Wave Motion

1. Wave Motion

2. In this chapter, we consider only mechanical waves, such as sound waves, water waves, and the waves transmitting in a guitar’s strings. • Elastic mediums are needed for the travel of mechanical waves. • Mechanical waves can appear when an initial disturbance is made to the mediums. On a microscopic level, the forces between atoms in the mediums are responsible for the propagation of the waves. 1-1 Mechanical waves

3. The particles of the medium do not experience any net displacement in the direction of the wave-as the wave passes, the particles simply move back and forth through small distance about their equilibrium position. What is a wave? It is the process of propagating oscillation in space. What are transmitted by a wave? Energy, momentum, phase…, but the particles are not.

4. Waves can be classified according to their properties as following. • According to direction of particle motion • “Transverse waves”: If the motion of the particle is perpendicular to the direction of propagation of the waves itself. (b)“Longitudinal wave”: If the motion of the particle is parallel to the direction of propagation of the waves. 1-2 Types of waves

5. 2. According to number of dimensions 1-D Waves moving along the string or spring 2-DSurface waves or ripple on water 3-DWaves traveling radially outward from a small source, such as sound waves and light waves. 3 According to periodicity pulse waves or periodic wave. The simplest periodic wave is a “simple harmonic wave’’ in which each particle undergoes simple harmonic motion.

6. The simplest periodic wave Other kinds of periodic waves: Square wave Triangle wave modulated wave Sawtoothed wave

7. 4. According to shape of wavefronts (a)The definitions of ‘wave surface’and ‘wavefront’? (b) The definition of ‘a ray’: A line normal to the wavefronts, indicating the direction of motion of the waves. Wavefronts are always  direction of Ray

8. Plane wave: The wavefronts are planes, and the rays are parallel straight lines. Spherical wave: The wavefronts are spherical, and the rays are radial lines leaving the point source in all directions. ★ Two different types of wavefronts: Plane waves Spherical waves

9. Wave front Wave Front * ray Spherical wave Plane wave Ray Wave surface Wavefront

10. 5. Waves in different fields in physics • Sound waves • Water waves • Earthquake waves • Light waves • Electromagnetic waves • Gravitational waves • Matter waves • Lattice waves

11. 1-3 Traveling waves • All the waves would travel or propagate, why here • say ‘traveling waves’? • (with respect to ‘standing wave’ • Definition of traveling waves: • The waves formed and traveling in an open • medium system. • Description of traveling waves • We use a 1-D simple harmonic, transverse, plane • wave as an example • Mathematics expressions • The vibration displacement y as a function of t and x.

12. The difference between vibration and wave motion: Vibration y(t): displacement as a function of time Wave y(x,t): displacement as a function of both time and distance 1. Equation of a sine wave What we want to know: y t = 0 t = t υ x Fig 1-6

13. If there is initial phase constant in the sinusoidal waves, the general equation of the wave at time t is: Several important concepts about waves: 1) The period Tof the wave is the time necessary for point at any particular x coordinate to undergo one complete cycle of transverse motion. During this time T, the wave travels a distance that must correspond to one wavelength . 2) The wavelength : the length of a complete wave shape.

14. 3) The frequency of the wave : 4) The wave number: 5) The angular frequency: (18-16)

15. The equation of a sine wave traveling in • direction is • The equation of a sine wave traveling in the • direction is Note that: speed of the wave

16. 2. Transverse velocity of a particle Note that v is the speed of wave transmitting. What is the velocity of particle oscillating? ---- It is called transverse velocity of a particle for transverse wave Transverse velocity: Tansverse acceleration:

17. 3. Phase and phase constant If the equation of the wave is: Phase phase constant Above equation can be written in two equivalent forms:

18. y (a) x B A y (b) t A B Fig 1-7 Two waves A and B: y = ymsin(kx – ωt–) wave A lead y = ymsin(kx – ωt ) wave B lag In y-x, wave A is ahead of wave B by a distance  /k In y-t, wave A is ahead of wave B by a time  /ω

19. Phase velocity vs group velocity • Phase velocity • Definition:The velocity of the motion of certain phase in a wave (for monochromatic wave(单色波,单一频率的波)) : source of the wave : the medium (non-dispersive) • Wave speed on a stretched string From dimensional analysis: 18-4 Wave velocity (speed) From mechanical analysis:

20. Wave velocity R o (1-19) F --- tension force exerted between neighboring elements μ --- mass density (mass/unit length)

21. When a wave passes from one medium to another medium, the frequency keeps the same, namely λand v may vary. 2) Group velocity For a group of waves with different In non- dispersive medium, All the waves with different moves with same speed . , determined only by the medium time = 0 Shape keeps x time = t υ x

22. time 0 x time t x In dispersive medium, All the waves with different moves with different speeds. Shape does not keeps!!! Group speed is needed to describe the waves. In this chapter, all the mediums met is assumed to be nondispersive.

23. 1-5* The wave equation

24. 1-6 Energy in wave motion A B (a) y Wave transmits energy. • Energy in wave motion Fig18-11a shows a wave traveling along the string at times and ( a time later ). x time time (b) dl dy dx Fig 1-11

25. What do we want to calculate? • dK/dt – the rate at which kinetic energy is transported • by wave. • dU/dt – the rate at which potential energy is transported. dK/dt For :

26. For : dU/dt The quantity is the slope of the string, and if the amplitude of the wave is not too large this slope will be small.

27. Note that: (a) dK and dU are both zero when the element has its maximum displacement ( the element at relaxed length ). (b) The mechanical energy is not constant, because the mass element is not an isolated system—neighboring mass elements are doing work on it to change its energy.

28. 2. Power and intensity • Power：the rate at which mechanical energy • is transmitted. Average power : • Intensity I: For spherical wave:

29. The principle of superposition: Two or more waves travel simultaneously through the same region of space, the superposition principle holds. 1-7 The principle of superposition See动画库\波动与光学夹\2-03波的叠加原理

30. When two or more waves combine at a particular point, they are said to “interfere”, and the phenomenon is called “interference.” We consider a general case, the equation of the two waves are Using the principle of superposition, 1-8 Interference(干涉) of waves

31. where , This resultant wave corresponds to a new wave having the same frequency but with an amplitude 1. If (in phase) , the resultant amplitude is , this case is known as constructive Interference ( 2. If (out of phase) the resultant amplitude is nearly zero, this is destructive interference

32. The resultant amplitude is shown in Fig1-16. x x Fig 1-16

33. In previous section, we consider the effect of superposing two component waves of equal amplitude and frequency moving in the same direction on a string. What is the effect if the waves are moving along the string in opposite direction? 1. We represent the two waves by 18-9 Standing waves

34. Hence the resultant wave is: (a) Eq(18-42) is the equation of a standing wave. It is nota traveling wave, because x and t do not appear in the combination or , required for a traveling wave. or (b) Nodes and antinodesof standing waves In a standing wave, the amplitude is not the same for different particles. The behavior is different from that of a traveling wave.

35. Antinodes The positions where the amplitude has a maximum value. if n=0,1,2,……. or Nodes The positions where the amplitude has a minimum value of zero. n=0,1,2,…, if or

36. —— Forward wave —— Backward wave —— Resultant wave To form a standing wave a a n n n

37. (c) Energy of standing waves For standing waves, the energy can not be transported along it, because the energy cannot flow past the nodes, which are permanently at rest. Fig 1-18 k k k k U U U U

38. Let us discuss the case when a transverse pulse wave travels along a string and reaches an end (boundary). • What will happen when it is reflected at the boundary? • If the reflection end is a fixed on, • the reflected pulse is inverted (changes a phase of • 180o), loses half wave at the boundary. • (b) If the reflection end is a free one, • the reflected pulse is unchanged, no half wave loss at • the boundary. 2. Reflection at a boundary

39. (a) (b) • Suppose a pulse travels along a string and reaches an end (a) Reflection from a fixed end, a transverse wave undergoes a phase change of 180o (b) At a free end, a transverse wave is reflected without change of phase. Fig 1-19

40. (a) n=1 (b) n=2 (c) n=3 (d) n=4 1) Standing waves in a string fixed at both ends 18-10 Standing waves and Resonance Fig 1-20

41. Thus the condition for a standing wave to be set up in a string of length Lfixed at both ends is is the nth wavelength in this infinite series. n is the number of half-wavelengths in the patterns. is the frequency of the allowed standing waves, (natural frequencies).

42. 2) Resonance in the stretched string • In Fig1-20, a student begins to shake the string. If the frequency of the driving forcematches one of the natural frequencies, we get a resonancein the string. (b) If the student shakes the string at a frequency that differs from one of the natural frequencies, the reflected wave returns to the student’s hand out of phase with the motion of the hand. No fixed standing wave pattern is produced.