Wave vector in spherical coordinates

As a general rule, having a wave vector $vec k$ implies that you have a travelling wave, i.e. one which transports momentum and energy.

This is particularly relevant to part of your question,

would this mean we can find troughs and crests moving along $(r,theta,phi)$ directions respectively,

because you cannot have a travelling wave along the $theta$ direction $-$ the energy would accumulate at the north axis and you'd need a source along the south. A travelling wave along the $phi$ axis is less of a problem (you can have energy circling around indefinitely) but it's still a problem (you would have energy circling around indefinitely).

That said, the general answer to your question is: yes. The way to get at it is to rephrase the Cartesian wavevector into saying that we want to solve the wave equation, $$ nabla^2 f = frac{1}{c^2}partial_t^2 f, $$ first by separating out $f(mathbf r,t)=f(mathbf r)e^{-iomega t}$ into a space- and time-dependent parts, and then handling the resulting Helmholtz equation for the spatial part by using Cartesian coordinates, $$ frac{partial^2 f}{partial x^2} + frac{partial^2 f}{partial y^2} + frac{partial^2 f}{partial z^2} = k^2 f, $$ and using separation of variables. Here the wavenumber arises as a straight definition $k^2=omega^2/c^2$, and once you solve the ODE of each individual coordinate into individual exponentials, the result can be folded back into $f(mathbf r)=e^{imathbf kcdot mathbf r}$, with a wavevector $mathbf k$ whose norm coincides with $k$.

Now, if you want the wave behaviour of $f(mathbf r)$ to go along the spherical coordinate surfaces, then you do the same, but now you address the Helmholtz equation $$ nabla^2 f = k^2 f $$ by expressing $nabla^2$ in spherical coordinates, and you separate the variables as $f(mathbf r)=R(r)Theta(theta)Phi(phi)$. The resulting procedure is standard, so I won't reproduce it here (try e.g. Jackson's Classical Electrodynamics for a thorough solution, or any book on advanced mathematical methods for physics). The upshot is that:

• The solution for the angular variables is that $Theta(theta)Phi(phi) = Y_{lm}(theta,phi)$ must be one of the family of spherical harmonics, which are functions which "wave" exclusively in the angular directions, with nodes along the $theta=rm const$ and $phi=rm const$ directions. However, because of the problems mentioned at the start, particularly in the $theta$ direction, they are not travelling waves $-$ they are standing waves.

  Moreover, the spherical-harmonics solutions do not have any associated "wave vector". Instead, the description of the wave moves over to the two indices $l$ and $m$, which describe the total number of nodes and how they're split into the azimuth and altitude directions.

• For the radial variable, the Helmholtz equation becomes the spherical Bessel equation, $$ r^{2}{frac {d^{2}R}{dr^{2}}}+2r{frac {dR}{dr}}+left(k^2r^{2}-l(l+1)right)R=0, $$ and its solutions are the spherical Bessel functions. As a general rule, all of these are spherical waves, with crests and troughs along the radial direction, but there's several different types:

  • The spherical Bessel functions of the first kind, $j_l(kr)$, describe standing waves, and are regular at the origin.
  • The spherical Bessel functions of the second kind, $y_l(kr)$, describe standing waves, but they have a singularity at the origin, so they can only be used if the origin is not part of the region of interest.
  • The spherical Hankel functions, $h_l^{pm}(kr)$, describe travelling waves, with a source (or sink) at the origin, where they have a singularity.

begin{align*} &text{with} vec{R}&=r,left[ begin {array}{c} cos left( phi right) sin left( theta right) sin left( theta right) sin left( phi right) cos left( theta right) end {array} right] vec{dR}&=left[ begin {array}{c} cos left( phi right) sin left( theta right) sin left( theta right) sin left( phi right) cos left( theta right) end {array} right] ,dr&+left[ begin {array}{c} -rsin left( theta right) sin left( phi right) rcos left( phi right) sin left( theta right) 0end {array} right] ,dphi&+ left[ begin {array}{c} rcos left( phi right) cos left( theta right) rcos left( theta right) sin left( phi right) -rsin left( theta right) end {array} right] ,dtheta end{align*}

thus $~vec{dR}ne (dr~,dphi~,dtheta)$