Standing Wave Simple Theory

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1 UIUC Physics 406POM Acustical Physics f Music/Musical Instruments Standing Wave Simple Thery In this example, we shw plts f acustic quantities assciated with the superpsitin f tw cunter-prpagating 1-D mnchrmatic plane traveling waves i.e. a simple standing wave in free air. Please refer t Physics 406 Lecture Ntes 1 p fr details. The individual right and left-traveling cmplex time-dmain ver-pressure amplitudes are: i( t kx p A ( x, t = Ae ω i( t kx and p B ( x, t = Be ω + with A B {necessarily}, where i A i A A Ae ϕ ϕ = Ae i B i B and B Be ϕ ϕ i( ωt kx i( ωt+ kx = Be p x, t = p x, t + p x, t = Ae + Be Pascals.. Thus: ( ( ( tt A B ( The individual right and left-traveling cmplex time-dmain 1-D particle velcity amplitudes A i( ωt kx i( ωt kx B i( ωt+ kx i( ωt+ kx are: ua( x, t = e u A e and: u (, B x t = e u B e (using c= ω k. ρc ρc i( t kx i( t kx A i( t kx B ω ω + ω i( ωt+ kx Thus: u (, (, (, tt x t = ua x t + ub x t = ua e + u ( B e = e e m s. ρc ρc iϕ B B Be B i( ϕ ( B ϕa i ϕb ϕa i BA Defining: R Δϕ = = e = Re = Re where: Δϕ iϕ A BA ϕb ϕa, then: A Ae A and: i( ωt kx i( ωt+ kx ( i( kx BA BA i t kx p (, i Δϕ ω +Δϕ tt x t = A e + R e e = Ae 1 + R e A i( t kx i( t kx i A ω ω + Δϕ ( i( kx BA BA i ωt kx +Δϕ u tt ( xt, = e Re e = e 1 Re ρ c ρ c The magnitudes f the cmplex ttal/resultant ver-pressure ptt ( x particle velcity u ( x are: tt and lngitudinal A u x = R kx+δ + R ρ c p tt ( x = A 1+ R cs( kx+δ ϕba + R and: ( 1 tt cs( ϕba The phases assciated with the cmplex ttal/resultant ver-pressure ( x lngitudinal particle velcity ϕ u tt are: ϕ ϕ ptt utt ( x ( x = tan = tan Prfessr Steven Errede, Department f Physics, University f Illinis at Urbana-Champaign, Illinis All rights reserved. ϕ and ( + ( +Δ ϕba + ( +ΔϕBA + ( +ΔϕBA +ΔϕBA ptt sin kx 1 R cs kx R cs kx sin kx cs kx 1 R cs kx R sin kx sin kx ( ( ( ( +ΔϕBA ( +ΔϕBA ( +Δ ϕba + +ΔϕBA sin kx 1 R cs kx R cs kx sin kx cs kx 1 R cs kx R sin kx sin kx ( (

2 UIUC Physics 406POM Acustical Physics f Music/Musical Instruments The cmplex frequency-dmain lngitudinal specific acustic impedance, it s magnitude and phase are, nting {here} that ω = ck : { 1 R } + i R sin( kx+δϕ (, BA z att x ω = z { 1+ R } R cs( kx+δϕ BA att (, ω z x = z { } 1 R + 4 R sin ( kx+δϕ BA { 1+ R } R cs( kx+δϕ BA R sin 1 ( kx ϕ +Δ BA ϕz ( x, ω = tan,,, att x =Δ ϕ ω = ϕ p p x ω ϕ x ω tt utt tt utt { 1 R } ( ( ( The cmplex frequency-dmain lngitudinal acustic intensity, it s magnitude and phase are: 1 A I a x R i R kx tt z (, ω = { 1 } + sin( +ΔϕBA 1 A ( { } I a x, ω = 1 R + 4 R sin ( kx+δϕ tt BA z R sin( kx ϕ 1 +Δ BA ϕi ( x, ω = tan (, (, (, (, att = ϕz x ω =Δ ϕ x ω = ϕ att p p x ω ϕ x ω tt utt tt utt { 1 R } The cmplex frequency-dmain ptential, kinetic and ttal energy densities assciated with cunter-prpagating 1-D mnchrmatic traveling plane waves (n.b. all purely real quantities: A wptl ( x, ω p tt ( x, 1 R R cs ( kx BA 4 c ω = + + +Δ ρ 4 ρc ϕ 1 1 A w ( x, ω ρ u ( x, ω = 1 + R R cs( kx+δϕ kin tt BA 4 4 ρc 1 A 1 A tt (, ω ptl (, ω + kin (, ω = 1+ = 1+ ρc zc w x w x w x R R We cded up the abve frmulas using MATLAB and shw plts in the figures belw f the magnitudes and phases f cmplex ttal ver-pressure, particle velcity, lngitudinal specific p θ u θ z θ I θ vs. acustic impedance and lngitudinal acustic intensity (, (, ( and ( θ kx fr 0 R 1in steps f 0.1, with Δ ϕ BA = 0.0. Please nte the suppressed zeres in sme f the fllwing plts! -- Prfessr Steven Errede, Department f Physics, University f Illinis at Urbana-Champaign, Illinis All rights reserved. tt tt att a tt

UIUC Physics 406POM Acustical Physics f Music/Musical Instruments Figure 1. R = 0.

UIUC Physics 406POM Acustical Physics f Music/Musical Instruments Figure. R = 0.

UIUC Physics 406POM Acustical Physics f Music/Musical Instruments Figure 3. R = 0.

4-7- Prfessr Steven Errede, Department f Physics,

5-8- Prfessr Steven Errede, Department f Physics,

UIUC Physics 406POM Acustical Physics f Music/Musical Instruments Figure 7. R = 0.

9-1- Prfessr Steven Errede, Department f Physics,

0-13- Prfessr Steven Errede, Department f Physics,

14 Listing f the MATLAB cde: UIUC Physics 406POM Acustical Physics f Music/Musical Instruments %========================================================================== % Simple_Standing_Wave.m % % Study f the acustical physics assciated with the linear superpsitin % f tw cunter-prpagating 1-D mnchrmatic plane/traveling waves, % = a standing wave. % %========================================================================== % Authr: Prf. Steven Errede 4/5/014 07:50 hr % % Last Update: 4/7/014 10:0 hr {SME} %========================================================================== clear all; clse all; npts = 40000; thetar = zers(1,npts; MgPtt = zers(1,npts; MgUtt = zers(1,npts; MgZtt = zers(1,npts; MgItt = zers(1,npts; PhPtt = zers(1,npts; PhUtt = zers(1,npts; PhZtt = zers(1,npts; PhItt = zers(1,npts; Wapt = zers(1,npts; Wakin = zers(1,npts; Watt = zers(1,npts; rh0 = 1.04; % Density f NTP (kg/m^3 c = 344.0; % Lngitudinal speed f NTP (m/s z0 = rh0*c; % Lngitudinal specific acustic impedance f free air (Rayls p0 = 1.000; % Over-pressure amplitude (RMS Pascals % ~R = ~B / ~A %MgR = e-7; MgR = 0.100; %MgR = e-7; delphi0_ba = 0.0; % = phi0b - phi0a (degrees delphi0_bar = (pi/180.0*delphi0_ba; % (radians thetar_l = -.0*pi + 1.0e-7; % = kx_l thetar_hi =.0*pi; % = kx_hi dthetar = (thetar_hi - thetar_l/npts; fr j = 1:npts; thetar(j = thetar_l + (j-1*dthetar; % = kx MgPtt(j = p0 *sqrt(1.0 + (.0*MgR*cs(.0*thetar(j + delphi0_bar + (MgR*MgR; MgUtt(j = (p0/(rh0*c*sqrt(1.0 - (.0*MgR*cs(.0*thetar(j + delphi0_bar + (MgR*MgR; MgZtt(j = MgPtt(j/MgUtt(j; MgItt(j = MgPtt(j*MgUtt(j/.0; Pnum = sin(thetar(j*(1.0 + (MgR*cs(.0*thetar(j + delphi0_bar + cs(thetar(j* (MgR*sin(.0*thetar(j + delphi0_bar; Pden = cs(thetar(j*(1.0 + (MgR*cs(.0*thetar(j + delphi0_bar sin(thetar(j* (MgR*sin(.0*thetar(j + delphi0_bar; PhPtt(j = (180.0/pi*atan(Pnum,Pden; Unum = sin(thetar(j*(1.0 - (MgR*cs(.0*thetar(j + delphi0_bar cs(thetar(j* (MgR*sin(.0*thetar(j + delphi0_bar; -14- Prfessr Steven Errede, Department f Physics, University f Illinis at Urbana-Champaign, Illinis All rights reserved.

15 UIUC Physics 406POM Acustical Physics f Music/Musical Instruments Uden = cs(thetar(j*(1.0 - (MgR*cs(.0*thetar(j + delphi0_bar + sin(thetar(j* (MgR*sin(.0*thetar(j + delphi0_bar; PhUtt(j = (180.0/pi*atan(Unum,Uden; Znum = (.0*MgR*sin(.0*thetar(j + delphi0_bar; Zden = (1.0 - (MgR*MgR; PhZtt(j = (180.0/pi*atan(Znum,Zden; Inum = (.0*MgR*sin(.0*thetar(j + delphi0_bar; Iden = (1.0 - (MgR*MgR; PhItt(j = (180.0/pi*atan(Inum,Iden; end Wapt(j = 0.5 *(MgPtt(j*MgPtt(j/(z0*c; Wakin(j = 0.5*rh0*(MgUtt(j*MgUtt(j; Watt(j = Wapt(j + Wakin(j; figure (01; subplt(,,1; plt (thetar,mgptt,'m',thetar,mgptt,'m.'; grid n; xlabel ('{\theta} = kx (radians'; ylabel (' p_{tt}({\theta} (RMS Pascals'; title (' p_{tt}({\theta} vs. {\theta}'; subplt(,,; plt (thetar,mgutt,'g.',thetar,mgutt,'g.'; grid n; xlabel ('{\theta} = kx (radians'; ylabel (' u_{tt}({\theta} (RMS m/s'; title (' u_{tt}({\theta} vs. {\theta}'; subplt(,,3; %plt (thetar,mgztt,'b',thetar,mgztt,'b.'; semilgy(thetar,mgztt,'b',thetar,mgztt,'b.'; grid n; xlabel ('{\theta} = kx (radians'; ylabel (' Z_{tt}({\theta} (Rayls'; title (' Z_{tt}({\theta} vs. {\theta}'; subplt(,,4; plt (thetar,mgitt,'k',thetar,mgitt,'k.'; grid n; xlabel ('{\theta} = kx (radians'; ylabel (' I_{tt}({\theta} (RMS Watts'; title (' I_{tt}({\theta} vs. {\theta}'; figure (0; subplt(,,1; plt (thetar,phptt,'m',thetar,phptt,'m.'; grid n; xlabel ('{\theta} = kx (radians'; ylabel ('{\phi}_{ptt}({\theta} (degrees'; title ('{\phi}_{ptt}({\theta} vs. {\theta}'; subplt(,,; plt (thetar,phutt,'g',thetar,phutt,'g.'; grid n; xlabel ('{\theta} = kx (radians'; ylabel ('{\phi}_{utt}({\theta} (degrees'; title ('{\phi}_{utt}({\theta} vs. {\theta}'; subplt(,,3; plt (thetar,phztt,'b',thetar,phztt,'b.'; %semilgy(thetar,phztt,'b',thetar,phztt,'b.'; -15- Prfessr Steven Errede, Department f Physics, University f Illinis at Urbana-Champaign, Illinis All rights reserved.

16 UIUC Physics 406POM Acustical Physics f Music/Musical Instruments grid n; xlabel ('{\theta} = kx (radians'; ylabel ('{\phi}_{ztt}({\theta} (degrees'; title ('{\phi}_{ztt}({\theta} vs. {\theta}'; subplt(,,4; plt (thetar,phitt,'k',thetar,phitt,'k.'; grid n; xlabel ('{\theta} = kx (radians'; ylabel ('{\phi}_{itt}({\theta} (degrees'; title ('{\phi}_{itt}({\theta} vs. {\theta}'; figure (03; plt (thetar,wapt,'m.',thetar,wakin,'g.',thetar,watt,'b.'; grid n; xlabel ('{\theta} = kx (radians'; ylabel ('w_{a}({\theta} (RMS Jules/m^{3}'; title ('w_{a}({\theta} vs. {\theta}'; legend ('w_{a}^{ptl}','w_{a}^{kin}','w_{a}^{tt}'; -16- Prfessr Steven Errede, Department f Physics, University f Illinis at Urbana-Champaign, Illinis All rights reserved.

Pitch vs. Frequency:

Pitch vs. Frequency: Pitch = human ear s perceptin f frequency f a sund vibratin Lw pitch lw frequency f vibratin/scillatin High pitch high frequency f vibratin/scillatin Q: Is the relatin between {perceived}