Question

A sinusoidal wave is propagating along a streched string that lies along the x-axis. The displacement of the string as a function of time is graphed in for particles at x-0 and at x=0.0900 m. (a) what is the amplitude of the wave? (b)what is the period of the wave? (c ) you are told that the two points x=0 and x=0.0900 m are within one wavelength of each other. if the wave is moving in the +x-direction, determine the wavelength and the wave speed. (d) if instead the wave is moving in the -x-direction, determine the wavelength and the wave speed. (e) would it be possible to derermine definitely the wavelength in parts (c ) and (d) if you were not told that the two points were within one wavelength of each other? why ot why not?

Answer 1

Let equation of wave is `y=A sin (omegat-kx+theta)`
`V_(p)=(dy)/(dt)=Aomegacos(omegat-kx+theta)`
`atx=0,t=0`,we have `y =0,v_(p)gt0`
`implies0= A sin thetaimplies theta=0` or `pi`
But `v_(p)gt0`,so `theta=0`
at `t=0, x=0.090 m`
`y=A=4 mmrarr amplitude`
`A=A sin (-k(0.090))`
`implies -k(0.090)=2npi+(pi)/(2)`
`implies -(2pi)/(lambda)(0.090)=2npi+(pi)/(2)`
`implies lambda=(-0.36)/(4n+1)`
for `n=-1,lambda=0.12 m`
from graph, time period `=T=0.04 s`
wave speed: `v=(lambda)/(T)=(0.12)/(0.04)=3 m//s`
if wave is moving in negativex-direction: let wave equation is
`y=A sin(omegat+kx+theta)`
Processing as above, we will get `lambda=(0.36)/(4n+1)`
`for `n=0,lambda=0.36 m`
wave speed: `v=(lambda)/(T)=(0.36)/(0.04)=9 m//s`