12.2 SoundComplex Sound WavesBeatsDoppler Effect.Complex waves• In general sound waves are a combination of different frequencies. • The different frequencies can be determined by mathematical procedure called a Fourier Transform. TimedisplacementFrequencyrelativeamplitudeComplex waves consist of different frequencycomponents , i.e. harmonics.Frequency dependence of HearingBeatsSuperposition of two waves with different frequencies produce oscillation in amplitude. Due to constructive and destructive interference.Tbb21b1fffT==−Beat FrequencyTuning musical instrumentsThe beat frequency for two musical instruments is zero when the two are in tune. (have the same frequency)2Doppler EffectDoppler effect- the shift in frequency of a wave where the source and observer are moving relative to one another.Two different cases:Observer moving – source stationarySource moving- observer stationary.Observer moving toward a Stationary source (Relative Velocity Increases)++== =+λoo oosssvv vv vff(1)fvv•Relative velocity of wave (vo+ v ) increases. •Frequency increasessosofvvff =−V = speed of soundVo=speed of observerObserver moving away from a stationary source (Relative velocity decreases) −−== =−λoo oosssvv vv vff(1)fvvVososofvvff −=−Relative velocity of wave (v-vo) decreases. Frequency decreases.Source moving toward a stationary observer(wavelength in the medium decreases)•When the source is moving the wavelength of the wave in the media is changed• source approaches observer A•Wavelength decreases and frequency heard by observer A increases• Source moves away from observer B.•Wavelength increases and frequency heard by observer B decreases.vSource Moving Toward observer A•Wavelength decreasesosss sssvvfvT vT vT==λ− −ossvffvv=−Moving Source Observerλ=λ −ssvT•Frequency increasesDistance traveled bysource in one periodSource Moving Away from observer B•Wavelength increasesosss sssvvfvT vT vT==λ+ +ossvffvv=+Moving SourceObserverλ=λ +ssvT•Frequency decreasesλ3Observer and source movingoossvvffvv⎛⎞+=⎜⎟−⎝⎠vsvovsourceobserver• The frequency increases when the source and observer are moving towardeach other. Observer and source moving⎛⎞−=⎜⎟+⎝⎠oossvvffvvvsvovsourceobserver• The frequency decreases when the source and observer are moving awayeach other. ExampleA fire truck is approaching an observer with a speed of 30 m/s. The siren has a frequency of 700 Hz. What frequency does the observer hear as the truck approaches? What frequency is heard after the truck passes. speed of sound 340 m/sApproaching sourceoos sssvvvff fvv vv⎛⎞⎛⎞+==⎜⎟⎜⎟−−⎝⎠⎝⎠()340700 700 1.09 763Hz340 30⎛⎞===⎜⎟−⎝⎠Departing sourceoos sssvvvff fvv vv⎛⎞⎛⎞+==⎜⎟⎜⎟−−⎝⎠⎝⎠()340700 700 0.92 643Hz340 30⎛⎞===⎜⎟+⎝⎠vsis negativevspositiveTwo trains are approaching each other each moving at 34 m/s. One train sounds a whistle at a frequency of 1000 Hz. Find the frequency of sound heard by an observer on the other train.ssssoof22.1f3434034340fvvvvf =−+=++=vsvovHz10x22.1f3o=Note each motion increases the frequency by about vtrain/v =10%.The net increase is about 20% This is true when vsource<< v.Approximate solution at low speeds.ssssssofvv11f)vv1(vvfvvvf−=−=−=ssof)vv1(f +≈At low speed vs<<vSource moving toward observer.Using the relation x1x11+≈−When x<<1Approximate solution for twotrains approachingssossossoofvv1vv1fvvvvvvfvvvvf⎟⎠⎞⎜⎝⎛+⎟⎠⎞⎜⎝⎛+≅⎟⎟⎠⎞⎜⎜⎝⎛−⎟⎠⎞⎜⎝⎛+=−+=soss2ososofvvvv1fvvvvvvv1f⎟⎠⎞⎜⎝⎛++≅⎟⎠⎞⎜⎝⎛+++=sossofvvvvff⎟⎠⎞⎜⎝⎛+≅−• The shift in frequency is approximately proportional to the ratio of the train velocities to speedof sound as we found in the previous example.• This is a good approximation when the train velocities are slow comparedto the speed of sound.• This is a good approximation for the Doppler shift of electromagnetic waves.negligible4Doppler shift of Electromagnetic waves• Electromagnetic waves are also shifted by the Doppler effect.• Since EM waves travel in a vacuum the equations governing the shift are different.• The same shift is observed for moving source or moving observer.• For motion with speeds less than the speed of light the relation is the same as for the approximate shift for sound waves when u<<v.suff(1 )c=±u = relative velocities of source and observe.c = speed of lightPositive sign when approachingNegative sign when moving away.Doppler Radarf1f2fDoppler radar is used to determine the speed of a car.The beat frequency between the Doppler shifted frequency and the initial frequency is measured to determine the speed of the car.f1 = fs(1+u/c)f2= f1(1+u/c) = fs(1+u/c)2 = fs(1+2u/c + (u/c)2)negligiblebeat frequency =