CHAPTER 10: Mechanical Waves (4 Hours)

1. CHAPTER 10: Mechanical Waves(4 Hours)

2. Learning Outcome: 10.1 Waves and energy (1/2 hour) At the end of this chapter, students should be able to: • Explain the formation of mechanical waves and their relationship with energy.

3. Water waves spreading outward from a source.

4. 10.1 Waves and energy • Waves is defined as the propagation of a disturbance that carries the energy and momentum away from the sources of disturbance. Mechanical waves • is defined as a disturbance that travels through particles of the medium to transfer the energy. • The particles oscillate around their equilibrium position but do not travel. • Examples of the mechanical waves are water waves, sound waves, waves on a string (rope), waves in a spring and seismic waves (Earthquake waves). • All mechanical waves require • some source of disturbance, • a medium that can be disturbed, and • a mechanism to transfer the disturbance from one point to the next point along the medium. (shown in Figures 10.1a and 10.1b)

5. Figure 10.1a Figure 10.1b

6. Learning Outcome: 10.2 Types of waves (1/2 hour) At the end of this chapter, students should be able to: • Describe • transverse waves • longitudinal waves • State the differences between transverse and longitudinal waves.

7. 10.2 Types of waves mechanical wave progressive or travelling wave stationary wave transverse progressive wave longitudinal progressive wave

8. direction of vibrations direction of the propagation of wave particle 10.2 Types of waves Progressive wave • is defined as the one in which the wave profile propagates. • The progressive waves have a definite speed called the speed of propagation or wave speed. • The direction of the wave speed is always in the same direction of the wave propagation . • There are two types of progressive wave, a. Transverse progressive waves b. Longitudinal progressive waves. 10.2.1 Transverse waves • is defined as a wave in which the direction of vibrations of the particle is perpendicular to the direction of the wave propagation (wave speed) as shown in Figure 10.3. Figure 10.3

9. particle direction of the propagation of wave direction of vibrations • Examples of the transverse waves are water waves, waves on a string (rope), e.m.w. and etc… • The transverse wave on the string can be shown in Figure 10.4. 10.2.2 Longitudinal waves • is defined as a wave in which the direction of vibrations of the particle is parallel to the direction of the wave propagation (wave speed) as shown in Figure 10.5. Figure 10.4 Figure 10.5

10. Examples of longitudinal waves are sound waves, waves in a spring, etc… • The longitudinal wave on the spring and sound waves can be shown in Figures 10.6a and 10.6b. Figure 10.6a

11. Sound as longitudinal waves • Longitudinal disturbance at particle A resulting periodic pattern of compressions (C) and rarefactions (R). Figure 10.6b

12. (a) (b) A (c) -A P(pressure) Pm P’ P0 (d) -Pm

13. Figure (a) and (b)

14. Figure (c) Figure (d) – graph of pressure against distance

15. Differences between transverse and longitudinal waves

16. Learning Outcome: 10.3 Properties of waves (2 hours) At the end of this chapter, students should be able to: • Define amplitude, frequency, period, wavelength, wave number . • Analyze and use equation for progressive wave, • Distinguish between particle vibrational velocity, and wave propagation velocity, . • Sketch graphs of y-t andy-x

17. 10.3 Properties of waves 10.3.1 Sinusoidal Wave Parameters • Figure 10.7 shows a periodic sinusoidal waveform. Figure 10.7

18. Amplitude, A • is defined as the maximum displacement from the equilibrium position to the crest or trough of the wave motion. Frequency, f • is defined as the number of cycles (wavelength) produced in one second. • Its unit is hertz (Hz) or s1.

19. Period,T • is defined as the time taken for a particle (point) in the wave to complete one cycle. • In this period, T the wave profile moves a distance of one wavelength, . Thus = Period of the wave Period of the particle on the wave and Its unit is second (s).

20. Wavelength, • is defined as the distance between two consecutive particles (points) which have the same phase in a wave. • From the Figure 10.7, • Particle B is in phase with particle C. • Particle P is in phase with particle Q • Particle S is in phase with particle T • The S.I. unit of wavelength is metre (m). Wave number, k • is defined as • The S.I. unit of wave number is m1.

21. Wave speed, v • is defined as the distance travelled by a wave profile per unit time. • Figure 10.8 shows a progressive wave profile moving to the right. • It moves a distance of  in time T hence Figure 10.8 and

22. The S.I. unit of wave speed is m s1. • The value of wave speed is constant but the velocity of the particles vibration in wave is varies with time, t • It is because the particles executes SHM where the equation of velocity for the particle, vy is Displacement, y • is defined as the distance moved by a particle from its equilibrium position at every point along a wave.

23. 10.3.2 Equation of displacement for sinusoidal progressive wave • Figure 10.9 shows a progressive wave profile moving to the right. • From the Figure 10.9, consider x = 0 as a reference particle, hence the equation of displacement for particle at x = 0 is given by Figure 10.9

24. Since the wave profile propagates to the right, thus the other particles will vibrate. • For example, the particles at points O and P. • The vibration of particle at lags behind the vibration of particle at O by a phase difference of  radian. • Thus the phase of particle at P is • Therefore the equation of displacement for particle’s vibration at P is • Figure 10.10 shows three particles in the wave profile that propagates to the right. Figure 10.10

25.  Phase difference () distance from the origin (x) • From the Figure 10.10, when  increases hence the distance between two particle,x also increases. Thus and

26. The wave propagates to the right : • Therefore the general equation of displacement for sinusoidal progressive wave is given by The wave propagates to the left : where

27. The wave propagates to the right : • Some of the reference books, use other general equations of displacement for sinusoidal progressive wave: The wave propagates to the left :

28. 10.3.3 Displacement graphs of the wave • From the general equation of displacement for a sinusoidal wave, The displacement, y varies with time, t and distance, x. Graph of displacement, y against distance, x • The graph shows the displacement of all the particles in the wave at any particular time, t. • For example, consider the equation of the wave is At time, t = 0 , thus

29. Thus the graph of displacement, y against distance, x is

30. Graph of displacement, y against time, t • The graph shows the displacement of any one particle in the wave at any particular distance, x from the origin. • For example, consider the equation of the wave is • For the particle at x = 0, the equation of the particle is given by hence the displacement-time graph is

31. Example 10.1 : A progressive wave is represented by the equation where y and xare in centimetres and t in seconds. a. Determine the angular frequency, the wavelength, the period, the frequency and the wave speed. b. Sketch the displacement against distance graph for progressive wave above in a range of 0 x   at time, t = 0 s. c. Sketch the displacement against time graph for the particle at x = 0 in a range of 0 t  T. d. Is the wave traveling in the +x or –x direction? e. What is the displacement y when t=5s and x=0.15cm

32. Solution : a. By comparing thus i. ii. iii. The period of the motion is with

33. Solution : a. iv. The frequency of the wave is given by v. By applying the equation of wave speed thus b. At time, t = 0 s, the equation of displacement as a function of distance, x is given by

34. Solution : b. Therefore the graph of displacement, y against distance, xin the range of 0 x   is

35. Solution : c. The particle at distance, x = 0 , the equation of displacement as a function of time, t is given by Hence the displacement, y against time, t graph is

36. d) e)

37. Figure 10.11 Example 10.2 : Figure 10.11shows a displacement, y against distance, x graph after time, t for the progressive wave which propagates to the right with a speed of 50 cm s1. a. Determine the wave number and frequency of the wave. b. Write the expression of displacement as a function of x and t for the wave above.

38. Solution : a. From the graph, By using the formula of wave speed, thus b. The expression is given by

39. 10.3.4 Equation of a particle’s velocity in wave and • By differentiating the displacement equation of the wave, thus • The velocity of the particle, vy varies with time but the wave velocity ,v is constant thus where

40. 10.3.5 Equation of a particle’s acceleration in wave • By differentiating the equation of particle’s velocity in the wave, thus • The equation of the particle’s acceleration also can be written as and where The vibration of the particles in the wave executes SHM.

41. Example 10.3 : A sinusoidal wave traveling in the +x direction (to the right) has an amplitude of 15.0 cm, a wavelength of 10.0 cm and a frequency of 20.0 Hz. a. Write an expression for the wave function, y(x,t). b. Determine the speed and acceleration at t = 0.500 s for the particle on the wave located at x = 5.0 cm. Solution : a. Given The wave number and the angular frequency are given by

42. Solution : By applying the general equation of displacement for wave,

43. Solution : b.i. The expression for speed of the particle is given by and the speed for the particle at x = 5.0 cm and t = 0.500 s is and where vy in cm s1 and x in centimetres and t in seconds

44. Solution : b. ii. The expression for acceleration of the particle is given by and the acceleration for the particle at x = 5.0 cm and t = 0.500 s is and where ayin cm s2and x in centimetres and t in seconds

45. Exercise 10.1 : 1. A wave travelling along a string is described by wherey in cm, x in m and t is in seconds. Determine a. the amplitude, wavelength and frequency of the wave. b. the velocity with which the wave moves along the string. c. the displacement of a particle located at x = 22.5 cm and t = 18.9 s. ANS. : 0.327 cm, 8.71 cm, 0.433 Hz; 0.0377m s1; 0.192 cm

46. Learning Outcome: 10.4 Superposition of waves (1 hour) At the end of this chapter, students should be able to: • State the principle of superposition of waves and use it to explain the constructive and destructive interferences. • Explain the formation of stationary wave. • Use the stationary wave equation : • Distinguish between progressive waves and stationary wave.

47. 10.4 Interference of waves 10.4.1 Principle of superposition • states that whenever two or more waves are travelling in the same region, the resultant displacement at any point is the vector sum of their individual displacement at that point. • For examples,

48. 10.4.2 Interference • is defined as the interaction (superposition) of two or more wave motions. Constructive interference • The resultant displacement is greater than the displacement of the individual wave. • It occurs when y1 and y2 have the same wavelength, frequency and in phase.

49. Destructive interference • The resultant displacement is less than the displacement of the individual wave or equal to zero. • It occurs when y1 and y2 have the same wavelength, frequency and out of phase

50. 10.4.2 Stationary (standing) waves • is defined as a form of wave in which the profile of the wave does not move through the medium. • It is formed when two waves which are travelling in opposite directions, and which have the same speed, frequency and amplitude are superimposed. • For example, consider a string stretched between two supports that is plucked like a guitar or violin string as shown in Figure 10.16. Figure 10.16