Question

A fisherman notices that his boat is moving up and down periodically, owing to waves on the surface of the water. It takes 2.5 s for the boat to travel from its highest point to its lowest, a total distance of 0.62 m. The fisherman sees that the wave crests are spaced 6.0 m apart. (a) How fast are the waves traveling? (b) What is the amplitude of each wave? (c) If the total vertical distance traveled by the boat were 0.30 m, but the other data remained the same, how would the answers to parts (a) and (b) be affected?

Answer 1

(a) How fast are the waves traveling?

from the data given:
Time for wave to travel from highest point to lowest = 2.50 sec.
This corresponds to half of one wavelength.
Therefore, time for one whole wavelength =2.50*2 = 5.00 sec.

The wave crests are spaced 6.00 m. apart.
This corresponds to the distance from one peak to the next peak of the waves.
By definition, this is one whole wavelength.
Hence, one wavelength = 6.00 m.

Since the speed of the wave

= distance / time,

we have speed = 6.00 m. divided by 5.00 sec

= 6.00 / 5.00. = 1.2 m/sec
And that is 1.2 m / sec.; the speed of the wave in a horizontal direction

.
(b) What is the amplitude of each wave?

As for the amplitude of the wave, the highest point of the wave to its lowest point is 0.62 m (The boat always sits on the wave - I assume it isn't sinking!)
This distance corresponds to the peak - peak vertical displacement of the wave.

The boat travels a total distance of 0.62 m from its highest point to its lowest.
Half this distance is the amplitude = 0.31 m

If the total vertical distance traveled by the boat were 0.30 m, but the other data remained the same, how would the answers to parts (a) and (b) be affected?

Wave speed: no change. Amplitude: same, half the displacement, but 0.15 metres this time.