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159.

Problem 20.65 (RHK)

A device that contains a transmitter and a receiver of waves in a single instrument is used to measure the speed V of a target object (idealised as a flat plate) that is moving directly toward the unit, by analysing waves reflected from it. (a) We have to apply the Doppler equations twice, first with the target as observer and with the target as a source, and have to show that the frequency of the reflected waves at the receiver is related to their source frequency by

where v is the speed of the waves. (b) In great many practical applications, in this case, we have to show that the equation above becomes

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(a)

Let the speed of the target which is moving toward the instrument be V.

Let the source frequency be and be the wavelength of the waves emitted by the transmitter of the instrument.

Frequency of the waves received by the target, , by the Doppler effect will be

Now the moving target becomes a moving source. We will now calculate the frequency measured by the stationary detector.

Let be the wavelength measured by the detector. As the source is moving toward the detector with speed V and is emitting waves of frequency the wavelength will be

Substituting for , we get

(b)

In great many practical situations , then

We thus find