(c) If the wave has amplitude 1.80 m when it speed is 200 m/s, what will be its amplitude where the water is 9.00 m deep?
To find the amplitude of the wave at a given water depth, you first need to understand the relationship between wave amplitude and water depth. The amplitude of a wave decreases as the water depth increases, according to the principle of wave shoaling.
The formula for wave shoaling is as follows:
A2 = A1 * (h2 / h1)
Where:
A1 = amplitude of the wave at the initial depth (given as 1.80 m)
A2 = amplitude of the wave at the final depth (which we need to find)
h1 = initial water depth (not given)
h2 = final water depth (given as 9.00 m)
Since we do not have the initial water depth, we cannot directly calculate the amplitude at the final depth. However, we have another piece of information: the wave speed in the water. We can use this information to calculate the initial water depth.
The formula to calculate the initial water depth is:
V = sqrt(g * h1)
Where:
V = wave speed (given as 200 m/s)
g = acceleration due to gravity (approximately 9.8 m/s²)
h1 = initial water depth (unknown)
Rearranging the formula, we can solve for h1:
h1 = (V²) / g
Substituting the given values:
h1 = (200²) / (9.8) = 4081.63 m²
Now that we have the initial water depth, we can substitute all the values into the shoaling formula to find the amplitude at the final depth:
A2 = 1.80 * (9.00 / 4081.63)
Calculating the above expression, we find:
A2 ≈ 0.00395 m
Therefore, the approximate amplitude of the wave where the water is 9.00 m deep will be 0.00395 m.