CHAPTER 10: Mechanical Waves (4 Hours) SF017 CHAPTER 10: Mechanical Waves (4 Hours)

Learning Outcome: 10.1 Waves and energy (1/2 hour) SF017 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.

Water waves spreading outward from a source.

10.1 Waves and energy Mechanical waves SF017 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)

SF017 Figure 10.1a Figure 10.1b

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.

transverse progressive wave longitudinal progressive wave SF017 10.2 Types of waves mechanical wave progressive or travelling wave stationary wave transverse progressive wave longitudinal progressive wave

10.2 Types of waves Progressive wave 10.2.1 Transverse waves SF017 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. direction of vibrations direction of the propagation of wave particle Figure 10.3

SF017 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 particle direction of vibrations direction of the propagation of wave Figure 10.5

SF017 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

SF017 Sound as longitudinal waves Longitudinal disturbance at particle A resulting periodic pattern of compressions (C) and rarefactions (R). Figure 10.6b

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

Figure (a) and (b) When the tuning fork is struck, its prongs vibrate, disturbing the air layers near it. When the prongs vibrate outwards, it compresses the air directly in front of it. This compression causes the air pressure to rise slightly. The region of increased pressure is called a compression. When the prongs move inwards, it produces a rarefaction, where the air pressure is slightly less than normal. The region of decreased pressure is called a rarefaction. As the turning fork continues to vibrate, the “compression” and “rarefaction” are formed repeatedly and spread away from it.

Figure (c) The figure shows the displacement of the air particles at particular time, t . At the region of maximum compression and rarefaction, the particle does not vibrate at all where the displacement of that particle is zero. Figure (d) – graph of pressure against distance Compression region The particles are closest together hence the pressure at that region greater than the atmospheric pressure (P0). Rarefaction region The particles are furthest apart hence the pressure at that region less than the atmospheric pressure (P0).

Differences between transverse and longitudinal waves Transverse wave Longitudinal wave Particles in the medium vibrate in directions perpendicular to the directions of travel of the wave. Particles in the medium vibrate in directions parallel to the directions of travel of the wave. Crest and trough are formed in the medium. Compression and rarefaction occur in the medium.

Learning Outcome: 10.3 Properties of waves (2 hours) SF017 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 and y-x

10.3 Properties of waves 10.3.1 Sinusoidal Wave Parameters SF017 10.3 Properties of waves 10.3.1 Sinusoidal Wave Parameters Figure 10.7 shows a periodic sinusoidal waveform. Figure 10.7

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.

SF017 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).

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.

is defined as the distance travelled by a wave profile per unit time. SF017 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

The S.I. unit of wave speed is m s1. SF017 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.

10.3.2 Equation of displacement for sinusoidal progressive wave SF017 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

For example, the particles at points O and P. SF017 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

SF017 From the Figure 10.10, when  increases hence the distance between two particle, x also increases. Thus  Phase difference ( ) distance from the origin (x) and

The wave propagates to the right : SF017 Therefore the general equation of displacement for sinusoidal progressive wave is given by The wave propagates to the right : The wave propagates to the left : where

The wave propagates to the right : SF017 Some of the reference books, use other general equations of displacement for sinusoidal progressive wave: The wave propagates to the right : The wave propagates to the left :

10.3.3 Displacement graphs of the wave SF017 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

Thus the graph of displacement, y against distance, x is SF017 Thus the graph of displacement, y against distance, x is

Graph of displacement, y against time, t SF017 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

Example 10.1 : A progressive wave is represented by the equation SF017 Example 10.1 : A progressive wave is represented by the equation where y and x are 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

iii. The period of the motion is with SF017 Solution : a. By comparing thus i. ii. iii. The period of the motion is with

a. iv. The frequency of the wave is given by SF017 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

b. Therefore the graph of displacement, y against distance, x in SF017 Solution : b. Therefore the graph of displacement, y against distance, x in the range of 0 x   is

c. The particle at distance, x = 0 , the equation of displacement as SF017 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

d) e)

SF017 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. Figure 10.11

By using the formula of wave speed, thus SF017 Solution : a. From the graph, By using the formula of wave speed, thus b. The expression is given by

10.3.4 Equation of a particle’s velocity in wave SF017 10.3.4 Equation of a particle’s velocity in wave 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 and where

10.3.5 Equation of a particle’s acceleration in wave SF017 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.

SF017 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

By applying the general equation of displacement for wave, SF017 Solution : By applying the general equation of displacement for wave,

b. i. The expression for speed of the particle is given by SF017 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

b. ii. The expression for acceleration of the particle is given by SF017 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 ay in cm s2 and x in centimetres and t in seconds

Exercise 10.1 : 1. A wave travelling along a string is described by SF017 Exercise 10.1 : 1. A wave travelling along a string is described by where y 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; 0.0377 m s1; 0.192 cm

Learning Outcome: 10.4 Superposition of waves (1 hour) SF017 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.

10.4 Interference of waves 10.4.1 Principle of superposition SF017 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,

SF017 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.

Destructive interference SF017 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

10.4.2 Stationary (standing) waves SF017 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

10.5.1 Characteristics of stationary waves SF017 When the string is pluck, the progressive wave is produced and travel in both directions along the string. At the end of the string, the waves will be reflected and travel back in the opposite direction. After that, the incident wave will be superimposed with the reflected wave and produced the stationary wave with fixed nodes and antinodes as shown in Figure 10.16. Node (N) is defined as a point at which the displacement is zero where the destructive interference occurred. Antinode (A) is defined as a point at which the displacement is maximum where the constructive interference occurred. 10.5.1 Characteristics of stationary waves Nodes and antinodes are appear at particular time that is determined by the equation of the stationary wave.

 = 2  (the distance between adjacent nodes or antinodes) SF017 From the Figure 10.17, The distance between adjacent nodes or antinodes is The distance between a node and an adjacent antinode is  = 2  (the distance between adjacent nodes or antinodes) The pattern of the stationary wave is fixed hence the amplitude of each particles along the medium are different. Thus the nodes and antinodes appear at particular distance and determine by the equation of the stationary wave. Figure 10.17

10.4.3 Equation of stationary waves SF017 10.4.3 Equation of stationary waves By considering the wave functions for two progressive waves, And by applying the principle of superposition hence and where

A cos kx Explanation for the equation of stationary wave SF017 Explanation for the equation of stationary wave A cos kx Determine the amplitude for any point along the stationary wave. It is called the amplitude formula. Its value depends on the distance, x Antinodes The point with maximum displacement = A where and

The point with minimum displacement = 0 Antinodes are occur when SF017 Therefore Nodes The point with minimum displacement = 0 Antinodes are occur when where and Nodes are occur when

SF017 sin t Determine the time for antinodes and nodes will occur in the stationary wave. Antinodes The point with maximum displacement = A Therefore where and Antinodes are occur when the time are

The point with minimum displacement = 0 SF017 Nodes The point with minimum displacement = 0 Therefore At time , t = 0, all the points in the stationary wave at the equilibrium position (y = 0). where and Nodes are occur when the time are

Graph of displacement-distance (y-x) SF017 Graph of displacement-distance (y-x)

Production of stationary wave SF017 Production of stationary wave

10.4.4 Differences between progressive and stationary waves SF017 10.4.4 Differences between progressive and stationary waves Progressive wave Stationary wave Wave profile move. Wave profile does not move. All particles vibrate with the same amplitude. Particles between two adjacent nodes vibrate with different amplitudes. Neighbouring particles vibrate with different phases. Particles between two adjacent nodes vibrate in phase. All particles vibrate. Particles at nodes do not vibrate at all. Produced by a disturbance in a medium. Produced by the superposition of two waves moving in opposite direction. Transmits the energy. Does not transmit the energy.

SF017 Example 10.4 : Two harmonic waves are represented by the equations below where y1, y2 and x are in centimetres and t in seconds. a. Determine the amplitude of the new wave. b. Write an expression for the new wave when both waves are superimposed. Solution : a. b. By applying the principle of superposition, thus

SF017 Example 10.5 : A stationary wave is represented by the following expression: where y and x in centimetres and t in seconds. Determine a. the three smallest value of x (x >0) that corresponds to i. nodes ii. antinodes b. the amplitude of a particle at i. x = 0.4 cm ii. x = 1.2 cm iii. x = 2.3 cm

a. i. Nodes particles with minimum displacement, y = 0 with SF017 Solution : By comparing thus a. i. Nodes particles with minimum displacement, y = 0 with

a. ii. Antinodes particle with maximum displacement, y = 5 cm SF017 Solution : a. ii. Antinodes particle with maximum displacement, y = 5 cm b. By applying the amplitude formula of stationary wave, i. ii. iii.

SF017 Example 10.6 : An equation of a stationary wave is given by the expression below where y and x are in centimetres and t in seconds. Sketch a graph of displacement, y against distance, x at t = 0.25T for a range of 0 ≤ x ≤. Solution : By comparing thus and

The particles in the stationary wave correspond to Antinode SF017 Solution : The particles in the stationary wave correspond to Antinode Node The displacement of point x = 0 at time, t = 0.25(2) = 0.50 s in the stationary wave is where and where and

Therefore the displacement, y against distance, x graph is SF017 Solution : Therefore the displacement, y against distance, x graph is

Exercise 10.2 : 1. The expression of a stationary wave is given by SF017 Exercise 10.2 : 1. The expression of a stationary wave is given by where y and x in metres and t in seconds. a. Write the expression for two progressive waves resulting the stationary wave above. b. Determine the wavelength, frequency, amplitude and velocity for both progressive waves. ANS. : 4 m, 30 Hz, 0.15 m, 120 m s1 2. A harmonic wave on a string has an amplitude of 2.0 m, wavelength of 1.2 m and speed of 6.0 m s1 in the direction of positive x-axis. At t = 0, the wave has a crest (peak) at x = 0. a. Calculate the period, frequency, angular frequency and wave number. ANS. : 0.2 s, 5 Hz, 10 rad s1 ,5.23 m1

SF017 THE END… Next Chapter… CHAPTER 11 : Sound wave