Mechanical Waves

1. Chapter 15 Mechanical Waves The propagation of an vibration In an elastic medium

2. Wave:Any disturbance propagates with time from one region of space to another. The types of waves 1. Mechanical waves: earthquake waves, sound wave, water wave, …

3. 2. Electromagnetic waves: light, microwaves, x-rays...

4. 3. Gravity waves: 4. Matter waves: electron, atom, molecule…….

5. Ogle :

6. §15-1 Generation & Propagation of a Mechanical Wave 机械波的产生和传播 §15-2 Wave Speed & Elasticity of the Medium 机械波的传播速度 媒质的弹性 §15-3 Wave Function of a Plane SHW 平面简谐波的函数 §15-4 Energy Energy Flow and Wave Intensity 波的能量 波动强度 §15-5 Huygen’s Principle 惠更斯原理

7. §15-6 Principle of Superposition of Waves Interference of Waves 波的叠加原理 波的干射 §15-7 Standing Waves 驻波 §15-8 The Doppler Effect 多普勒效应

8. §15-1 Generation & Propagation of a mechanical Wave I. Mechanical wave The propagation of a mechanical vibration in an elastic medium. The conditions of generation and propagation of a M-waves: 1. There must be a vibrating center called source of wave. 2. There must be an elastic medium propagating wave.

9. II. Classification Transverse wave：the individual particles of medium vibrate in the direction that is perpendicular to the direction of wave propagating. Longitudinal wave：the individual particles of medium vibrate in the direction that is parallel to the direction of wave propagating.

10. propagate Continuous wave Pulse wave Moving wave: The vibration states “travel” in some direction. Standing wave

11. y v v y0 y x o P x III.The mathematical description of moving wave:---Wave function： • Choose o –source of wave ，x--the direction of wave propagating，v--wave speed Assume, at time t, that the displacement of the particle locating in o is

12. y v v y0 y = the displacement of the particle locating in o at time , x o P x Then , at time t, the displacement of the particle locating in P --Wave function propagating in +x-direction i.e.

13. y o x u u y0 y Wave function propagating in - x-direction is

14. 波 线 波 面 波 面 波 线 IV.The geometric description of moving wave: wave front and ray Spherical wave Plane wave

15. §15-2 Wave Speed & Elasticity of the Medium I. Deformation of elastic objects tension or compression Basic deformation shear

16. p V0+  V V0 p+  p 1.Elastic modulus （1）Bulk modulus An object has volume V0 at the pressure p . The volume =V0+  V at pressure p+ p. 体 变 --Volumetric strain(体应变) Let Definition --bulk modulus

17. （2）Tensile elastic modulus The bar is pulled by f, 长 变 Stress(应力) Tensile strain(线应变) Definition --Young’s modulus

18. d D （3）Shear modulus S A shearing force is exerted on the faces of the object. Shear stress(剪切应力) 切 变 Shear strain(切应变) Definition ----shear modulus

19. Note liquid and gas produce bulk strain only. Solid can produce bulk, tensile, and shear strain. 2.Wave speed in elastic medium Wave speed(phase speed相速)v： -- the propagating speed of the phases of a vibration. v depends only on the density and elasticity of the medium in which wave propagates.

20. In some kind of uniform solid, Transverse wave: Longitudinal wave: or In some kind of uniform gas or liquid, longitudinal wave can be propagated only. Longitudinal wave:

21. :The ratio of molar capacity  In ideal gas,  In a stretched string , Transverse wave: T ：the tension in string  ：the mass per unit length of the string

22. §15-3 Wave Function of a Plane SHM Harmonic wave： The propagation of simple harmonic motion in elastic medium. The source of wave and every segment of medium are moving in harmonic motion with same frequency. Wave function：The moving function of any particle of medium when there is a wave in the medium.

23. I. Wave function of plane harmonic wave Assume that the source of point O is moving in SHM i.e. That the state ( phase ) of point O is propagated by the wave from O to P needs time :

24. Sothe displacement of point P at time t =the displacement of point O at time i.e. -- Wave function of plane harmonic wave

25. x --the vibrating state propagates the distance x along the wave ray during the time interval Discussion  Wave function describes the propagating of the vibrating state.

26. vibrating curve  If x =x = a constant, then Let then --the motion of the particle at point x is SHM

27. the wave pattern at time t If t =t = a constant, then Let Then --the displacement of each particle on the wave ray at any timet’

28.  Other forms of the wave function of plane harmonic wave

29.  Wave function of plane harmonic wave traveling in the –x direction Wave function of a spherical harmonic wave Wave function of a cylindrical harmonic wave

30. Assume a general wave function is Let II.The dynamic equation of mechanical waves Then and -- The dynamic equation of mechanical waves

31. can describe all kinds of waves.

32. [Example] A plane harmonic wave is travelingin +xdirection. Amplitude=A，T=18s , =36m. The particle at origin O is at its equilibrium position and is moving toward the +y direction when t=0. Find the wave function, the oscillation equation at x=9m,  the wave pattern equation and the coordinates of all crests at t =3s.

33. Solution  Assume the wave function is then the vibration equation of the particle at point O is： According to initial conditions: We get

34. At x=9m， the oscillation equation is

35. At t =3s， the wave pattern equation is The coordinates x of crests should satisfy We get

36. §15-4 Energy, Energy flow and Wave intensity Wave propagating The particles in medium vibrating The medium deformation Kinetic energies Potential energies Energies propagating

37. I. Energy of wave • Take a transverse wave generating and propagating along a streched string as an example. • Assume the wave function is • Take any segment AB as a particle.

38. Y v x A B

39. Kinetic energy The vibrating speed of the particle (AB) is If the mass of AB is m=x，its vibrating K-energy is

40. Potential energy When AB  A B, it possesses elastic P-energy, the tension in string T=v2

41. i.e. • The total energy of AB When wave propagates in a volume medium,  ，x  V

42. Remarks TheEk, Ep andE change with same phase for any particle of the medium.  The Ek, Ep andE has maximum when the particle locate in its equilibrium position. Ek=Ep =E= 0 when the particle locate in its maximum displacement.  E is not conservative for any particle. A particle absorbs energy continuously from its contiguous particle in the front of it and gives off the energy to the particle behind it.----energy propagation

43. II. Energy density of wave • Energy density：the wave energy stored in a unit volume. • The average energy density: the average value over one period.

44. Energy flow: the amount of energy passing a given area in unit time. III. Energy flow density of wave (wave intensity) • Average energy flow(平均能流)

45. Energy flow density of wave (wave intensity) ：

46. IV. Sound wave 1. SW is a mechanical longitudinal wave 20~~20000HZ，-----(audible)sound Frequency range: <20HZ-----infrasonic wave >20000HZ-----supersonic wave Acoustics：study the generation, propagation, receiving of SW and the reactions between SW and matters..

47. 2. sound intensity： 3. sound intensity level (声强级) : The range of audible wave： --reference intensity Let Then the sound intensity level of I is Decibel (分贝) Or

48. 4. Characters and applications of sound wave noise audible：music supersonic ： larger , smaller , less diffraction, good direction--sonar Easily reflected by the boundary surfaces--B超 Larger energy, easily focused --weld (焊接), drill(钻孔) infrasonic ：less decline，longer propagating distance --study the motions of earth, ocean…

49. §15-5 Huygens’ Principle I. Huygens’ principle (hypothesis) All points on a wave front serve as point sources of spherical secondary wavelets. After a time t, the envelope of these secondary wavelets will be the new wave front. It provides a geometrical method to find the new wave front at later instant from known shape of a portion of a wave front at some instant. It can be used to discuss various problems concerning the propagation of waves.

50. spherical wave plane wave