We are interested in analyzing the velocity of a wave in water. By drawing a sketch of the wave, and labeling it, we decide that the velocity depends on the wavelength (l), the depth of the water (H), the density of the water (r), and the effect of gravity (g). We have measured wave speeds in many situations; the data are given below.
Water Depth (H) [m] | Wave Length (L) [m] | Velocity (v) [m/s] |
1.0 | 10.0 | 7.4 |
1.0 | 20.0 | 7.7 |
2.0 | 30.0 | 10.8 |
9.0 | 40.0 | 18.7 |
0.2 | 13.0 | 3.5 |
4.0 | 24.0 | 13.6 |
18.0 | 20.0 | 14.0 |
0.3 | 2.6 | 4.0 |
33.0 | 30.0 | 17.1 |
5.0 | 15.0 | 11.9 |
(a) Construct a plot of wave velocity versus either wave length or water depth. You will see substantial scatter.
(b) Using your plot, estimate the wave velocity for a wave length of 25 meters in water that is 25 meters deep. Is this even possible using this plot?
(c) Perform a dimensional analysis on the parameters (v, ρ, H, g, and λ). After calculating the new dimensionless ratio values, make a dimensionless plot.
(d) Recalculate the answer to question (b).
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