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The apparent depth of a swimming pool

When you look vertically downwards into a swimming pool full of water the bottom of the pool seems closer to you than it really is. The apparent depth at this point can be worked out using the formula:

Apparent depth = Real depth/Refractive index of water

However as you look towards the end of the pool the bottom seems to curve upwards we are thinking about a pool of constant depth to start with. You can see why this is by looking at the following diagram. The red crosses mark the positions of the images of points on the pool further and further from the observer who is standing on one end.


The curved black line shows the apparent position of the bottom of the pool.

You can also see that the pool will appear to be shorter than it really is.

If the depth of the pool really does change then the situation becomes more complex and you would need to allow for that when drawing the diagram. The table shows the positions of points on the curve for the apparent position of the bottom of the pool for a 2.0 m deep pool and with the observer's eye 1.33 m above the water surface.

 

Apparent depth
of pool (D)
Distance from near
end of pool (X)
Apparent depth
of pool (D)
Distance from near
end of pool (X)
1.503 0 0.213 3.865
1.488 0.353 0.141 4.422
1.445 0.695 0.097 5.006
1.290 1.312 0.068 5.606
1.086 1.814 0.050 6.225
0.878 2.211 0.038 6.855
0.544 2.816 0.029 7.493
0.335 3.337 0.023 8.135

 

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© Keith Gibbs