Finding the correct formula for the speed of waves in shallow water

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SUMMARY

The correct formula for the speed of waves in shallow water is derived from the relationship between gravitational acceleration (g) and water depth (h). The established formula is (gh)^(1/2), which indicates that wave speed is proportional to the square root of the product of gravity and depth. Dimensional analysis confirms this relationship, validating the formula's accuracy. Other options presented, such as (1/2)gh^2, gh, and h/g, do not correctly represent the wave speed in shallow water.

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lostpersson
Question: The speed of waves in shallow water depends only on the acceleration due to gravity (g) and the water depth (h). Which of the following formulas describes the speed of the wave?
a) (1/2)gh^2
b) gh
c) h/g
d)(gh)^(1/2)

Attempt:

Change in V = at
t= ΔV/a

Δx=vt+(1/2)at2
Δx = vt+(1/2)(v/a)
Δx = v[t+1/(2a)]
aΔx = V2 + V

Not sure if my calculations are correct, and if they are, I'm stuck here.
 
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Try dimensional analysis.
 
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