Calculata’ s Mission
AQUiLA MAGAZINE
THE OCEAN
DOLPHIN MATHS
MATHS AND ECHOLOCATION
Dolphins use echolocation to help‘ see’ underwater. They fire off a series of rapid-fire clicks, sending soundwaves through the water. As the soundwaves bounce off nearby objects, the returning echoes tell the dolphins information about the objects, including their size, shape and distance.
The loudness, or amplitude, of these clicks differs. In a series of clicks, the clicks don’ t sound the same – some are quiet, some are loud. Professor Timothy Leighton, at the University of Southampton and his team wondered if they could use this variation in amplitude to see if they could spot the fish from the bubbles. They came up with a model called biased pulse summation sonar( BiaPSS).
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Calculata’ s Mission
Today we’ re going to take a simplified version of their findings( the real maths is much too complicated even for a mathematician like me!) and do some dolphin maths!
DIFFICULTY
Method 1 – subtraction
Let’ s focus on just a tiny part of the clicks the dolphin sends out: two quick clicks, one after the other: click-click.
Suppose the second click is one-third of the amplitude of the first, so it’ s one-third as loud as click one.
Click 1 = amplitude 1
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Click 2 = amplitude 1 / 3
If conditions in the water remain the same, the echoes will come back in the same proportions – so the echo from click 1 will be three times as loud as the echo from click 2. So the fish scatter the first pulse back as amplitude 1, the second pulse as amplitude 1 / 3.
Suppose the dolphin processes the returning signals differently, depending on their amplitude. So, let’ s multiply the second pulse by 3, and subtract it from the first pulse:
1-( 3 x 1 / 3) = 1- 1 = 0
Question: Can you see why we multiply by 3, rather than 2 or 4 or any other number?
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