Question

John is playing with a laundry line. He is holding one end of the tense line and moves it up and down. In that way, he will get sinus shaped waves moving along the line. It takes 0.5 seconds from one of the ends highest point until it returns to that point. The distance between the highest point to the lowest is 18 centimeter. The wave propagates with the velocity of 12.0 meters/second. At time t=0 [s], Johns end of the line is located at its lowest point. Assume that the waves are not reflected at the other end of the line and therefore not affecting the wave. State the amplitude, angular frequency, wavenumber and the wave length. Also, state the equation of the wave if we assume that the wave is moving in the x-direction. Give also an equation that shows the location of a particle on the line 3.0 meters from Johns hand, as a function of time

Answer 1

from the given data

Time period, T = 0.5 s
Amplitude, A = 18/2 = 9.0 cm
wave speed, v = 12.0 m/s

a) Amplitude, A = 9.0 cm or 0.09 m

b) angular frequency, w = 2*pi/T

= 2*pi/0.5

= 12.6 rad/s

c) wave number, k = w/v

= 12.6/12

= 1.05 rad/m

d) lamda = v/f

= v*T

= 12*0.5

= 6.0 m

e) y(x,t) = A*sin(k*x - w*t + phi)

at x = 0, t = 0, y = -A

-A = A*sin(0 - 0 + phi)

==> phi = -pi/2 radians

so,

y(x,t) = A*sin(k*x - w*t + phi)

= 0.09*sin(1.05*x - 12.6*t - pi/2) <<<<<<-----Answer

f) at x = 3.0 m

y(3,t) = 0.09*sin(1.05*3 - 12.6*t - pi/2)

y(3,t) = 0.09*(1.58 - 12.6*t) <<<<<<-----Answer