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Mechanical Waves

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

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Mechanical Waves

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  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

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