Lecture 6 2D 3D waves Sound and light Power and Intensity Doppler effect for i mechanical waves e g sound ii EM waves Example A 2 31 kg rope is stretched between supports 10 4 m apart If one end of the rope is tweaked how long will it take for the resulting disturbance to reach the other end Assume that the tension in the rope is 42 8 N 2D 3D waves 2D circular waves wavefronts lines locating crests small section appear as straight lines far away 3D spherical waves appear as planes far away described by D x t same at every point in yz plane 2D 3D waves D r t A r sin kr t 0 with A r decreasing with r Phase and phase difference phase kx t 0 D x t A sin wavefronts are surfaces of same displacement constant phase x 2 phase difference 2 between adjacent wavefronts separated by Sound waves vsound in air at 20 343 m s larger in liquid solid human ears 20 Hz to 20 k Hz ultrasound 20 k Hz Electromagnetic EM waves oscillations of EM field can travel in vacuum e g light from stars vlight c 3 108 m s in vacuum vsound visible spectrum 400 nm violet blue to 700 nm orange red sound flight fsound EM spectrum visible higher frequencies UV X rays lower frequencies IR micro radio waves index of refraction light slowed down speed of light in vacuum n speed of light in material frequency does not change e g sound wave hitting water c v c fvac vac mat vac fmat vmat mat Power and Intensity Power is rate of transfer of energy by wave Brightness loudness depends also on area receiving power intensity I Pa power to area ratio SI units W m2 Uniform spherical wave I from energy conservation total energy crossing wavefront is same r22 I1 I2 r 2 Psource 4 r 2 1 2 I A energy of oscillations E 12 kA2 Decibels for wide range of human hearing 10 dB log I I0 dimensionless 10 12 W m2 threshold 0 at threshold increasing by 10 dB factor of 10 I increases by a Example The planet Pluto s average distance from the sun is 5 9 10 12 m Calculate the sun s intensity at the distance of Pluto assuming the sun radiates a total power of 4 0 1026 W Doppler effect relative motion between observer and wave source modifies frequency e g pitch of ambulance siren drops as it goes past moving source Pablo detects f Nancy detects f vs 0 f0 if source at rest Doppler effect derivation motion of wave crest once leaves source governed by medium not affected by source moving wave crests bunched up in front stretched out behind 0 speed v f f0 f In time t 3T source moves 3vs T wave crest 0 moves 3vT 3 wave crests in 3vT 3vs T 3 f v v v vs Doppler effect for moving source Doppler effect moving observer not same as source moving motion relative to medium not just source vs observer matters Doppler effect for EM waves no medium use Einstein s theory of relativity 1 vs c 1 vs c red receding source longer wavelength red shift 1 vs c 1 vs c blue approaching source shorter wavelength blue shift where vs is the speed of the source relative to the observer
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