# Physics

The speed of ocean waves depends on their wavelength Î» (in meters) and the gravitational field strength g (in m/sË†2) in this way:

V=KÎ»Ë†p GË†q

Where K is a dimentionless constant. Find the value of the exponents p and q.

This is a dimensional analysis problem. If g and Î» are the only quantities that can affect the speed (ocean depth is actually another, but we will assume a very deep ocean), then some combination
V * Î»^-p * g^-q must be a dimensionless constant, K. The length dimension exponent of the product is 1 -p -q = 0 and the time dimension exponent is -1 +2q = 0.
This leads to -1/2 + q = 0
1/2 - p = 0
p = 1/2 q = 1-p = 1/2
V = K Î»^1/2 g^1/2

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1. The crests of some swells or waves in the ocean are 20 m apart. As they pass, a boat at anchor bobs up and down every 5 seconds. The frequency and wavelength of the water wave are

5 Hz and 20 m

0.2 Hz and 20 m

20 Hz and 5 m

5 Hz and 4 m.

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