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Have you ever noticed how the sound of a passing car changes pitch as it passes? Small children often imitate this shift as they play with toy cars. This change in frequency is called The Doppler Shift, and it occurs when the sound's source or the observer move relative to one another. When they move toward each other, the observed frequency is higher than the source frequency. When they move away from each other, the observed frequency is lower than the source frequency. Consider an observer at rest listening to the sound of an idling car. The observed frequency, f sub o, is equal to v sub e over lambda, where lambda is the wavelength of the sound and v sub e is the effective velocity of the sound perceived by the observer. In this case v sub e is simply the sound speed v. The observed frequency, f sub o, is thus identical to the source frequency, f sub s If the observer is moving toward the source with a velocity v sub o, the effective velocity of the sound increases. It is equal to the sum of the sound's speed v, and the observer's velocity, v sub o. Since the wavelength lambda of the sound produced by the car remains constant, the observed frequency equals the sum of the sound's speed v, and the observer's velocity, v sub o, divided by lambda. We can use the relation between lambda and the source frequency, f sub s, to express the observed frequency as a function of the source frequency. This expression shows that the observed frequency increases by the ratio of the observer's velocity divided by the sound's speed. If the observer moves away from the source, the effective velocity is equal to the difference between the sound's speed and the observer's velocity. It is straightforward to show that the decrease in the observed velocity is proportional to the ratio of the observer's velocity and the sound's speed Next let's consider the case in which the source moves with a velocity v sub s and the observer is at rest. Since the speed of the sound depends only on the properties of the medium, the effective velocity of the sound measured by the observer is not affected by the motion of the source. However, the movement does affect the sound's effective wavelength. This is because, when the source is moving, it compresses the wave-fronts in front of the source and the wave-fronts behind it become farther apart. The distance between two wave fronts, or the effective wavelength lambda sub e, is thus equal to lambda - the wavelength of the sound at rest - plus or minus the distance traveled by the source during one period, v sub s, divided by the source's frequency, f sub s. The minus sign is used for the effective wavelength in front of the source, and the plus sign for the effective wavelength behind the source. By expressing the wavelength lambda as a function of the source frequency, f sub s, and the sound speed, v, we can rewrite the effective wavelength lambda sub e. We then substitute that expression into the definition of the observed frequency to obtain the relation between the observed frequency and the frequency of a moving source. Thus, the formula for the Doppler effect depends on whether or not the source is approaching or moving away from the observer: the observed frequency increases when the source approaches the observer and decreases when the source moves away from the observer. We can combine the preceding equations to get a single equation for the observed frequency when both the observer and the source are moving. As before, f sub s is the frequency of the source at rest, v the sound's speed, v sub o the velocity of the observer, and v sub s the velocity of the source. The lower part of the two signs is used for the case in which the observer and the source are moving away from each other, and the lower part is used when they are moving toward each other When both the observer and the source are moving much slower than the speed of sound, moving either the source or the observer has about the same effect on the frequency, and thus we can further simplify the equation. Here v sub r is the speed at which the observer and the source are moving towards or away from each other; the sign for v sub r is positive when the two are moving toward each other and negative when they are moving away from each other. If the source nears or exceeds the speed of sound, the effect of the Doppler shift is extreme. As the source approaches the speed of sound, the waves are increasingly compressed by the plane until, at length, all the waves lie together. If the source continues to increase in speed, new sound waves will be positioned in front of earlier ones, and a cone-shaped shock wave will form. This is the source of the sonic boom heard when a super-sonic plane flies over. fobserved > fsource fobserved < fsource Observer and source at rest ... Observer and source at rest ... Observer and source at rest ... Observer and source at rest ... Observer and source at rest ... fo = = ... Observer and source at rest ... fo = = = fs ... Observer and source at rest Observer approaching and source at rest ... Observer approaching and source at rest ... Observer approaching and source at rest ve = v + vo Observer approaching and source at rest ... ve = v + vo Observer approaching and source at rest ... fo = = ... ve = v + vo Observer approaching and source at rest ... fo = = ... ve = v + vo Observer approaching and source at rest ... fo = = ... ve = v + vo Observer approaching and source at rest ... fo = = ... ve = v + vo ... v + vo v / fs Observer approaching and source at rest ... fo = = ... ve = v + vo fo = = fs(1 + ) v + vo v / fs ... Observer approaching and source at rest ... fo = = ... ve = v + vo fo = = fs(1 + ) v + vo v / fs ... Observer approaching and source at rest ... fo = = ... ve = v + vo fo = = fs(1 + ) v + vo v / fs ... Observer approaching and source at rest Observer receding and source at rest ... Observer receding and source at rest ... Observer receding and source at rest ... Observer receding and source at rest ve = v - vo Observer receding and source at rest ... ve = v - vo fo = = fs(1 - ) v - vo v / fo ... Observer receding and source at rest ... ve = v - vo fo = = fs(1 - ) v - vo v / fo ... Observer receding and source at rest Source moving and observer at rest ... Source moving and observer at rest Source moving and observer at rest ... Source moving and observer at rest ... Source moving and observer at rest ... Source moving and observer at rest le = l ± vs Source moving and observer at rest le = l ± vs / Source moving and observer at rest le = l ± vs / fs Source moving and observer at rest le = l - vs / fs ... Source moving and observer at rest le = l - vs / fs le = l + vs / fs ... Source moving and observer at rest le = l - vs / fs le = l + vs / fs Source moving and observer at rest le = l - vs / fs Source moving and observer at rest le = l ± vs / fs Source moving and observer at rest ... lfs ± vs fs le = l ± vs / fs Source moving and observer at rest ... le = = (1 ± ) lfs ± vs fs ... le = l ± vs / fs Source moving and observer at rest Source moving and observer at rest Source moving and observer at rest Source moving and observer at rest Source moving and observer at rest ... fo = fs ( ) v 1 - vs / v Source approaching and observer at rest ... fo = fs ( ) v 1 - vs / v Source approaching and observer at rest fo = fs ( ) v 1 + vs / v Source receding and observer at rest ... fo = fs ( ) v 1 - vs / v fo = fs ( ) v 1 + vs / v Source approaching and observer at rest Source receding and observer at rest ... fo = fs ( ) v 1 - vs / v fo = fs ( ) v 1 + vs / v Source approaching and observer at rest Source receding and observer at rest Source and observer are moving ... Source and observer are moving ... Source and observer are moving ... Source and observer are moving ... Source and observer are moving ... Source and observer are moving away ... Source and observer are moving towards ... fo = fs ( 1 ± ) ... Observer and source are moving much slower than the speed of sound ... Observer and source are moving much slower than the speed of sound ... Observer and source are moving much slower than the speed of sound ... fo = fs ( 1 + ) ... Observer and source are moving much slower than the speed of sound ... fo = fs ( 1 + ) ... fo = fs ( 1 - ) ... Observer and source are moving much slower than the speed of sound

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