Complex waves 2.2 Sound Complex Sound Waves Beats Doppler Effect. Complex waves consist of different frequency components , i.e. harmonics. • In general sound waves are a combination of different frequencies. • The different frequencies can be determined by mathematical procedure called a Fourier Transform. Frequency dependence of Hearing displacement Time relative amplitude Frequency Beats Superposition of two waves with different frequencies produce oscillation in amplitude. Due to constructive and destructive interference. Beat Frequency Tb fb = Tuning musical instruments The beat frequency for two musical instruments is zero when the two are in tune. (have the same frequency) 1 = f2 − f1 Tb 1 Observer moving toward a Stationary source (Relative Velocity Increases) Doppler Effect Doppler 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 stationary Source moving- observer stationary. V = speed of sound Vo =speed of observer •Relative velocity of wave (vo + v ) increases. •Frequency increases fo = v + vo v + vo v = fs = (1 + o )fs λs v v fo − fs = Observer moving away from a stationary source (Relative velocity decreases) vo fs v Source moving toward a stationary observer (wavelength in the medium decreases) Vo v Relative velocity of wave (v-vo) decreases. Frequency decreases. fo = v − vo v − vo v = fs = (1 − o )fs λs v v fo − fs = − •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. vo fs v Source Moving Toward observer A Source Moving Away from observer B λ Moving Source •Wavelength decreases •Frequency increases fo = v fs v − vs Observer Distance traveled by source in one period λ = λ s − v sT v v fo = = λ s − v s Ts vTs − v s Ts Moving Source Observer •Wavelength increases •Frequency decreases fo = λ = λ s + v sT fo = v v = λ s + v s Ts vTs + v s Ts v fs v + vs 2 Observer and source moving Observer and source moving v source v source vs vo vs observer vo ⎛ v + vo ⎞ fo = fs ⎜ ⎟ ⎝ v − vs ⎠ • The frequency increases when the source and observer are moving toward each other. Example A 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/s ⎛ v − vo ⎞ fo = fs ⎜ ⎟ ⎝ v + vs ⎠ • The frequency decreases when the source and observer are moving away each other. Two 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. v vs Approaching source vo vs positive ⎛ v + vo ⎞ ⎛ v ⎞ ⎛ 340 ⎞ fo = fs ⎜ ⎟ = fs ⎜ ⎟ = 700 ⎜ ⎟ = 700 (1.09 ) = 763Hz ⎝ 340 − 30 ⎠ ⎝ v − vs ⎠ ⎝ v − vs ⎠ Departing source fo = ⎛ v + vo ⎞ ⎛ v ⎞ fo = fs ⎜ ⎟ = fs ⎜ ⎟ ⎝ v − vs ⎠ ⎝ v − vs ⎠ ⎛ 340 ⎞ = 700 ⎜ ⎟ = 700 ( 0.92 ) = 643Hz ⎝ 340 + 30 ⎠ 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 for two trains approaching Approximate solution at low speeds. Source moving toward observer. v v 1 f fo = fs = f = v s v s v − vs v(1 − s ) 1− s v v At low speed vs <<v vs )fs v fo = v + vo ⎛ v + v o ⎞⎛ v ⎞ ⎛ v ⎞⎛ v ⎞ ⎟⎟fs ≅ ⎜1 + o ⎟⎜1 + s ⎟fs fs = ⎜ ⎟⎜⎜ v − vs v ⎠⎝ v ⎠ ⎝ v ⎠⎝ v − v s ⎠ ⎝ negligible v v v ⎞ v ⎞ ⎛ v ⎛ v fo = ⎜1 + s + o + s 2 o ⎟fs ≅ ⎜1 + s + o ⎟fs v v v ⎠ v v ⎠ ⎝ ⎝ v ⎞ ⎛v fo − fs ≅ ⎜ s + o ⎟fs v ⎠ ⎝ v Using the relation 1 ≈ 1+ x 1− x v + vo 340 + 34 fs = fs = 1.22fs v + vs 340 − 34 fo = 1.22 x10 3 Hz vs is negative fo ≈ (1 + observer When x<<1 • 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 compared to the speed of sound. • This is a good approximation for the Doppler shift of electromagnetic waves. 3 Doppler 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. u f = fs (1 ± ) c u = relative velocities of source and observe. c = speed of light Positive sign when approaching Negative sign when moving away. Doppler Radar Doppler radar is used to determine the speed of a car. f1 f u f2 The beat frequency between the Doppler shifted frequency and the initial frequency is measured to determine the speed of the car. negligible f1 = fs (1+u/c) f2 = f1(1+u/c) = fs (1+u/c)2 = fs (1+2u/c + (u/c)2) beat frequency = f2 – fs = 2 u fs c 4