiniion o he zero of ime, and also physically corresponds o a (simulaneous) roaion of boh of he (muually-perpendicular) real and imaginary axes in he complex plane by an angle, (. The physical meaning of he remaining phase afer his redefiniion of ime/shif in overall phase is a phase difference beween he second complex ampliude, Z( relaive o he firs, Z( is ( ( (. Thus, a he (newly) redefined ime * = (/( = 0 (and hen subsiuing * he resuling overall, ime-redefined ampliude is: Zo( = (exp{i(} + (exp{i(}exp{i(} or: Zo( = (exp{i(} + (exp{i(( + ()}. The so-called phasor relaion for he wo individual complex ampliudes, Z(, Z( and he resuling overall ampliude, Zo( in he complex plane is shown in he figure below. Zo(0) Z(0) (0) (0) Z(0) The magniude of he resuling overall ampliude, Zo( can be obained from Zo( = Zo(Z*o( = Z( + Z( = (Z( + Z()(Z( + Z()* = (Z( + Z()(Z*( + Z*() = Z(Z*( + Z(Z*( + Z(Z*( + Z(Z*( = Z( + Z( + Re[Z(Z*(] = ( + ( + ((Re[exp{i(((() + ()}] = ( + ( + (({((() + (} 3 Professor Seven Errede, Deparmen of Physics, Universiy of Illinois a Urbana- Champaign, Illinois ll righs reserved.

4 The phase angle, ( of he overall/resulan ampliude, Zo( relaive o he firs ampliude, Z( can be obained from he relaion Tan{(} Im{Zo(}/Re{Zo(, hus: Tan{(}= (Sin{((() + (}/[( + ({((() + (}] Beas Phenomenon Linearly superpose (i.e. add) wo signals wih ampliudes ( and (, and which have similar/comparable frequencies, ( ~ (, wih insananeous phase of he second signal relaive o he firs of (: 0 ( ( ( ) 0 ( o 0 0 Noe ha a he ampliude level, here is nohing explicily over and/or obvious in he above mahemaical expression for he overall/oal/resulan ampliude, o( ha easily explains he phenomenon of beas associaed wih adding ogeher wo signals ha have comparable ampliudes and frequencies. However, le us consider he (insananeous) phasor relaionship beween he individual ampliudes for he wo signals, ( and ( respecively. Their relaive iniial phase difference a ime = 0 is (=0) and he resulan/oal ampliude, o(=0) is shown in he figure below, for ime, = 0: o(0) (0) (0) (0) (0) From he law of cosines, he magniude of he oal ampliude, o( a an arbirary ime, is obained from he following: Thus, o o o ( ( ) ( ( ) 0 0 For equal ampliudes, 0 = 0 = 0, zero relaive iniial phase, = 0 and consan (i.e. ime-independen frequencies, and, his expression reduces o: 4 Professor Seven Errede, Deparmen of Physics, Universiy of Illinois a Urbana- Champaign, Illinois ll righs reserved. ( 0 ( ) 0 ( ) o ( ) ( 0

5 The phase of he oal ampliude, o( relaive o ha of he firs ampliude (, a an arbirary ime, is ( and is obained from he projecions of he oal ampliude phasor, o( ono he y- and x- axes of he -D phasor plane: ( ( Sin( Tan ( ( ( ( The oal ampliude, o( = ( + ( vs. ime, is shown in he figure below, for imeindependen/consan frequencies of f = 000 Hz and f = 980 Hz, equal ampliudes of uni srengh, 0 = 0 =.0 and zero relaive phase, = 0.0 Clearly, he beas phenomenon can be seen in he above waveform of oal ampliude, o( = ( + ( vs. ime,. From he above graph, i is obvious ha he bea period, bea = /fbea = sec = /0 h sec, corresponding o a bea frequency, fbea = /bea = 0 Hz, which is simply he frequency difference, fbea f f beween f = 000 Hz and f = 980 Hz. Thus, he bea period, bea = /fbea = / f f. When f = f, he bea period becomes infiniely long, and no beas are heard. 5 Professor Seven Errede, Deparmen of Physics, Universiy of Illinois a Urbana- Champaign, Illinois ll righs reserved.

6 In erms of he phasor diagram, as ime progresses, he individual ampliudes ( and ( acually precess a (angular) raes of = f and = f radians per second respecively, compleing one revoluion in he phasor diagram, for each cycle/each period of = / = /f and = / = /f, respecively. If a ime = 0 he wo phasors are precisely in phase wih each oher (i.e. wih iniial relaive phase = 0.0), hen he resulan/oal ampliude, o( = 0) = ( = 0) + ( = 0) will be as shown in he figure below. o(=0) = (=0) + (=0) (=0) (=0) s ime progresses, if, (phasor wih angular frequency = f = *000 = 000 radians/sec and = f = *980 = 960 radians/sec in our example above) phasor, wih higher angular frequency will precess more rapidly han phasor (by he difference in angular frequencies, = ( ) = ( ) = 40 radians/second). Thus, as ime increases, phasor will lead phasor ; evenually (a ime = ½bea = 0.05 = /40 h sec in our above example) phasor will be exacly = radians, or 80 degrees behind in phase relaive o phasor. Phasor a ime = ½bea = 0.05 sec = /40 h sec will be oriened exacly as i was a ime = 0.0 (having precessed exacly N = / = f/ = f = 5.0 revoluions in his ime period), however phasor will be poining in he opposie direcion a his insan in ime (having precessed only N = / = f/ = f = 4.5 revoluions in his same ime period), and hus he oal ampliude o( = ½bea = ( = ½bea + ( = ½bea will be zero (if he magniudes of he wo individual ampliudes are precisely equal o each oher), or minimal (if he magniudes of he wo individual ampliudes are no precisely equal o each oher), as shown in he figure below. o( = ½bea = ( = ½bea + ( = ½bea = 0 ( = ½bea = ( = ½bea s ime progresses furher, phasor will coninue o lag farher and farher behind, and evenually (a ime = bea = sec = /0 h sec in our above example) phasor, having precessed hrough N = 49.0 revoluions will now be exacly = radians, or 360 degrees (or one full revoluion) behind in phase relaive o phasor (which has precessed hrough N = 50.0 full revoluions), hus, he ne/overall resul is he same as being exacly in phase wih phasor! his poin in ime, o( = bea = ( = bea + ( = bea = ( = bea = ( = bea, and he phasor diagram looks precisely like ha a ime = 0. Thus, i should (hopefully) now be clear o he reader ha he phenomenon of beas is manifesly ha of ime-dependen alernaing consrucive/desrucive inerference beween wo periodic signals of comparable frequency, a he ampliude level. This is by no means a rivial poin, as ofen he beas phenomenon is discussed in physics exbooks in he conex of inensiy, Io( = o( = ( + (. From he above discussion, he physics origin of he beas phenomenon has absoluely nohing o do wih inensiy of he overall/ resulan signal. 6 Professor Seven Errede, Deparmen of Physics, Universiy of Illinois a Urbana- Champaign, Illinois ll righs reserved.

7 The primary reason ha he phenomenon of beas is discussed more ofen in erms of inensiy, raher han ampliude is ha he physics is perhaps easier o undersand from he inensiy perspecive a leas mahemaically, hings appear more obvious, physically: I I o o o Le us define: ) ( ) ( ) ( ( I o nd hen le us use he mahemaical ideniy: Thus: I o The le us define: We hen obain: Using he ideniy: ( 0 ( ( ) ( ) ( ) ( ) ( We hen obain he addiional relaion, which is no usually presened and/or discussed in physics exbooks: ( ( ) I o ( ( ) ( ) 0 This laer formula shows ha here are a.) DC (i.e. zero frequency) componens/consan erms associaed wih boh ampliudes, 0 and 0, b.) nd harmonic componens wih f and f, as well as c.) a componen associaed wih he sum of he wo frequencies, = f + f, and d.) a componen associaed wih he difference of he wo frequencies, f = f f. This is a remarkably similar resul o ha associaed wih he oupu response from a sysem wih a quadraic non-linear response o a pure/single-frequency sine-wave inpu! 7 Professor Seven Errede, Deparmen of Physics, Universiy of Illinois a Urbana- Champaign, Illinois ll righs reserved.

8 Special/Limiing Case mpliude Modulaion: When 0 >> 0 and f >> f, hen he exac expression for he oal ampliude, () () () () () () () ( () ()) () o 0 0 () () () ( () ()) () 0 0 can be approximaed by he following expression(s), neglecing erms of order m (0/0) << under he radical sign, and, defining ( (( (), and noing ha for f >> f, ( (( () (: () o 0 () 0 0 () () () () ( () ()) () () 0 0 () () () () () () () () () () () However, when f >> f, he relaive phase difference ( changes by radians (essenially) for each cycle of he frequency, f. Hence we can safely se o zero his phase difference, i.e. ( = 0 because is (ime-averaged) effec in he f >> f limi is negligible. Using he Taylor series expansion for he case when <<, he expression for he oal ampliude hen becomes: () () () () () o () () () () () () () () () m 0 The raio m (0/0) << is known as he (ampliude) modulaion deph of he highfrequency carrier wave (, wih ampliude 0 >> 0 and frequency f >> f, modulaed by he low frequency wave (, wih ampliude 0 and frequency f. This is he underlying principle of M radio M sands for mpliude Modulaion 8 Professor Seven Errede, Deparmen of Physics, Universiy of Illinois a Urbana- Champaign, Illinois ll righs reserved.

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