A standing wave pattern on a string is described by y(x, t) = 0.082 sin (6x)(cos

Question

A standing wave pattern on a string is described by y(x, t) = 0.082 sin (6x)(cos 48t), where x and y are in meters and t is in seconds. For x 0, what is the location of the node with the (a) smallest, (b) second smallest, and (c) third smallest value of x? (d) What is the period of the oscillatory motion of any (nonnode) point? What are the (e) speed and (f) amplitude of the two traveling waves that interfere to produce this wave? For t 0, what are the (g) first, (h) second, and (i) third time that all points on the string have zero transverse velocity?

Explanation / Answer

Here, k = 6

=> =   1/3 m

since nodes occur at x=0,/2,,3/2

a)   smallest x = 0 m

b) second smallest x = 1/6 m

c)   Third smallest x = 1/2 m

d)    Here, = 48

=>    T = 1/24 sec

e)    v =f = 24 * 1/3   = 8 m/sec

f) Here,   2y0 = 0.082 m

=>   amplitude of the two traveling waves = 0.041 m

g)    first time = T/4   = 1/96 sec

h)    second time = 3T/4 = 1/32 sec

i) Third time = 5T/4 = 5/96 sec

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