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General Doppler |
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One peculiarity of the scientific literature is that the Doppler effect is almost invariably defined as something that uniquely affects waves, sometimes even specializing further to light and sound waves. Without the specialization, one could arguably interpret the word “wave” to include any sequence of moving entities, such as a sequence of bullets from a machine gun, but the typical person would not classify a sequence of bullets as a wave. Nevertheless, it is self-evident that the Doppler effect applies just as much to a sequence of bullets as it does to a sequence of wave crests, or a sequence of anything else. |
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For example, if a fighter aircraft is directly approaching us at speed V = 400 mph, and shooting bullets with speed U = 1400 mph (relative to the jet, so 1800 mph relative to us) at a frequency of 5 bullets per second, the bullets will strike us at a frequency of 5[1/(1 – V/U)] = 6.42 bullets per second. (Here we neglect relativistic effects, since they are negligible in this situation.) This is the formula for the Doppler effect, where V is the speed of the approaching source and U is the speed of the transmitted entity, whether it is a wave crest or a bullet or anything else. This is illustrated in the figure below. |
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The source (machine gun) is approaching the origin at speed V = Δx/Δt1, and two consecutive bullets are emitted from the gun separated by the time interval Δt1, moving at speed U = Δx/Δt2. These bullets arrive at the origin separated by the time interval Δt3 given by |
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The emission and reception frequencies are the reciprocals of the time intervals, so the relationship between those frequencies in this situation is |
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For another example, suppose an enemy army is steadily approaching our outpost, and we dispatch two scouts, one hour apart, to ride at constant speed of 21 mph until they sight the army, and then immediately return at the same speed. They arrive back at our outpost 45 minutes apart. How rapidly is the army approaching? The situation is as depicted in the figure below. |
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Letting U be the speed of the scouts and V the speed of the approaching army, we have |
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and also |
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As expected, this is the Doppler formula squared (because it is applied in both directions), and since we have t2 − t1 = 1 hour and t4 − t3 = 3/4 hour, it follows that the army is approaching at the speed V = 3 mph. Notice that we have not provided enough information to know the distance of the approaching army, but we have enough to infer the rate of change of the distance. |
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In essence, the Doppler effect is nothing but a direct consequence of the obvious fact that, at a given speed, an object takes less time to move a shorter distance. This elementary fact applies to any sequence of things, whether it is wave crests or bullets or photons or carrier pigeons. To isolate the pure Doppler effect from the effects of varying speeds we might wish to focus on entities that all share a fixed speed, but strictly speaking even this is an unnecessary restriction. |
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As an aside, we note that there is one kind of frequency to which the Doppler effect does not apply, and that is any intrinsic oscillation of individual entities. For example, if a rifle bullet is spinning at a certain rotational frequency (revolutions per second), it will be spinning at this frequency regardless of the relative speed of encountering the bullet (aside from the negligible relativistic effects). In contrast, the frequency of arrival of a sequence of entities, passing a given location at different times, and being at different locations at the same time, exhibits the Doppler effect depending on the relative speed of encounter, as explained above. |
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