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lecture6 - Lecture 6 Notes 07 06 The Doppler effect The...

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Lecture 6 Notes: 07 / 06 The Doppler effect The Doppler effect is a shift in frequency that results from motion of the source or the observer (or both). Consider first the case of a stationary source and a moving observer. Let f S be the frequency of the source, and f O be the frequency measured by the observer. The source creates a bunch of sound waves with wave fronts separated by a wavelength λ . If the observer were stationary, the number of wave fronts passing him every second would be equal to the frequency of the source, f S = c / λ . However, if the observer is moving towards the source with speed v , he will pass more wavefronts per second: f O = (c+v) / λ . Dividing the frequency of the observer by that of the source, we get f O / f S = (c+v) / c , or If the observer moves away from the source, rather than towards it, the sign of v is simply reversed. Now consider a moving source and a stationary observer. Suppose the source is moving towards the observer with speed v :
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If the source was stationary, it would emit wave crests a distance λ apart, but now, over one period, the source moves a distance vT , so the next wave crest is emitted a distance of only λ - vT = λ - v / f S from the first. The observer therefore sees a wavelength of l O = λ - v / f S and a frequency of f O = c / λ O = c / ( λ - v / f S ) . Multiplying both the numerator and the denominator by f S and using λ f S = c , we obtain Again, if the source is receding from the observer rather than approaching, the sign of v is reversed. The rule of thumb is that the frequency increases if the source and the observer are moving closer with time, and decreases if the source and the observer are getting farther and farther away from each other. Example: Suppose the source is moving towards the observer with speed v S , and the observer is simultaneously moving towards the source with speed v O . What is the frequency measured by the observer, in terms of the source frequency? We can derive the equation for both the source and the observer moving by breaking up the problem into two steps. First, introduce an additional observer A , located between the source S and the observer O , who is stationary with respect to the medium through which the sound is propagating: The stationary observer A measures the frequency of the moving source to be The sound wave passes through A at this frequency, and continues on the observer O . Therefore, A can be treated as a stationary source with frequency f A . The frequency measured by the moving observer O is then
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Example: Suppose that an active sonar on a stationary ship is used to determine a receding ship's speed. A sonar ping is emitted at a frequency of 1400 hz , and the echo returns with a frequency of 1390 hz . The speed of sound in water is approximately 1500 m/s . How fast is the ship moving away?
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