Bats and the Doppler Shift

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What compensation for doppler shifts do bats perform to keep their echoes within their hearing range?

Big Ideas: 
  • Bats use returning echoes from their cries to gather information about their environment.

Bats perform echolocation by sending out cries and gathering information about their surroundings based on the echoes that return back to them. The incoming direction of the echoes helps determine where things are located. The time it takes for the echoes to return relates to how far various objects are away. By the intensity of the returning signal they can tell information about the size of the objects and distance as well1,2.

When bats are flying they experience a Doppler-shift in the frequency of the cries they initially sent out compared to the frequency of echoes that returns to them. Horseshoe bats change the frequency of their cries to keep the returning echoes centered within a narrow frequency range where they have very sensitive hearing. This is called Doppler-shift compensation and allows them to hear the beating wings of insects as slight deviations of frequency within their most sensitive hearing range3.

If the bat is moving at vb = 6 m/s we can calculate what frequency they have to cry at so that the echoes they hear are at 76.5 kHz (where they are most sensitive in their hearing).

Assumptions:

Most of the echoes the bats hear are from trees and its surroundings which are stationary.

Calculations:

In determining the frequency that the bat hears returning back to him two Doppler shifts take place4. First the bat is a moving source of frequency f0, and the speed of sound in air is v = 343 m/s, so that the frequency that hits a stationary object is

$ f' = \dfrac{f_o}{1 - \dfrac{v_b}{v}} $

The bat is then an observer moving toward the stationary object that acts like a source of f '.

The bat observes

<br />
\begin{eqnarray}<br />
f'' & =& (1 + \dfrac{v_b}{v}) f' \nonumber \\<br />
& =& \dfrac{(1 + \dfrac{v_b}{v})}{(1 - \dfrac{v_b}{v})} f_o \nonumber<br />
    \end{eqnarray}<br />

If the bat wants to hear f ''= 76.5 kHz, we can solve for the frequency the bat cries at, f0, as

$ f_o = \dfrac{1 - \dfrac{v_b}{v}}{1 + \dfrac{v_b}{v}} f'' = 73.9 \textnormal{ kHz} $

Interpretation:

Compensating for two Doppler shifts, Horseshoe bats change their cry so that the returning echoes they hear are in their sensitive range. A calculation with representative numbers gives what frequency they would have to cry at to hear echoes from stationary objects within their sensitive hearing range.

  • 1. Tuszynski, Dixon, "Biomedical Applications of Introductory physics", p. 201, John Wiley & Sons, Inc. (2002).
  • 2. Ahlborn, "Zoological Physics", p. 355-356, Springer (2004).
  • 3. Thomas, Moss, Vater (Ed.), "Echolocation in Bats and Dolphins", p.17, The University of Chicago Press. (2004)
  • 4. Knight, Jones, Field, "College Physics: a strategic approach", 1st edition, p. 636-637, Pearson Addison-Wesley (2007)

Comments

How does the bat interpret

How does the bat interpret the small amount of energy that is returned from beating insect wings? Does it show up as a weak signal at a slightly different frequency from the main echo, or is it a weak signal with a highly different frequency because the insect wings are moving rapidly, but over a very small distance. Given the small amount of energy in the return from an insect it seems the bat must use some other detection method than amplitude because that would be incredibly small. Perhaps some sort of spectrum analysis?

Dean, you are thinking about

Dean,
you are thinking about the right thing: it is indeed the difference in frequency that matters here. The buzzing sound of insects is in a range that we can hear (think flies or mosquitos, for example). By contrast a bat's cry and hearing is at a much higher frequency. The frequency difference is on the order of a hundred times.
So the answer to your question is that the bat's hearing is not sensitive to the sound the insect makes itself but to the sound waves it bounces off the insect, i.e. its own echolocation cries. It also has a way of being extra-sensitive to the Doppler-shifted sound.
If you want to know more, I found an interesting summary on the hearing of bats here:
http://chargedmagazine.org/2013/12/badass-biology-bat-echolocation/

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