1 Chapter 12 Waves Wave motion is the most universal physical phenomenon, we discover the wave motion all around us. Mechanical waves—acoustic wave, water wave Electromagnetic waves—radio, light wave Matter waves—electron wave We will concentrate to the mechanical waves. Wave is a propagating disturbance that transfers energy, momentum and information as its own characteristic speed from one region of space to another. 2 § 12.1 mechanical waves 1. What is mechanical waves? Source of wave—a disturbance(oscillation) Medium—for propagation of waves Longitudinal waves: waves whose oscillation or jiggling is along the line the wave propagates. According to the direction of particle motion 2. Classification of mechanical waves Plane of polarization-- the plane in which the jiggling of a transverse waves disturbance occurs. Pulse wave: Periodic train of waves: § 12.1 mechanical waves Transverse waves: the waves whose oscillation or jiggling is perpendicular to the line along which the waves propagates. 3 Wavefronts—the surfaces on them all the points are in the same state of motion. Wave ray—the direction which are perpendicular to the wavefronts or parallel to the velocity of the waves § 12.1 mechanical waves According to the shape of the wavefronts Plane waves: the wavefronts are plane Cylindrical wave: the wavefronts are cylindrical Spherical wave: the wavefronts are spherical § 12.1 mechanical waves 4 Seismic waves— earthquakes produce both longitudinal and transverse waves Sand scorpion— located its prey by using longitudinal and transverse waves. § 12.1 mechanical waves m/s150= m/s50= td v d v d t tl ?=??=? 75 § 12.2 one-dimensional waves and the equation of waves 1. Wavefunction of one-dimensional waves moving at constant velocity x y o x(x′) yy′ oo′ x ?? vt x′ P P )()0,( xΨxy = )( )(),( vtxΨ xΨtxy ?= ′ = vv ?→ )( )(),( vtxΨ xΨtxy += ′ = 5 x y o x(x′) yy′ oo′ x ?? vt x′ P P ),()0,( txyxy PP = =?vtx constant phase t x vv t x d d 0 d d =?=? Phase speed § 12.2 one-dimensional waves and the equation of waves 2. The classical wave equation vtxu uΨvtxΨtxy ?= =?= )()(),( 2 2 2 2 )( u Ψ x u u Ψ ux Ψ u Ψ x u u Ψ x Ψ ? ? = ? ? ? ? ? ? = ? ? ? ? = ? ? ? ? = ? ? 2 2 2 2 2 )( u Ψ v t u u Ψ v ut Ψ u Ψ v t u u Ψ t Ψ ? ? = ? ? ? ? ? ? ? = ? ? ? ? ?= ? ? ? ? = ? ? 2 2 2 2 2 x Ψ v t Ψ ? ? = ? ? 0 1 2 2 22 2 = ? ? ? ? ? t Ψ vx Ψ or § 12.2 one-dimensional waves and the equation of waves 6 3. Periodic waves x ),( 0 txΨ λ o λ (a) Space period A snapshot at some instant t 0 t o ),( 0 txΨ T T (b) Time period An oscillation at any given position x 0 § 12.2 one-dimensional waves and the equation of waves In one time period, some oscillatory state propagate one space period. Phase speed: νλ λ ?== T v λ T § 12.2 one-dimensional waves and the equation of waves 7 x ),( txΨ v r o vT=λ tv? (c) Traveling wave νλλν === vvT T 1 4. Sinusoidal (harmonic) waves )](cos[),( vtxkAtxΨ ?= x ),( txΨ v r o λ § 12.2 one-dimensional waves and the equation of waves 4. The physical meaning of the harmonic waves Model: One-dimensional harmonic waves x v r o If the oscillatory equation at point o is )cos( 0 φω += tAΨ ?),( == txΨΨ --harmonic motion § 12.2 one-dimensional waves and the equation of waves 8 Method 1 if )cos( 0 φω ′ += tAΨ v OP(x) x The propagating time of the state from point o to point P(x) v x t =? Choose a ray of wave as x axis, construct a one-dimensional coordinate system. O is the origin, v is the velocity along the +x direction. The phase of point P(x) at instant t is the same as the point o at instant t-?t. § 12.2 one-dimensional waves and the equation of waves ])(cos[),( φω +?= t v x AtxΨ (1) ),0(),( 0 ttΨtxΨ p ??= ])(cos[ φω ′ +?= v x tA therefore u OP(x) x Method 2: The phase at point P lag relative to point o π λ 2? x The phase difference between two points of space interval of λ is 2π. § 12.2 one-dimensional waves and the equation of waves 9 Equations (1) and (2) is identical. )2cos(),( π λ φω ?? ′ += x tAtxΨ p )2cos(),( φωπ λ +??= t x AtxΨ or (2) ω π λ 2 vvT == Due to § 12.2 one-dimensional waves and the equation of waves (1) ])(cos[),( φω +?= t v x AtxΨ λ ππ ωλ 22 === k T vT where § 12.2 one-dimensional waves and the equation of waves ])(cos[),( φω +?= t v x AtxΨ )2cos( φω λ π +?= t x A ])(2cos[ φ λ π +?= T tx A LL= +?= ])(cos[ φvtxkA )cos( φω +?= tkxA 10 § 12.2 one-dimensional waves and the equation of waves 6. The comparison of graphs of oscillation and waveform T, A, φ, the moving direction of point (a) The graphs of oscillation A, λ, determine φ from the graph of waveform at t=0, the moving direction of point (b) The graphs of waveform ?? a b )m(x 04.0 )0,(xΨ o 2.0 P m/s08.0=v Example: figure depicts the waveform of a traveling sinusoidal wave at instants t=0 s, find (a) the oscillatory equation of point o; (b) the wavefunction ;(c) the oscillatory equation of point P; (d) the moving directions of points a and b. ),( txΨ § 12.2 one-dimensional waves and the equation of waves 11 (a) )srad(4.02 )m(4.0)(04.0 1? ?== == π λ πω λ v mA ? ? a b )m(x 04.0 ),( txΨ o 2.0 P m/s08.0=v Solution: ) 2 4.0cos(04.0),0( π π += ttΨ 2 0 π ? = The initial phase angle of point o: ? ? ? ? ? ? +?= 2 ) 08.0 (4.0cos04.0),( π π x ttxΨ (b) the wavefunction § 12.2 one-dimensional waves and the equation of waves Ψ a b x 04.0 ),( txΨ o 2.0 p ?? ? ? ? ? ? ? += ? ? ? ? ? ? +?= 2 4.0cos04.0 2 ) 08.0 4.0 (4.0cos04.0),( π π π π t ttxΨ P (c) the oscillatory equation of point P (d) the moving directions of points a and b. § 12.2 one-dimensional waves and the equation of waves 12 § 12.3 Energy transport via mechanical waves 1. The velocity of wave on a string μ μ θμ θθ / )/( /, )2/sin(2)/( 2 2 Tv r l Trvl rllm TTrvm = ? =?∴ ?=?=? ≈=?Q The wave is viewed from a reference frame moving with the wave so it is seen to be at rest. The disturbance is motionless but the string is observed to move to the left at a constant speed v. x y T r T′ r θ 2/θ l? r C 2/θ § 12.3 Energy transport via mechanical waves The general relationship for the speed of mechanical waves in material media: 2/1 factormass factorforceaofmagnitude ? ? ? ? ? ? ? ? ∝v 2/1 solid ? ? ? ? ? ? ? ? = ρ E v 2/1 liquid ? ? ? ? ? ? ? ? = ρ B v Sound wave in solid: Sound wave in liquid: young′s modulus bulk modulus Generalize to the other waves: 13 PEKEE ddd += For a string wave: xxd y ldyd xm yxl dd )d()d(d 22 μ= += § 12.3 Energy transport via mechanical waves 2. The mechanical energy of a small string element of medium ? ? ? ? ? ? ? ? = ? ? ? ? ? ? ?+≈ ? ? ? ? ? ? ?+=?+= 22 222 )( 2 1 d1) d d ( 2 1 1d 1) d d (1dd)d()d( x y x x y x x y xxyxlδ x x y TPE d)( 2 1 d 2 ? ? = then x t y t y mKE d)( 2 1 )(d 2 1 d 22 ? ? = ? ? = μKinetic energy: )dd(d xlTlTPE ?== δ 22 )d()d(d yxl += Potential energy: § 12.3 Energy transport via mechanical waves 14 The mechanical energy: x x y Tx t y PEKEE d)( 2 1 d)( 2 1 ddd 22 ? ? + ? ? = += μ For a harmonic wave [] []φω ω φω φωωφω +??= ? ? ? ? ? ? +? ? ? = ? ? +?= ? ? ? ? ? ? +? ? ? = ? ? )(sin)(cos )(sin)(cos tkxA v t v x A xx y tkxAt v x A tt y § 12.3 Energy transport via mechanical waves []xtkxA v TE d)(sin)( 2 1 d 2222 φω ω μω +? ? ? ? ? ? ? += μ T v =due to [ ] VtkxAE d)(sind 222 φωρω +?= generalize [ ] xtkxAE d)(sind 222 φωμω +?= we have § 12.3 Energy transport via mechanical waves 15 An non-isolated differential medium Comparison: An isolated harmonic oscillator The mechanical energy conserved KE and PE is changed with out of phase The mechanical energy is not conserved KE and PE is changed in phase § 12.3 Energy transport via mechanical waves Equilibrium position: Transverse waves： x xd y ld yd with maximum distortion, maxmax d,)( PE x y ? ? oscillating speed maxmax d, KEu Position of the maximum displacement: with minimum distortion, minmin d,)( PE x y ? ? oscillating speed minmin d, KEu § 12.3 Energy transport via mechanical waves 16 Longitudinal waves: Equilibrium position(the center of compression and rarefaction): with maximum distortion, maxmax d,)( PE x y ? ? oscillating speed maxmax d, KEu Position of the maximum displacement: with minimum distortion, minmin d,)( PE x y ? ? oscillating speed minmin d, KEu § 12.3 Energy transport via mechanical waves 3. The power of wave [ ] xtkxAE d)(sind 222 φωμω +?= []φωωμ +?== )(sin d d 222 tkxAv t E P []zytkxAv t E P dd)(sin d d 222 φωωρ +?== [ ] VtkxAE d)(sind 222 φωρω +?=or § 12.3 Energy transport via mechanical waves 17 VtkxAPEKEE d])[(sinddd 222 ?+?=+= φωρω 4. The density of energy ])[(sin d d 222 φωωρ +?== tkxA V E w the density of energy: 22 2 1 ωρA= ∫ +?= T ttkxA T w 0 222 av d])[(sin 1 φωωρ The average density of energy: § 12.3 Energy transport via mechanical waves vAI r r 22 2 1 ωρ= --wave intensity vA vw s P I av 22 av av 2 1 ωρ ? = = = 5. Wave intensity []zytkxAv t E P dd)(sin d d 222 φωωρ +?== The direction of energy transportation is same as that of wave § 12.3 Energy transport via mechanical waves v tv? 18 The power through the area s 1 and s 2 is same 2 22 21 22 1 2 1 2 1 svAsvA ?=? ωρωρ 1 2 1 2 1 2 2 1 2 2 r r hr hr s s A A === π π 1 2 2 2 2 1 2 1 r r A A I I == Example: a cylindrical plane wave propagates in a homogeneous medium, find the relationship of wave intensity with respect to the amplitude and the distance to the wave source. Solution: § 12.3 Energy transport via mechanical waves The wave function of cylindrical plane waves: ])cos[( 0 φω +?= tkr r A Ψ ])cos[( 0 φω +?= tkr r A Ψ Prove the wave function of spherical plane waves is § 12.3 Energy transport via mechanical waves 19 § 12.4 Reflection and transmission of waves 1. Tie one end of the rope to a rigid wall The pulse experiences a change of phase of π. 2. the one end of the rope is free There is no phase change on reflection. 3. Two different ropes with different mass densities A pulse is transmitted to the second rope and a pulse is reflected. P546 § 12.5 Sound wave and the acoustic Doppler effect 1. What is a sound wave? Sound waves in air or other gases represent such rapid oscillations in density that the inertial properties of the air make it quite elastic. 20 2. Sound intensity and sound level Threshold for hearing: ~10 -12 W/m 2 at ~1 kHz Cause deafening pain: ~1 W/m 2 12log10log 12 =? αα The sound level: Let 212 0 W/m10 ? ≡I 0 log)dB10( I I ≡β Unit: decibels Each factor of 10 increase in sound intensity I produces an additive change in sound level of 10dB.(P551 table 12.1) § 12.5 Sound wave and the acoustic Doppler effect 3. The acoustic Doppler effect νν = ′ § 12.5 Sound wave and the acoustic Doppler effect 21 1motion of the observer with the source and medium at rest )1()1( obsobsobs v vvv ±=±=±= ′ ν λν ν λ νν § 12.5 Sound wave and the acoustic Doppler effect 2motion of the sound source with the observer and medium at rest )( / sss s s vv v v v v v Tv v Tv mmm m ν λν ν νλλ ν λλ === ? = ′ = ′ § 12.5 Sound wave and the acoustic Doppler effect 22 3the motion of both observer and sound source )( obs s vv vv m ± = ′ νν 4motion of the medium with the source and observer at rest νν = ′ § 12.5 Sound wave and the acoustic Doppler effect )( smed obsmed vvv vvv m± ±± = ′ νν 5motion of the observer and/or source with a wind 4. Shock waves )( s vv v ? = ′ νν Mach cone: )(sin s s vv v v >=φ Mach number: v v s numberMach = § 12.5 Sound wave and the acoustic Doppler effect 23 § 12.6 the superposition of waves 1. The linear principle of superposition When two waves encounter together, if the amplitudes is not too large, the total wave distance at any point x and time t is the sum of the individual wave disturbances. L+++= ),(),(),(),( 321total txΨtxΨtxΨtxΨ )cos(),( 11 φω +?= tkxAtxΨ )cos(),( 22 φω +?= tkxAtxΨ )cos()cos( ),(),(),( 21 21total φωφω +?++?= += tkxAtkxA txΨtxΨtxΨ 2. Interference of waves ) 2 cos() 2 cos(2 ] 2 )()( cos[ ] 2 )()( cos[2),( 2112 21 21 total φφ ω φφ φωφω φωφω + +? ? = +??+? ? +?++? = tkxA tkxtkx tkxtkx AtxΨ § 12.6 the superposition of waves 24 ) 2 cos() 2 cos(2),( 2112 total φφ ω φφ + +? ? = tkxAtxΨ ),2,1,0(0,)12( ),2,1,0(2,2 total total L L ==+=? ===? nAn nAAn πφ πφ constructive destructive if 21 AA ≠ then ),2,1,0(,)12( ),2,1,0(,2 21total 21total L L =?=+=? =+==? nAAAn nAAAn πφ πφ § 12.6 the superposition of waves 0=?φ πφ =? 32πφ =? ),( and ),( 2 1 txΨ txΨ ),( 1 txΨ ),( 2 txΨ ),( 1 txΨ ),( 2 txΨ ),( txΨ ),( txΨ ),( txΨ Ψ Ψ Ψ Ψ ΨΨ Interference of waves § 12.6 the superposition of waves 25 )cos(),( 11 φω +?= tkxAtxΨ )cos(),( 22 φω ++= tkxAtxΨ )cos()cos( ),(),(),( 21 21total φωφω ++++?= += tkxAtkxA txΨtxΨtxΨ ) 2 cos() 2 cos(2 ] 2 )()( cos[ ] 2 )()( cos[2),( 1221 21 21 total φφ ω φφ φωφω φωφω ? + + += ++?+? ? ++++? = tkxA tkxtkx tkxtkx AtxΨ 3. Standing waves § 12.6 the superposition of waves ) 2 cos() 2 cos(2 1212 total φφ ω φφ ? + + += tkxAΨ Special case： 0 21 ==φφ tkxAΨ ωcoscos2 total = § 12.6 the superposition of waves Standing wave is a special phenomenon of interference. 26 § 12.6 the superposition of waves The characteristic of the standing waves: Some points keep zero disturbance: a,c,e,g,… Some points have maximum amplitudes: o,b d f, … 1 20 AA<< The other points 0) 2 cos( 12 = + + φφ kx )( 44 )12( 12 φφ π λλ +?+= kx 2 )12( 2 12 πφφ += + + kkx ),2,1,0( L±±=k The position of nodes: § 12.6 the superposition of waves 27 )( 42 12 φφ π λλ +?= kx 1) 2 cos( 12 = + + φφ kx π φφ kkx = + + 2 12 ),2,1,0( L±±=k The position of antinodes: 2 λ ? =x The distance between the successive nodes or antinodes: § 12.6 the superposition of waves 0) 2 1 ( 2 1 22 1 22 1 =?+= vAvAI rr r ωρωρ The wave intensity of standing waves: The energy is switched between the nodes and the antinodes. § 12.6 the superposition of waves 28 The characteristic of energy for standing wave :2/,0 Tt = PEE = Concentrate near the nodes. :4/3,4/ TTt = KEE = Concentrate near the antinodes. § 12.6 the superposition of waves 4. The musical instruments with strings μ ν μ νλ λλ T L T T vL 2 1 2 1 1 = ==== --fundamental frequency § 12.6 the superposition of waves μ ν T L n n 2 1 = --eigenfrequency 2 n nL λ = 29 § 12.6 the superposition of waves 1. Wave groups and beats )cos( )cos( ),(),(),( )cos(),( )cos(),( 22 11 21 222 111 txkA txkA txΨtxΨtxΨ txkAtxΨ txkAtxΨ ω ω ω ω ?+ ?= += ?= ?= § 12.7 Wave groups and beats ] 22 cos[ ] 22 cos[2),( 2121 2121 tx kk tx kk Atx ωω ωω Ψ ? ? ? ? + ? + = 30 ] 22 cos[]cos[2),( tx k tkxAtx ω?? ωΨ ??= )] 2 () 2 cos[(2 tx k A ω?? ?The other factor if 21 or2/ ωωω? < This wave represents a slowly varying amplitude factor for superposition, which travels at its own group speed. § 12.7 Wave groups and beats 1 1 1 21 21 v kkkk v =≈ + + == ωωωω if , then the phase speed 21 ωω ≈ Let x=0 ) 2 cos(cos2 ttAΨ ω ω ? = k v kvkv kk v k v k v k v d d )( d d d d , d d phase phasephasegroup phasegroup group +=== == ? ? = ω ωω ω dispersion The superposition as a function of time: § 12.7 Wave groups and beats 31 ) 2 cos(cos2 ttAΨ ω ω ? = § 12.7 Wave groups and beats 2. Fourier analysis § 12.7 Wave groups and beats From sinusoidal waves, any manner of more complex periodic waves can be built. )7cos 7 1 5cos 5 1 3cos 3 1 (cos 4 ???+?+?= tttt A x ωωωω π For instance: A square wave 32 tAtAxxx ωω 3coscos 2121 +=+= 1 x 2 x t t O 1 T O 2 T tO x 1 T 1 x 2 x t t t O O 1 T O 2 T x 1 T )3cos( 3 1 cos 21 πωω ++=+= ttxxx § 12.7 Wave groups and beats § 12.7 Wave groups and beats A flute An oboe A saxophone