A long rope lies along the x-axis on a frictionless horizontal table. A transverse pulse train travels toward the negative x direction. At time t=0, the following observations are made:
• The point at x=0.25m is at its maximum positive displacement of +0.12m and is moving downward.
• The next point to the right that is also at maximum positive displacement is located at x=0.65m.
A stroboscopic light shows that each element of the rope completes 3.5 full oscillations every second.
(a) Determine the wavelength and the frequency of the wave. Show all reasoning.
(b) Write a complete mathematical expression y(x,t) for the displacement of the rope, choosing upward as the positive y direction. Your equation must be consistent with every detail in the observations above.
(c) Using the expression obtained in part (b), calculate the vertical displacement of the rope at position x=0.40m at time t=0.10s.
(d) Explain how the sign of the phase constant you introduced in part (b) ensures that the element at x=0.25m is moving downward at t=0.