A wave of frequency f =1000 Hz, propagates at a velocity `v=700 m//s` along the x-axis.
(a). What is the wavelength of the wave?
(b). Find all the points at a given time, at which the phase of the wave exceeds the phase at the origin by `pi//3` radian.
(c ) Find the phase gained at a given point x during a time interval `Deltat 0.5xx10^(-3)s`.

Explanation

The function form of the wave is
`phi=A sin(kx-omegat)`..`(i)` where `omega`=2pif=2000 pi Hz and k =(2pi)/(lambda)`, as `lambda` is given by
`lambda=v/f=(700m//s)/(1000Hz)=0.7 m`
(b). Let the plane at `x_(0)` be shifted by `pi//3` relative to the phase at `x=0`. equation (i) then yields
`Asin((pi)/(3)-omegat)=A sin((2pi)/(0.7m)x_(0)-omegat)`...`(ii)`
Hence, `-omegat+(pi)/(3)+2pin=(2pi)/(0.7 m)x_(0)-omegat,n=0,+- 1`,...
That is `x_(0)=(0.7)/(6)+0.7n)m~~(0.11+0.7n)m,n=0,+-1`,...`(iii)`
(c ). The phase at a point `x`, at a momement `t`, is given by
`phi(x,t)=kx-omegat`
`Deltaophi=omegaDeltat`
`impliesDeltaphi pi-2pixx1000 Hzxx0.5xx10^(-3)s`
`i.e.`, `Deltaphi=-pi rad`
which corresponds to one half cycle...`(iv)`