The Doppler Effect

You have probably heard of the Doppler Effect in conjunction with recent improvements in radar technology: "Doppler Radar" is capable of measuring the velocities of winds, and is instrumental in the identification of tornados. The basic principle is familiar to you when driving as well: the pitch (frequency) of the horn or siren of an approaching vehicle is higher than when it passes you and recedes. This "Doppler Shift" in the frequency also has an important medical usage in the measurement of the speeds of moving fluids inside the body.

When an object which emits a wave (sound or light), the frequency is determined by the object itself. The wavelength, however, is a function of the speed of propagation in the medium as well as the motion of the object within the medium. When the source of the wave approaches you at a speed u s, the wavelength is shortened by an amount u s T, where T is the period of the wave. This is simply due to the motion of the source: when the wave cycle started, the source was at point "a", but when the cycle ends the source has moved u s T closer. (See a 37K quicktime animation). Since the "received" wavelength is related to the "source" wavelength by

l r = l s - u s T

= l s - u s l s / c

= l s (c - u s) / c,

the received frequency is related to the source frequency by

n r = n s c / (c - u s).

Hence the frequency you hear is higher than the frequency emitted by the approaching source. As the source passes you and recedes, the "speed of approach" u s becomes negative, and the frequency you hear becomes lower than the frequency emitted by the now receding source. This shift in frequency is the Doppler Shift.

The same principle applies when the source is stationary but you are approaching it at a speed u r. Now the received wavelength is related to the source wavelength by

l r = l s - u r l r / c

= l s c / (c + u r)

(since the moving receiver now determines the period of the wave) and the received frequency is related to the source frequency by

n r = n s (c + u r) / c.

If both the source and receiver are moving and u s and u r are the speeds with which they are approaching each other (respectively), the Doppler Shift is

n r = n s (c + u r) / (c - u s).

How can this be used to measure (for instance) the speed of flowing blood in vivo? If u is the speed of the blood and n s is the frequency of an ultrasonic source, two Doppler Shifts occur as the source is reflected from the moving blood. First the frequency is shifted as the blood "receives" it, since the blood is moving toward the source. Then, when the echo is "sent" from the blood to the monitor the frequency is shifted again, this time due to the motion of the source. The resultant shift is

n r = n s (c + u) / (c - u).

This allows us to determine the speed of the blood as

u = c (n r - n s) / (n r + n s).

So if the speed of sound in blood is 1500 m/s and a transmission of 1 MHz results in a 1.05 MHz echo, the speed of the blood is 36.6 m/s. Since the measurement is instantaneous, the speed of blood during pulsatile flow can be measured as a function of time. Comparing such measurements to the results of the Pulsatile Flow Mathematica notebook, we find the simulation to reproduce the correct orders of magnitude in terms of both maximum blood velocity and time of flow for blood leaving the heart.

Problems

The next section is about the physics of hearing.



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1996, Kenneth R. Koehler. All Rights Reserved. This document may be freely reproduced provided that this copyright notice is included.

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