Question

A plane wave of frquency `omega` propagates so that a certain phase of oscillation moves along the `x,y,z` axes with velocities `upsilon_(1), upsilon_(2), upsilon_(3)`. Respectively. Find the wave vector`k`, assuming the unit vetors `e_(x), e_(y), e_(z)` of the coordinate axes to be assigned.

Answer 1

The phase of the oscillation can be written as
`Phi=omegat-vec(k). vec(r)`
When the wave moves alont the `x-` axis
`Phi=omegat-k_(x)x(` On putting `k_(y)=k_(z)=0).`
Since the velocity associated with this wave is `v_(1)`
We have
`k_(x)=(omega)/( v_(1))`
Similarly `k_(y)=(omega)/( v_(2))` and `k_(z)=(omega)/(v_(3))`
Thus `vec(k)=(omega)/(v_(1))hat(e)_(x)+(omega)/(v_(2))hat(e)_(y)+(omega)/(v_(3))hat(e)_(3)`