Lecture 19. Waves. Types of waves. Transverse waves Longitudinal waves. Periodic Motion. Superposition Standing waves Beats

Transcription

1 Lecture 19 Waves Types of waves Transverse waves Longitudinal waves Periodic Motion Superposition Standing waves Beats

2 Waves: Transmit energy and information Originate from: source oscillating Mechanical waves Require a medium for their transmission Involve mechanical displacement Sound waves Water waves (tsunami) Earthquakes Wave on a stretched string Non-mechanical waves Can propagate in a vacuum Electromagnetic waves Involve electric & magnetic fields Light, X-rays Gamma waves, radio waves microwaves, etc

3 Mechanical waves Need a source of disturbance Medium Mechanism with which adjacent sections of medium can influence each other Consider a stone dropped into water. Produces water waves which move away from the point of impact An object on the surface of the water nearby moves up and down and back and forth about its original position Object does not undergo any net displacement Water wave will move but the water itself will not be carried along. Mexican wave

4 Transverse and Longitudinal waves Transverse Waves Particles of the disturbed medium through which the wave passes move in a direction perpendicular to the direction of wave propagation Wave on a stretched string Electromagnetic waves Light, X-rays etc Longitudinal Waves Particles of the disturbed medium move back and forth in a direction along the direction of wave propagation. Mechanical waves Sound waves

5 Transverse waves Pulse (wave) moves left to right Particles of rope move in a direction perpendicular to the direction of the wave Rope never moves in the direction of the wave Energy and not matter is transported by the wave

6 Transverse and Longitudinal waves Transverse Waves Motion of disturbed medium is in a direction perpendicular to the direction of wave propagation Longitudinal Waves Particles of the disturbed medium move in a direction along the direction of wave propagation.

7 Vibrational motion Object attached to spring. Spring compressed or stretched a small distance (x) and then released ; a force (F r ) is exerted on the object by the spring Motion of mass as a function of time traces out a sine wave Displacement m F r x x m time F r m Net restoring force F r proportional to x: object undergoes simple harmonic motion Fr x F kx r

8 Fr kx Hooke s law Not only applies to springs

9 Displacement Waves Object vibrating with single frequency Single frequency wave characteristics Displacement versus time (or distance) crest l amplitude Trough l Time or distance Wavelength : Distance between two successive identical points on the wave (λ) Amplitude : Max height of a crest or depth of a trough relative to the normal level Wave Velocity : Velocity at which the wave crests move. s v s vt l vt t 1 l v f v fl

10 Density of Air Waves Sound A plucked string will vibrate at its natural frequency and alternately compresses and rarefies the air alongside it. rarefaction compression Compressed air [increased pressure] Rarefied air [reduced pressure] Air molecules move away from high pressure region >>>>>> setting up longitudinal wave organised vibrations of air molecules>> sound

11 Sound waves-(variation in air pressure) can cause objects to oscillate Example: ear drum is forced to vibrate in response to the air pressure variation

12 Wave characteristics Frequency of waves Frequency (f) of a wave is independent of the medium through which the wave travels. it is determined by the frequency of the oscillator that is the source of the waves. Speed of waves The speed of the wave is dependent on the characteristics of the medium through which the wave is traveling. Wavelength The wavelength (l) is a function of both the oscillator frequency and the speed (v) of the wave such that v fl

13 Superposition Two or more waves travelling through same part of medium at the same time What happens? Adding waves sum of the disturbances of the combined waves If amplitude increases: constructive interference If amplitude decreases: destructive interference Vocal sounds combination of waves of different frequencies Voice individually recognisable

14 displacement Waves Superposition Simple case: Addition of two waves with same wavelength and amplitude In step: Added, crest to crest (trough to trough) wave 1 wave 2 time resultant Out of step: Added, crest to trough wave 1 wave 2 resultant

15 Standing waves Two waves (same frequency) travelling in opposite directions Waves reflected back from a fixed position Fundamental 1 st Overtone 2 nd Overtone 3 rd Overtone Nodes; positions of no displacement Antinodes; positions of maximum displacement Distance between successive nodes (antinodes) = l Applications 2 Microwave ovens Musical instruments

16 Standing waves Fundamental 1 st Overtone 2 nd Overtone 3 rd Overtone String held tightly at both ends Only certain modes of vibration allowed Only certain wavelengths allowed Standing wave must have node at either end Length of string may be changed to get other wavelengths Example: guitar fingering Changing the vibrating length Standing waves: organ pipes

17 Waves on a stretched string Consider a vibrating string; Wave speed is a function of tension of the string Mass per unit length Wave speed v T m/ L T is the tension m is the mass of the string L is the length of the string

18 Example What is the frequency of the fundamental mode of vibration of a wire of length 400mm and mass 3.00 g with a tension of 300N. Wave speed v f l T m/ L v N ( m) kg v (4 10 ) m s 200ms v fl l= 2L f 1 v 200ms 2L 2 0.4m 250Hz

19 Superposition Simple case: Addition of two waves with same frequency and amplitude Beats If the two waves interfering have slightly different frequencies (wavelengths), beats occur. In step (in phase) In step (in phase) Out of step (out of phase)

20 Beats If the two waves interfering have slightly different frequencies (wavelengths), beats occur. Wave 1 Wave 2 resultant Waves get in and out of step as time progresses Result- constructive and destructive interference occurs alternately Amplitude changes periodically at the beat frequency Beat frequency = f b = f 1 -f 2 Absolute value: beat frequency always positive

21 Beats f b = f 1 -f 2 If frequency difference = zero No beats occur Wave 1 Wave 2 resultant

22 Beats Beats can happen with any type of waves Sound waves Beats perceived as a modulated sound: loudness varies periodically at the beat frequency Application Accurate determination of frequency Example Piano tuning Adjust tension in wire and listen for beats between it and a tuning fork of known frequency The two frequencies are equal when the beats cease. Easier to determine than when listening to individual sounds of nearly equal frequencies f 1 = 264Hz f 2 = 266 Hz Beat frequency 2Hz

23 Multiple frequencies of different amplitudes added together complex resultant Sound waves Resultant tone Particular musical instrument Person s voice

24 Question Tuning a guitar by comparing sound of the string with that of a standard tuning fork. A beat frequency of 5 Hz is heard when both sounds are present. The guitar string is tighten the and the beat frequency rises to 8Hz. To tune the string exactly to the frequency of the tuning fork what should be done? a) continue to tighten the string b) loosen the string c) it is impossible to determine

25 Resonance Most objects have a natural frequency: Determined by size shape composition System is in resonance if the frequency of the driving force equals the natural frequency of the system Resonance: examples child being pushed on a swing. Opera singer -breaking glass Voice Air passages of the mouth, larynx and nasal cavity together form an acoustic resonator. Voiced sound depend on resonant frequencies of the total system depends on system s volume and shape

26 Resonance: examples Tacoma narrows Bridge, Washington State, US, 1940 Electrical Resonance: Example: Tuning in radio station Adjust resonant frequency of the electrical circuit to the broadcast frequency of the radio station To pick up signal

27 Question Frequency is constant. Its is determined by the source of the wave. Since f 1 period Period in constant Propagation speed depends on properties of string v fl T m/ L