重郵大學物理英文版PPT13

1、the net vibrationcosaxt12212221cos2aaaaa221122111coscossinsinaaaatgreviewsuperposition of vibrationchapter 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 medium.electromag

2、netic wavesdo not need a mediumwavestransverse 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 wavemechanical wavesthe propagation of a disturbance in a medium.the conditi

3、ons 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.the essence of mechanical waves:the disturbance is transferred through space, but the matter does not.the propagation of the

4、 disturbance also means a transfer of energy.waves on a string112-1 harmonic waves the characteristic of harmonic wavesevery medium element oscillates around the equilibriumposition in simple harmonic motion, but the wave propagatesaway from the source of disturbance. the propagation of simple harmo

5、nic motion in space2)the phase of the particle which oscillates later is smaller. mediumdisturbancev18y(x,t) = a cos(t kx)a = amplitude = angular frequencyk = wave number = 2/harmonic wave functionassuming: initial phase is zero at x=0 and t=0generally,the transverse displacement is not zero at x=0

6、and t=0( , )cos()y x tatkxphase constantcan be determined from the initial conditions.simple harmonic vibration function:( )cosay ttthe vibration y as a function of time t.0121 2 )(st)(cmxs1the harmonic wave function:the wave function y(x, t) represents the y coordinate of any point p located at pos

7、ition 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 that time.( , )cos()y x tatkxamplitude and wavelengthwavelengthwavelength : the distance between identical points

8、 on the wave.amplitude aamplitude a: the maximum displacement of a point on the wave.a19period and velocitylperiod t : the time for a point on the wave to undergo one complete oscillation.speed u: the wave moves one wavelength in one period, so its speed is u = / t.tu21tuwave properties.the speed of

9、 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 / or u / fexample 2-1-1xypuoxsuppose the harmonic vibration function of origin at t )cos()(00tatyfind: the harmonic wave function of point p at

10、tsolution: the time for the vibration to arrive point p is:uxt xypuoxthe vibration at point p at t is identical with that of point o at t-t)(cos),(),(0 uxtattoytxythen we have the wave function of point p:example 1-1-2xyp uoxsuppose the harmonic vibration function of origin at t )cos()(00tatyfind: t

11、he harmonic wave function of point p at tthe vibration at point p at t is identical with that of point o at t+t)(cos)(cos),(),(00 uxtauxtattoytxytherefore, the harmonic wave function can be written as:)(cos) ,(0 uxtatxyor:)(2cos) ,(0 xttatxy)(2cos) ,(0 uxtatxy)(2cos) ,(0 xutatxyif the wave travels l

12、eft, use x substitute x.ut2t = ut2 uthe parameters a， u of a certain planar cosine wave are 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 wh

13、en t=0.yxx pouy 8 83 example 2-1-3yxx 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 velocity of x=0 at t=0:0sin)0, 0(0 av?0121 2 )(st)(cmxs1the simple harmonic vibration curve:the velocity at a certain timeis the slope of

14、 the tangent line of that point.the harmonic wave curve (displacement 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)yxx pouy 8 83 solution: 1) taking o as the originthe vibration function of o is:

15、)cos(), 0(0 tatywhen t=00cos)0, 0(0 aythen20 the velocity of x=0 at t=0:0sin)0, 0(0 avthus02therefore, the vibration function of o is:)2cos(), 0( tatythe wave function of x taking o as origin is:2)(cos), 0(),( uxtattytxy1) taking p as the originyxx pouy 8 83 the vibration function of p is:) cos(), 0

16、( tatywhen t=0cos)0, 0( ay then anyone is ok, we choose )cos(), 0( tatythe wave function of x taking p as origin is:)(cos),( uxtatxythe 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 po

17、int clearly!the phase constants are different if we take various original points.2) 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:the velocity at x point a

18、t 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 axis2-2 wave speed / phase speed uthe speed of a wave is a constant that depends only on the medium.ut and

19、 t are related !note: the speed of the wave u is different from the vibration velocity of a certain medium element v.yvtthe 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 mediumb) wave propagati

20、ng in solid1) transverse wave gu g : shear elastic modulus2) longitudinal wave yu y : young modulus2-3 energy of harmonic wavesmechanical wave:the disturbance is propagating through a medium.disturbancevibration statephaseenergyenergy of traveling harmonic wavesthe wave function:)(cos0 uxtayxyoabx y

21、 the waveform (at t=t1):segment ab in the mediumthe mass of ab:mxthe mass density of the mediumthe kinetic energy of ab:221mvek xyoabx 2)( 21tyx )(sin210222 uxtxathe potential energy of ab:l t tyouy x x)(xltep t: tension22)()(yxl 2/12)(1 xyx )(2112 xyx)(212xyxtep 2)(21xyxt 2)(21xyxt )(sin2102222 uxtuxat)(sin210222 uxtxa tu )(sin210222uxtxaeepkthe magnitude and phase of kinetic energy and potentialenergy are identical at any time.note: the energy difference between wave and vibration!xyabwaveformmaxpemaximum deformationmaximu