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Lecture 19 Waves Types of waves Transverse waves Longitudinal waves Periodic Motion Superposition Standing waves Beats Waves 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 Waves 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 Waves 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 Waves 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 Waves 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. Waves Vibrational motion Object attached to spring. Spring compressed or stretched a small distance (x) and then released ; a force (Fr) is exerted on the object by the spring Motion of mass as a function of time traces out a sine wave Displacement m Fr x m time x Fr m Net restoring force F r Fxr proportional to x: object undergoes simple harmonic motion Fr kx Waves Hooke’s law Not only applies to springs Fr kx Waves Object vibrating with single frequency Single frequency wave characteristics Displacement versus time (or distance) crest l amplitude Time or distance Displacement Trough l 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 s crests move. v s vt t l vT 1 l v vf l f Waves Sound A plucked string will vibrate at its natural frequency and alternately compresses and rarefies the air alongside it. rarefaction compression Density of Air of Density 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 Waves 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 Waves 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 vf l Waves 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 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 displacement resultant Out of step: Added, crest to trough wave 1 wave 2 resultant Waves Standing waves Two waves (same frequency) travelling in opposite directions Waves reflected back from a fixed position Fundamental 1st Overtone 2nd Overtone 3rd Overtone Nodes; positions of no displacement Antinodes; positions of maximum displacement Distance between successive nodes (antinodes) = l Applications 2 Microwave ovens Musical instruments Waves Standing waves Fundamental 1st Overtone 2nd Overtone 3rd 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 Waves Waves on a stretched string Consider a vibrating string; Wave speed is a function of •tension of the string •Mass per unit length T Wave speed v mL/ T is the tension m is the mass of the string L is the length of the string Waves 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. T Wave speed vfl mL/ 300Nm (400 103 ) v 3 103 kg v(4 104 ) m 2 s 2 200 ms 1 l= 2L v200 ms1 f 250 Hz 2Lm 2 0.4 vf l Waves 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) Waves 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 = fb = f1-f2 Absolute value: beat frequency always positive Waves Beats fb = f1-f2 If frequency difference = zero No beats occur Wave 1 Wave 2 resultant Waves 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 f1 = 264Hz f2 = 266 Hz Beat frequency 2Hz Waves Multiple frequencies of different amplitudes added together •complex resultant Sound waves Resultant tone •Particular musical instrument •Person’s voice Waves 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 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 Resonance: examples Tacoma narrows Bridge, WashingtonElectrical State, Resonance: US, 1940Example: Tuning in radio station Adjust resonant frequency of the electrical circuit to the broadcast frequency of the radio station To “pick up” signal Waves Question Frequency is constant. Its is determined by the source of the wave. 1 Since f Period in constant period Propagation speed depends on properties of string T vfl mL/.
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