A transverse wave on a string is described by the following wave function. y = (

Question

A transverse wave on a string is described by the following wave function.
y = (0.090 m) sin [(x/9 + 3t)]

(a) Determine the transverse speed and acceleration at t = 0.260 s for the point on the string located at x = 1.70 m.

i)................. m/s
ii)................ m/s2

(b) What are the wavelength, period, and speed of propagation of this wave?
i)................. m
ii)................ s
ii)................ m/s

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Explanation / Answer

Equation of transverse wave on a string is
             y = (0.090 m) sin [(x/9 + 3t)]
compare this with y = A sin [kx +wt] amplitude A = 0.09 m propagation constant k = /9 rad / m angular frequency w = 3 rad / s
(a) the transverse speed v = dy / dt          v = 0.09 cos [(x/9 + 3t)] ( (3)             = 0.848 cos [(x/9 + 3t)]
speed at t = 0.26 s and x = 1.7 m is     v = 0.848 cos [ (*1.7 / 9 ) + 3*0.26 ]        = 0.848 cos [ 3.043 ]        = 0.848 *(-0.9952 )        = -0.8439 m / s transverse accleration a = dv / dt               a = 0.848 {- sin [(x/9 + 3t)]}* ( 3)                  = -7.994 sin [(x/9 + 3t)] acceleration at t = 0.260 s and at x = 1.70 m is a = -7.994 sin (3.043)                 a = -7.994 * 0.098                    = -0.7869 m / s^ 2
(b) the wavelength = 2 / k                                 = 18 m period T = 0.666667 s speed of propagation of this wave v = w / k                                                        = 27 m / s

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