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Doppler Effect

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Introduction In our everyday life, we are used to perceive sound by our sense of hearing. Sounds are the vibrations that travel through the air. It is characterized by the wave quantities which include frequency, wavelength, period and speed. The paper aims to present basic concepts on Doppler Effect in sound. It would help us understand the changes in the relative motion of the source and the observer.

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Questions? Why does the siren on a moving ambulance seem to produce sound with a higher pitch when it approaches an observer then decreases when it recede the observer. Is this simply because of the relative distance between the observer and the ambulance (sound)? Is it because of the loudness of the sound produced by the siren?

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Definition Doppler Effect is the change in the frequency (or wavelength) of any emitted waves, such as a wave of light or sound as the source of the wave approaches or moves away from an observer. This effect was named from the Austrian physicist, Christian Johann Doppler, who first stated the physical principle in 1842.

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Consider the Following Doppler’s principle explains why, if the source of waves and the observer are approaching each other, the sound heard by the observer becomes higher in pitch, whereas if the source and observer are moving apart the pitch becomes lower. For the sound waves to propagate it requires a medium such as air, where it serves as a frame of reference with respect to which motion of source and observer are measured.

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SITUATION 1 Stationary Source and Observers (NO DOPPLER EFFECT) A stationary sound source S emits a spherical wavefronts of one λ apart spread out at speed v relative to the medium air. A stationary sound source S emits a spherical wavefronts of one λ apart spread out at speed v relative to the medium air. In time t, the wavefronts move a distance vt toward the observers, O 1 & O 2. In time t, the wavefronts move a distance vt toward the observers, O 1 & O 2. The number of wavelengths detected by the observer infront and behind the source are the same and equal to vt/λ.The number of wavelengths detected by the observer infront and behind the source are the same and equal to vt/λ.

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So, the frequency f heard by both stationary observers is given by equation (1), f - frequency of sound source v - speed of sound waves t - time λ - wavelength

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Consider the Following What if both of the observers in figure 1 are moving, is there any change in the frequency and wavelength of the source? What if both of the observers in figure 1 are moving, is there any change in the frequency and wavelength of the source?

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SITUATION 2 Stationary Source; Moving Observers Observer 1 moves a distance v o t toward the source at speed v o Observer 1 moves a distance v o t toward the source at speed v o We know that wavefronts also move at speed v towards O 1 in time t at distance vt. The relative speed of the wavefronts with respect to O 1 becomes (v + v O )t. We know that wavefronts also move at speed v towards O 1 in time t at distance vt. The relative speed of the wavefronts with respect to O 1 becomes (v + v O )t. The number of wavelengths intercepted by O 1 at this distance is The number of wavelengths intercepted by O 1 at this distance is (v + v O )t/ λ.

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From equation (1), we have λ = v/ f, f’ becomes (3) This shows that there is an increase in the frequency f’ heard by O 1 as it goes nearer to the sound source as given by, (2)

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If observer 2 moves away from the sound source, the distance traveled by the wavefronts with respect to O 2 in time t, is vt – v o t. Consequently, there would be a decrease in the frequency heard by O 2 as given by, (4)

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Combining Equations (3) and (4), we have (5) (STATIONARY SOURCE; MOVING OBSERVER) In these situations only the frequency heard by the observers changes due to there motion relative to the source. However the wavelength of sound remains constant.

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SITUATION 3 Moving Source; Stationary Observers As the source moves a distance v S T (T=1/f period of wave) toward O 1 there is a decrease in the wavelength of sound by a quantity of v s T. The shortened wavelength λ’ becomes λ’ = λ – v s T

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The frequency f’ of sound wave heard by O 1 increases as given by, (6)

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With respect to observer 2, the wavelength of sound increases, where λ’ becomes λ + vsT. The frequency f’ of sound wave heard by O 2 decreases as given by, (7)

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Combining Equations (6) and (7), we have (8) (MOVING SOURCE; STATIONARY OBSERVER)

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SITUATION 4 Moving Source and Observer From the equations (5) and (8), we can now derive the equation of general Doppler Effect by replacing f in equation (5) with f’ of equation (8). This result to, (MOVING SOURCE AND OBSERVER) (9)

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The ± signs correspond to the direction of the source or observer when they are moving relative to the other. These would determine whether there is an increase or decrease on the frequency heard by the observer during the motion.

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(APPROACHING OBSERVER; RECEEDING SOURCE ) If v o > v s, increase in observed frequency If v o > v s, increase in observed frequency If v o < v s, decrease in observed frequency If v o < v s, decrease in observed frequency (RECEEDING OBSERVER; RECEEDING SOURCE ) Decrease in observed frequency Decrease in observed frequency

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(APPROACHING OBSERVER; APPROACHING SOURCE) Increase in observed frequency Increase in observed frequency (RECEEDING OBSERVER; APPROACHING SOURCE) If v o > v s, decrease in observed frequency If v o > v s, decrease in observed frequency If v o < v s, increase in observed frequency If v o < v s, increase in observed frequency

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