A transverse sinusoidal wave is moving along a string in the positive direction

Question

A transverse sinusoidal wave is moving along a string in the positive direction of an x axis with a speed of 80 m/s. At t = 0, the string particle at x = 0 has a transverse displacement of 4.0 cm from its equilibrium position and is not moving. The maximum transverse speed of the string particle at x = 0 is 16 m/s. What is the frequency of the wave? What is the wavelength of the wave? If y(x, t) = y_m sin(kx plusminus omega t + phi) is the form of the wave equation, what are the correct choice of sign in front of

Explanation / Answer

(a) The simple harmonic motion relation

= um/ym = 16/0.040 = 400 rad/s

f = / 2 = 400 / 2 = 64 Hz

(b) Using v = f , we find

= 80/64 = 1.26 m » 1.3 m

(c) The amplitude of the transverse displacement is

ym = 4.0 cm = 0.040 m

(d) The wave number is

k = 2p / = 2p / 1.3 = 5.0 rad/m

(e) The angular frequency, as obtained in part (a), is = 400 rad/s

(f) The function describing the wave can be written as

y = 0.040 sin (5x – 400t + )

where distances are in meters and time is in seconds. We adjust the phase constant f to satisfy the condition y = 0.040 at x= t= 0. Therefore, sin f= 1, for which the “simplest” root is = /2. Consequently, the answer is

y = 0.040 sin (5x – 400t + /2)

(g) The sign in front of is minus.

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