Why does a harmonic plane wave have that mathematical form?

[Tosh5457]

Hi, why does the harmonic plane wave have the form below:

$$V(r,t)= acos[\omega (t-\frac{r\cdot s}{v})+\delta ]$$

(r is the position vector, s is the vector that points to the direction the wave is propagating, v is the wave propagation velocity and delta is the phase constant).

 

[andrien]

it just comes out by solving the wave eqn
∇²v=∂_t²v,where c=1 I have put.
You can use spherical coordinates to get it,but more simple would be cartesian coordinate and then using the method of separation of variables, you can get it.

 

[pabloenigma]

More simply, any function that represents a traveling wave is a function of r.s-vt,ie f(r.s-vt)(r and s are vectors ofcourse). If a harmonic function travels as a wave,then the nature of the function is harmonic,ie sine or cosine.

 

[Francessca]

what is the definition of a plane harmonic wave?

 

[vanhees71]

Using spherical coordinates gives you the spherical harmonics and spherical Bessel functions, not the plane wave. Separation in Cartesian coordinates gives plane waves.

 

[sophiecentaur]

what is the definition of a plane harmonic wave?

It's just a posh way of describing a simple wave for which the displacement at any given time is the same over a plane (at right angles to the direction of propagation). It's the same expression as you get for a wave on a one dimensional string and is the limit for spherical wave at a great distance from the source. It's a very convenient approximation to use.