重郵大學(xué)物理英文版PPT-(13)

1、2021/8/21The net vibrationcosAxt12212221cos2AAAAA221122111coscossinsinAAAAtgReviewSuperposition of vibration2021/8/22Chapter 2 Mechanical wavesWaves: a disturbance travels away from its source.Water waves, sound waves, radio waves, X-rays WavesMechanical WavesThe disturbance is propagating through a

2、 medium.electromagnetic WavesDo not need a medium2021/8/23WavesTransverse WavesThe medium oscillates perpendicular to the direction the wave is moving. Longitudinal WavesWater waveThe medium oscillates in the same direction as the wave is movingsound wave2021/8/24Mechanical WavesThe propagation of a

3、 disturbance in a medium.The conditions all the mechanical waves require:1) Some source of disturbance2) A medium that can be disturbed3) Some physical mechanism through which particles can influence one another.2021/8/25The essence of mechanical waves:The disturbance is transferred through space, b

4、ut the matter does not.The propagation of the disturbance also means a transfer of energy.2021/8/26Waves on a String112021/8/272-1 harmonic waves The characteristic of harmonic wavesEvery medium element oscillates around the equilibriumposition in simple harmonic motion, but the wave propagatesaway

5、from the source of disturbance. The propagation of simple harmonic motion in space2021/8/282)The phase of the particle which oscillates later is smaller. mediumdisturbancev2021/8/2918y(x,t) = A cos(t kx)A = amplitude = angular frequencyk = wave number = 2/harmonic wave functionAssuming: initial phas

6、e is zero at x=0 and t=02021/8/210Generally,The transverse displacement is not zero at x=0 and t=0( , )cos()y x tAtkxPhase constantCan be determined from the initial conditions.2021/8/211Simple harmonic vibration function:( )cosAy ttThe vibration y as a function of time t.0121 2 )(st)(cmxs12021/8/21

7、2The harmonic wave function:The wave function y(x, t) represents the y coordinate of any point P located at position x at any time t.Two variables x and t.If t is fixed, the wave function y as a function of x, calledwaveform, defines a curve representing the actual geometric shape of the pulse at th

8、at time.( , )cos()y x tAtkx2021/8/213Amplitude and WavelengthWavelengthWavelength : The distance between identical points on the wave.Amplitude AAmplitude A: The maximum displacement of a point on the wave.A192021/8/214Period and VelocitylPeriod T : The time for a point on the wave to undergo one co

9、mplete oscillation.Speed u: The wave moves one wavelength in one period, so its speed is u = / T.Tu21Tu2021/8/215Wave Properties.The speed of a wave is a constant that depends only on the medium, not on amplitude, wavelength or period (similar to SHM) uTT = 2 / and T are related ! = u T or = 2 u / o

10、r u / f2021/8/216Example 2-1-1xypuOxSuppose the harmonic vibration function of origin at t )cos()(00tAtyFind: the harmonic wave function of point P at tSolution: the time for the vibration to arrive point P is:uxt 2021/8/217xypuOxThe vibration at point P at t is identical with that of point O at t-t

11、)(cos),(),(0 uxtAttoytxyThen we have the wave function of point P:2021/8/218Example 1-1-2xyp uOxSuppose the harmonic vibration function of origin at t )cos()(00tAtyFind: the harmonic wave function of point P at t2021/8/219The vibration at point P at t is identical with that of point O at t+t)(cos)(c

12、os),(),(00 uxtAuxtAttoytxy2021/8/220Therefore, the harmonic wave function can be written as:)(cos) ,(0 uxtAtxyOr:)(2cos) ,(0 xTtAtxy)(2cos) ,(0 uxtAtxy)(2cos) ,(0 xutAtxyIf the wave travels left, use x substitute x.2021/8/221uT2T = uT2 u2021/8/222The parameters A， u of a certain planar cosine wave a

13、re known. Calculating t=0 from the moment of the following figure, 1)write the wave function taking O and P as the origin respectively. 2) Find the magnitude and direction of the speed at x1= / 8 and x2= 3/ 8 when t=0.yxx Pouy 8 83 Example 2-1-32021/8/223yxx Pouy 8 83 Solution: 1) taking O as the or

14、iginThe vibration function of O is:)cos(), 0(0 tAtyWhen t=00cos)0, 0(0 Aythen20 The velocity of x=0 at t=0:0sin)0, 0(0 Av?2021/8/2240121 2 )(st)(cmxs1The simple harmonic vibration curve:The velocity at a certain timeis the slope of the tangent line of that point.2021/8/225The harmonic wave curve (di

15、splacement as a function of x):uyxot=t1t=t2, t2t1If the slope of a certain point of the curve y(x) 0, the velocity at this point 0 (the wave travels right wards)2021/8/226yxx Pouy 8 83 Solution: 1) taking O as the originThe vibration function of O is:)cos(), 0(0 tAtyWhen t=00cos)0, 0(0 Aythen20 The

16、velocity of x=0 at t=0:0sin)0, 0(0 Avthus022021/8/227Therefore, the vibration function of O is:)2cos(), 0( tAtyThe wave function of x taking O as origin is:2)(cos), 0(),( uxtAttytxy2021/8/2281) taking P as the originyxx Pouy 8 83 The vibration function of P is:) cos(), 0( tAtyWhen t=0cos)0, 0( Ay th

17、en Anyone is Ok, we choose )cos(), 0( tAtyThe wave function of x taking P as origin is:)(cos),( uxtAtxy2021/8/229The wave function of x taking O as origin is:( , )(0,)cos ()2xy x tyttAtuThe wave function of x taking P as origin is:( , )cos ()xy x tAtuWe must identify the origin point clearly!The pha

18、se constants are different if we take various original points.2021/8/2302) Find the magnitude and direction of the speed at x1= / 8 and x2= 3/ 8 when t=0.yxou8 83 The velocity at x point:( , )cos ()2xy x tAtu( , )sin ()2y x txvAttu 2sin2Atx Because the vibration is:2021/8/231The velocity at x point

19、at t moment:2( , )sin2v x tAtx Take x=/8, t=0 into the above equation: Av22)8, 0( yxx Pouy 8 83 Along the negative y axisTake x=3/8, t=0 into the above equation: Av22)38, 0( Along the positive y axis2021/8/2322-2 wave speed / phase speed uThe speed of a wave is a constant that depends only on the me

20、dium.uT and T are related !Note: the speed of the wave u is different from the vibration velocity of a certain medium element v.yvt2021/8/233The speed of a wave is a constant that depends only on the medium.A) Wave propagating in liquid, gas/ fluid Bu B : bulk elastic modulus : the density of the me

21、dium2021/8/234B) Wave propagating in solid1) Transverse wave Gu G : shear elastic modulus2) longitudinal wave Yu Y : Young modulus2021/8/2352-3 energy of harmonic wavesMechanical wave:The disturbance is propagating through a medium.disturbanceVibration statephaseenergy2021/8/236Energy of traveling h

22、armonic wavesThe wave function:)(cos0 uxtAyxyOABx y The waveform (at t=t1):Segment AB in the mediumThe mass of AB:mxthe mass density of the medium2021/8/237The kinetic energy of AB:221mvEk xyOABx 2)( 21tyx )(sin210222 uxtxA2021/8/238The potential energy of AB:l T TyOuy x x)(xlTEp T: tension22)()(yxl

23、 2/12)(1 xyx )(2112 xyx2021/8/239)(212xyxTEp 2)(21xyxT 2)(21xyxT )(sin2102222 uxtuxAT)(sin210222 uxtxA Tu 2021/8/240)(sin210222uxtxAEEpkThe magnitude and phase of kinetic energy and potentialenergy are identical at any time.2021/8/241Note: the energy difference between wave and vibration!xyabwaveformmaxpEMaximum deformationMaximum velocitymaxkE2021/8/242The mechanical