Question

5. Imagine a string that is fixed at both ends (e.g. a guitar string). When plucked, the string forms a standing wave. The vertical displacement u of the string varies with position r and time t. Suppose u(x,t) = 2 sin(nx) sin(mt/2), for 0 x 1 and t 0. Convince yourself of the following: If we freeze the string in time, it will form a sine curve. Alternatively, if we instead focus on a single position, we will see the point oscillate up and down in time. (See figure below.) u(x, t (Image from textbook) (a) What is the period of the motion in time? (b) Find the rate of change of the displacement with respect to time at a fixed position (This is vertical velocity of that point on the string.) (c) At a fixed time, what point on the string is moving fastest? (d) At a fixed position on the string, when is the string moving fastest? (e) Find the rate of change the displacement with respect to position at a constant time (This is the slope of the string.) (f) At a fixed time, where is the slope of the string the greatest?

Answer 1

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