A standing wave pattern on a string is described by y ( x, t ) = 0.048 sin (8 ?

Question

A standing wave pattern on a string is described by y(x, t) = 0.048 sin (8?x)(cos 64?t), 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

given y(x, t) = 0.048 sin (8*pi*x)(cos 64*pi*t)

omparing with standard stationary wave equation

y (x , t) = A*sin(kx) cos(wt)

Amplitude A = 0.048 m

wavenumber K = 2*pi/? = 8*pi

wavelength = ? = 1/4 = 0.25 m

w = 64*pi

w = 2*pi*f = 64*pi

f = 32 Hz

nodes will be at at x = 0 , x = ?/2 = 0.125 , x = ? = 0.25

a) x = 0

b) x = 0.125 m

c) x = 0.25

d) T = 1/f = 1/32 = 0.03125 s

e) speed v = ?*f = 8 m/s

f) A = 0.048/2 = 0.024 m

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the particle at nodes will have zero transverse velocity when they are at extreme positions

g)

v (x/t) = (d/dt)*y(x,t)

v(x,t) = -A*cos(8pix)sin(64pit)

if v = 0

sin64pit = 0

64pit = n*pi

t = n/64

g)

n = 0

t= 0

h) n = 1

t = 1/64 = 0.015625 s

i) n = 2

t = 2/64 = 1/32 = 0.03125 s

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