重邮大学物理英文版PPT (13)

1、The net vibration cosAxt 1221 2 2 2 1 cos2AAAAA 2211 2211 1 coscos sinsin AA AA tg Review Superposition of vibration Chapter 2 Mechanical waves Waves: a disturbance travels away from its source. Water waves, sound waves, radio waves, X-rays Waves Mechanical Waves The disturbance is propagating throu

2、gh a medium. electromagnetic Waves Do not need a medium Waves Transverse Waves The medium oscillates perpendicular to the direction the wave is moving. Longitudinal Waves Water wave The medium oscillates in the same direction as the wave is moving sound wave Mechanical Waves The propagation of a dis

3、turbance in a medium. The conditions all the mechanical waves require: 1) Some source of disturbance 2) A medium that can be disturbed 3) Some physical mechanism through which particles can influence one another. The essence of mechanical waves: The disturbance is transferred through space, but the

4、matter does not. The propagation of the disturbance also means a transfer of energy. Waves on a String 11 2-1 harmonic waves The characteristic of harmonic waves Every medium element oscillates around the equilibrium position in simple harmonic motion, but the wave propagates away from the source of

5、 disturbance. The propagation of simple harmonic motion in space 2)The phase of the particle which oscillates later is smaller. medium disturbance v 18 y(x,t) = A cos(t kx) A = amplitude = angular frequency k = wave number = 2/ harmonic wave function Assuming: initial phase is zero at x=0 and t=0 Ge

6、nerally, The transverse displacement is not zero at x=0 and t=0 ( , )cos()y x tAtkx Phase constant Can be determined from the initial conditions. Simple harmonic vibration function: ( )cosAy tt The vibration y as a function of time t. 0 1 2 1 2 )(st )(cmx s1 The harmonic wave function: The wave func

7、tion 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, called waveform, defines a curve representing the actual geometric shape of the pulse at that time. ( , )cos()y x tAtkx Amplitude

8、 and Wavelength Wavelength Wavelength : The distance between identical points on the wave. Amplitude A Amplitude A: The maximum displacement of a point on the wave. A 19 Period and Velocity lPeriod T : The time for a point on the wave to undergo one complete oscillation. Speed u: The wave moves one

9、wavelength in one period, so its speed is u = / T. T u 21 T u Wave Properties. The speed of a wave is a constant that depends only on the medium, not on amplitude, wavelength or period (similar to SHM) u T T = 2 / and T are related ! = u T or = 2 u / or u / f Example 2-1-1 x y p u O x Suppose the ha

10、rmonic vibration function of origin at t )cos()( 00 tAty Find: the harmonic wave function of point P at t Solution: the time for the vibration to arrive point P is: u x t x y p u O x The vibration at point P at t is identical with that of point O at t-t )(cos),(),( 0 u x tAttoytxy Then we have the w

11、ave function of point P: Example 1-1-2 x y p u O x Suppose the harmonic vibration function of origin at t )cos()( 00 tAty Find: the harmonic wave function of point P at t The vibration at point P at t is identical with that of point O at t+t )(cos )(cos),(),( 0 0 u x tA u x tAttoytxy Therefore, the

12、harmonic wave function can be written as: )(cos) ,( 0 u x tAtxy Or: )(2cos) ,( 0 x T t Atxy )(2cos) ,( 0 u x tAtxy )( 2 cos) ,( 0 xutAtxy If the wave travels left, use x substitute x. u T 2 T = uT 2 u The parameters A， u of a certain planar cosine wave are known. Calculating t=0 from the moment of t

13、he 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. y x x Po u y 8 8 3 Example 2-1-3 y x x Po u y 8 8 3 Solution: 1) taking O as the origin The vibration function of O is:)cos(),

14、0( 0 tAty When t=00cos)0, 0( 0 Aythen 2 0 The velocity of x=0 at t=0: 0sin)0, 0( 0 Av ? 0 1 2 1 2 )(st )(cmx s1 The simple harmonic vibration curve: The velocity at a certain time is the slope of the tangent line of that point. The harmonic wave curve (displacement as a function of x): u y x o t=t1

15、t=t2, t2t1 If the slope of a certain point of the curve y(x) 0, the velocity at this point 0 (the wave travels right wards) y x x Po u y 8 8 3 Solution: 1) taking O as the origin The vibration function of O is:)cos(), 0( 0 tAty When t=00cos)0, 0( 0 Aythen 2 0 The velocity of x=0 at t=0: 0sin)0, 0( 0

16、 Avthus0 2 Therefore, the vibration function of O is: ) 2 cos(), 0( tAty The wave function of x taking O as origin is: 2 )(cos), 0(),( u x tAttytxy 1) taking P as the origin y x x Po u y 8 8 3 The vibration function of P is:) cos(), 0( tAty When t=0cos)0, 0( Ay then Anyone is Ok, we choose )cos(), 0

17、( tAty The wave function of x taking P as origin is: )(cos),( u x tAtxy The wave function of x taking O as origin is: ( , )(0,)cos () 2 x y x tyttAt u The wave function of x taking P as origin is: ( , )cos () x y x tAt u We must identify the origin point clearly! The phase constants are different if

18、 we take various original points. 2) Find the magnitude and direction of the speed at x1= / 8 and x2= 3/ 8 when t=0. y x o u 8 8 3 The velocity at x point: ( , )cos () 2 x y x tAt u ( , ) sin () 2 y x tx vAt tu 2 sin 2 Atx Because the vibration is: The velocity at x point at t moment: 2 ( , )sin 2 v

19、 x tAtx Take x=/8, t=0 into the above equation: Av 2 2 ) 8 , 0( y x x Po u y 8 8 3 Along the negative y axis Take x=3/8, t=0 into the above equation: Av 2 2 ) 3 8 , 0( Along the positive y axis 2-2 wave speed / phase speed u The speed of a wave is a constant that depends only on the medium. u T and

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

21、opagating in solid 1) Transverse wave G u G : shear elastic modulus 2) longitudinal wave Y u Y : Young modulus 2-3 energy of harmonic waves Mechanical wave: The disturbance is propagating through a medium. disturbance Vibration state phase energy Energy of traveling harmonic waves The wave function:

22、 )(cos 0 u x tAy x y O AB x y The waveform (at t=t1): Segment AB in the medium The mass of AB:mx the mass density of the medium The kinetic energy of AB: 2 2 1 mvEk x y O AB x 2 )( 2 1 t y x )(sin 2 1 0 222 u x txA The potential energy of AB: l T T y O u y x x )(xlTE p T: tension 22 )()(yxl 2/12 )(1

23、 x y x )( 2 1 1 2 x y x )( 2 1 2 x y xTE p 2 )( 2 1 x y xT 2 )( 2 1 x y xT )(sin 2 1 0 2 2 2 2 u x t u xAT )(sin 2 1 0 222 u x txA T u )(sin 2 1 0 222 u x txAEE pk The magnitude and phase of kinetic energy and potential energy are identical at any time. Note: the energy difference between wave and vibration! x y a b waveform maxp E Maximum deformation Maximum velocit