Transverse waves on a string have wave speed 8.00 m/s, amplitude 0.0700 m, and w

Question

Transverse waves on a string have wave speed 8.00 m/s, amplitude 0.0700 m, and wavelength 0.310 m. The waves travel in the -x-direction, and at t = 0 the x = 0 end of the string has its maximum upward displacement. (a) Find the following properties of these waves, frequency f = Hz period T = s wave number k = rad/m (b) Write a wave function describing the wave. (Use the following as necessary: t and x.) y(x, t) = m (c) Find the transverse displacement of a particle at x = 0.360 m at time t = 0.150 s. (d) How much time must elapse from the instant in part (c) until the particle at x = 0.360 m next has maximum upward displacement?

Explanation / Answer

wave speed v = 8 m/s

amplitude A = 0.07 m

wave length = lambda = 0.31 m

(a)

v = lambda*f

f = v/ lambda = 8/0.31 = 25.8 Hz

T = 1/f = lambda/v = 0.31/8 = 0.3875 s

wave number k = 2pi/lambda = 2pi/0.31 = 20.3 rad/m

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(b)

y = A*cos(kx + wt)

angular frequency w = 2*pi*f = 2*pi*25.8 = 162.1 rad/s

y = 0.07*cos(20.3x + 162.2t)

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(c)

y = 0.07*cos((162.2*0.15) + (20.3*0.36) ) = 0.0683 m = 6.83 cm

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(d)

y = 0.07

0.07 = 0.07*cos((162.2*t) + (20.3*0.36) )

t = 0.167 s

time elapsed = 0.167 - 0.15 = 0.017 s

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