A plane harmonic wave with frequency `omega` propagates at a velocity `upsilon` in a direction forming angles `alpha, beta, gamma` and with `x,y, z` axes . Find the phase difference between the oscillations at the points of medium with coordinates `x_(1), y_(1), z_(1)` and `x_(2), y_(2), z_(2)`.

Explanation

Equation of plane wave is given by
`xi(r,t)=a cos ( omegat-vec(k). vec(r))`, where `'vec(k)=(omega)/( v)hat(n)` called the wave vector
and `hat(n)` is the unit vector normal to the wave surface in the direction of the propagation of wave.
or `c i( x,y,z) = a cos ( omegat-k_(x)x-k_(y)y-k_(z)z)`
`=a cos ( omegat-kx cos alpha-ky cos beta- kz cos gamma)`
Thus `xi(x_(1),y_(1), z_(1), t)= a cos ( omegat- k x_(1) cos alpha- k y_(1) cos beta-kz_(1) cos gamma)`
and ` xi(x_(2), y_(2), z_(2),t)=a cos ( omegat- kx_(2) cos alpha- ky_(2) cos beta- k z_(2) cos gamma)`
Hence the sought wave phase difference
`varphi_(2)-varphi_(1)=k[(x_(1)-x_(2))cos alpha + (y_(1)-y_(2))cos beta+(z_(1)-z_(2))cos gamma]`
or `Deltavarphi=|varphi_(2)-varphi_(2)|=l|[(x_(1)-x_(2))cos alpha+(y_(1)-y_(2))cos beta+(z_(1)-z_(2))cos gamma]|`
`=(omega)/( v)|[(x_(1)-x_(2))cos alpha+(y_(1)-y_(2)) cos beta +(z_(1)-z_(2)) cos gamma]|`