A sinusoidal transverse wave of wavelength 20.0 cm travels along a string in the

Question

A sinusoidal transverse wave of wavelength 20.0 cm travels along a string in the positive direction of an x axis. The displacement y of the string particle at x = 0 is given in the figure as a function of time t. The scale of the vertical axis is set by ys= 6 cm. The wave equation is to be in the form of

y = ym sin(kx-wt+?).

(a) At t = 0, is a plot of y versus x in the shape of a positive sine function or a negative sine function? What are (b) ym, (c) k, (d) ?, (e) ?, (f) the sign in front of ?, and (g) the wave speed (speed of the wave along the string) and (h)What is the transverse velocity of the particle at x = 0 when t = 3.00 s?

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Explanation / Answer

? = 22.0 cm, k = 2?/?, from the graph, period T = 14 s, ? = 2?/T

y = 6 sin(2?x/? - 2?t/T + ?)

note at x = 0, t = 0, dy/dt > 0 (from the graph)

dy/dt = -6*2?/T cos? > 0, ? = ?

so y = 6 sin(2?x/? - 2?t/T + ?) = 6 sin(?x/11 - ?t/7 + ?)

a) wave speed v = ?/T = 1.57 cm/s

b) dy/dt = -6?/7 cos(?x/11 - ?t/7 + ?)

at x = 0, t = 3.00 s

dy/dt = -6?/7 cos(-?*3.00/7 + ?) = 0.599 cm/s

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