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

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The speed of waves in shallow water is determined by the acceleration due to gravity (g) and the water depth (h). The correct formula for wave speed is (gh)^(1/2), which indicates that wave speed increases with greater depth and gravity. The discussion includes attempts to derive this relationship using kinematic equations and dimensional analysis. Participants express uncertainty about their calculations and seek clarification on the correct approach. Understanding the relationship between wave speed, gravity, and water depth is crucial for accurate predictions in shallow water wave dynamics.
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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The book claims the answer is that all the magnitudes are the same because "the gravitational force on the penguin is the same". I'm having trouble understanding this. I thought the buoyant force was equal to the weight of the fluid displaced. Weight depends on mass which depends on density. Therefore, due to the differing densities the buoyant force will be different in each case? Is this incorrect?

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