Question

A child playing with a long rope ties one end and holds the other. The rope is stretched taut along the horizontal. The child shakes the end he is holding , up and down, in a sinusoidal manner with amplitude $10cm$ and frequency $3Hz.$ Speed of the wave is $15m/s$ and at $t=0,$ displacement at the child's end is maximum positive. Assuming that there is no wave reflected from the fixed end, so that the waves in the rope are plane progressive waves, answer the following questions.
(Also assume that the wave propagates along the positive x-direction).
Equation of displacement of a point $2.5m$ from the child's end can be expressed as

• A. $y=-\left(0.1m\right)cos\left(18.8rad/s\right)t$
• B. $y=0.1mcos\left(12.5rad/s\right)t$
• C. $y=\left(0.1m\right)sin\left(4rad/s\right)t$
• D. $y=-\left(0.1m\right)sin\left(10rad/s\right)t$

Answer 1

Answer: (A)

$y=-\left(0.1m\right)cos\left(18.8rad/s\right)t$

$\omega =2\pi f=6\pi rad/s$
and $k=\frac{\omega }{v}=\frac{6\pi }{15}=1.26$
$y=0.1cos(1.26x-18.8t)$
At point $2.5m$ from child and equation of displacement
$y=0.1cos(3.15-18.8t)$
$=-0.1cos\left(18.8\right)t$