A shallow rectangular tank is fitted with a row of vibration paddles that generate periodic water waves. Marks are painted on the tank bottom every 5.0cm along the x-axis, starting from the paddle line at x=0.
High-speed video shows the following:
• At t=0, crests are located directly above the bottom marks numbered 2, 8, 14, and 20.
• At t=0.18s, the same pattern of crests is located above marks 5, 11, 17, and 23.
• Each crest rises 1.4cm above the undisturbed water level; each trough falls 1.4cm below that level.
Assume the wave pattern remains perfectly regular and that reflections from the tank walls are negligible for the times considered.
(1) Part 1: (a) On your own paper, sketch a graph of vertical displacement y versus position x for the water surface at t=0. Clearly label the x-coordinates of at least two consecutive crests and troughs.
(2) Part 2: (b) Using the video information at t=0, determine the wavelength λ of the water wave. Show how you extract this value from the crest positions.
(3) Part 3: (c) Using the change in crest position between t=0 and t=0.18s, calculate the period T and the frequency f of the wave. Explicitly state any ratio or proportion you employ.
(4) Part 4: (d) Compute the wave speed v in the tank. Compare the value obtained from part (c) with the result that would be obtained by applying the standard relationship between frequency and wavelength using your answers from parts (b) and (c). Comment on the consistency of the two methods.
(5) Part 5: (e) If the amplitude were doubled while all other parameters remained unchanged, describe qualitatively how the energy transported by the wave would change, and justify your answer with appropriate wave theory.