Fish Size in Water
Name: Laura L.
I was kayaking the other day and saw a rather large fish in no more than 2 1/2 ft
of water. My question is would the water magnify the size of the fish or are things only
magnified underwater while wearing a mask. If the water does magnify things at that depth
how much larger would they appear. I do not want to be telling any untrue fish tales!
Interesting question: the usual question is how deep does an object underwater appear.
(the answer is 3/4 of its actual depth; an object four feet underwater appears to be 3 feet
underwater, using the index of refraction of water to be 4/3).
To answer your question, though, look straight down at one end of the fish from a height h
above the water and at an angle a with a line perpendicular to the surface of the water at
the other end of the fish, which is at a depth d underwater. A ray of light refracts at the
water's surface to an angle b with the perpendicular where sin a = n sin b, where n=4/3 is the
index of refraction of the water. If
you know a little trigonometry and draw a diagram as I described, I am sure you can obtain:
w = (h + d) tan a,
s = h tan a + d tan b,
m = w/s = ((h+d) tan a)/(h tan a + d tan b)
Here w is the apparent length of the fish, s is the actual length and m is the magnification.
Using the approximation sin a = a (accurate at small angles, but good to 2% even at 20 degrees):
m = n(h+d)/(nh+d)
Notice that if the fish is at the surface (d=0), the magnification is 1 and
the fish is seen unmagnified.
At d = h, m = 2n/(n+1).
Using n = 4/3, this gives m=8/7.
But wait!!! The fish looks closer to the surface than it is and so you observe the fish to be
3/4 * 8/7 = 6/7 of its actual size if you correct for it seeming to be closer and therefore
larger than it actually is! At any rate, it looks somewhere between 6/7 and 8/7 of its actual
size, so you are not exaggerating that much (unless you are truly exaggerating!)
I hope this is clear, but I certainly find it confusing... If you would like more words from
me, feel free to write again.
Best, Dick Plano, Professor of Physics emeritus, Rutgers University
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Update: June 2012