One end of a long rope is tied to a fixed vertical pole. The rope is stretched horizontally with a tension `8 N` . Let us consider the length of the rope to be along x-axis. A simple harmonic oscillator at `x=0` generates a transverse wave of frequency `100 Hz` and amplitude `2 cm` along the rope. Mass of a unit length of the rope is `20 g//m`. ignoring the effect of gravity, answer the following questions.
which of the following is correct?

Options

A. The wave propagates with a fixed speed and any particle of the medium vibrates with the same fixed speed.
B. The wave propagates with a fixed speed but any particle of the medium vibrates with a varible speed.
C. The wave propagates with a varible speed but any particle of the medium vibrates with some fixed speed.
D. The wave propagates with a varible speed and any particle of the medium also vibrates with a varible speed.

Answer

B

Explanation

`v=sqrt((T)/(mu))=20 m//s`
`lambda=(v)/(f)=(20)/(100)=0.2 m=20 cm`
and `k=(2pi)/(lambda)=10pi, omega=2pif=200pi`
So `y=-0.2cos(10pix-200pit)`
wave velocity is constant for a medium but particle velocity keeps changing.
as `v=t'=4pi sin(10pix-200pit)`
`(d^(2)y)=-0.2x(200pi)^(2) cos(10pix-200pit)`
For `a_(max)=-aomega^(2)=-7888 m//s^(2)`
`|a_(max)|=7888 m//s^(2)`
As frequency doubles, `lambda` becomes half, speed of wave remains same.