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1、The net vibrationReviewSuperposition 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.electromagnetic WavesDo not need a mediumWavesTransverse Wav
2、esThe medium oscillates perpendicular to the direction the wave is moving. Longitudinal WavesWater waveThe medium oscillates in the same direction as the wave is moving sound waveMechanical WavesThe propagation of a disturbance in a medium.The conditions all the mechanical waves require:1) Some sour
3、ce 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 disturbance also means a transfer of energy.Wave
4、s 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 harmonic motion in space2)The phase of the particle wh
5、ich oscillates later is smaller. mediumdisturbancev 18y(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 and t=0Phase constantCan be determined from the
6、initial conditions.Simple harmonic vibration function:The vibration y as a function of time t.The 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
7、x, calledwaveform, defines a curve representing the actual geometric shape of the pulse at that time.Amplitude and WavelengthWavelengthWavelength : The distance between identical points on the wave.Amplitude AAmplitude A: The maximum displacement of a point on the wave.A19Period and VelocitylPeriod
8、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.21TuWave Properties.The speed of a wave is a constant that depends only on the medium, not on amplitude, wavelength or period (similar to SHM) and T
9、are related ! = u T or = 2 u / or =u / fExample 2-1-1Suppose the harmonic vibration function of origin at t Find: the harmonic wave function of point P at tSolution: the time for the vibration to arrive point P is:The vibration at point P at t is identical with that of point O at t-tThen we have the
10、 wave function of point P:Example 1-1-2Suppose the harmonic vibration function of origin at t Find: the harmonic wave function of point P at tThe vibration at point P at t is identical with that of point O at t+tTherefore, the harmonic wave function can be written as:Or:If the wave travels left, use
11、 x substitute x.The 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 when t=0.Example 2-1-3
12、Solution: 1) taking O as the originThe vibration function of O is:When t=0thenThe velocity of x=0 at t=0:?The simple harmonic vibration curve:The velocity at a certain timeis the slope of the tangent line of that point.The harmonic wave curve (displacement as a function of x):t=t1t=t2, t2t1If the sl
13、ope of a certain point of the curve y(x) 0, the velocity at this point 0 (the wave travels right wards)Solution: 1) taking O as the originThe vibration function of O is:When t=0thenThe velocity of x=0 at t=0:thusTherefore, the vibration function of O is:The wave function of x taking O as origin is:1
14、) taking P as the originThe vibration function of P is:When t=0thenAnyone is Ok, we choose The wave function of x taking P as origin is:The wave function of x taking O as origin is:The wave function of x taking P as origin is:We must identify the origin point clearly!The phase constants are differen
15、t 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.The velocity at x point:Because the vibration is:The velocity at x point at t moment:Take x=/8, t=0 into the above equation:Along the negative y axisTake x=3/8, t=0 into the above e
16、quation:Along the positive y axis2-2 wave speed / phase speed uThe speed of a wave is a constant that depends only on the medium. and T are related !Note: the speed of the wave u is different from the vibration velocity of a certain medium element v.The speed of a wave is a constant that depends onl
17、y on the medium.A) Wave propagating in liquid, gas/ fluidB : bulk elastic modulus : the density of the mediumB) Wave propagating in solid1) Transverse waveG : shear elastic modulus2) longitudinal waveY : Young modulus2-3 energy of harmonic wavesMechanical wave:The disturbance is propagating through
18、a medium.disturbanceVibration statephaseenergyEnergy of traveling harmonic wavesThe wave function:The waveform (at t=t1):Segment AB in the mediumThe mass of AB:the mass density of the mediumThe kinetic energy of AB:The potential energy of AB:T: tensionThe magnitude and phase of kinetic energy and po
19、tentialenergy are identical at any time.Note: the energy difference between wave and vibration!waveformMaximum deformationMaximum velocityThe mechanical energy of AB:Mechanical energy of wave changes with time periodicallyMechanical energy of simple harmonic vibration keeps constant.energy density o
20、f wave:Area mass density of the medium average energy density of wave:energy flow of wave:The energy passes through unit area in unit time.energy flow of wave changes with time periodicallyAverage energy flow of wave:The average energy flow in one period.energy flow density of wave / wave intensity The average energy flow in one period through the unit area which is perpendicular to the propagating direction of the wave.
