Wave vector in spherical coordinates

Question

I understand the wave vector for plane waves in Cartesian coordinates. Along the direction of propagation of wave, $$k = \sqrt{k_x^2+k_y^2+k_z^2} = \frac{2\pi}{\lambda}$$ If $(k_x, k_y, k_z) \neq 0$, this would mean we can find troughs and crests moving along $(x,y,z)$ directions respectively. The dot product in wave equation in Cartesian coordinates, $\vec{dr}=(dx,dy,dz)$ would be $$\vec{k}\cdot\vec{dr} = k_x dx + k_y dy + k_z dz$$

What would the interpretation be in case of spherical coordinates? Can we still have wave vector, $k = (k_{r},k_{\theta},k_{\phi})$ along $(r,\theta,\phi)$ directions such that $$k = \sqrt{k_r^2+k_{\theta}^2+k_{\phi}^2} = \frac{2\pi}{\lambda}$$ If $(k_{r},k_{\theta},k_{\phi}) \neq 0$, would this mean we can find troughs and crests moving along $(r,\theta,\phi)$ directions respectively. Is the following dot product in wave equation in spherical coordinates, $\vec{dr}=(dr,d\theta,d\phi)$ correct? $$\vec{k}\cdot\vec{dr}= k_{r}dr + k_{\theta}rd\theta + k_{\phi}r\sin\theta d\phi$$

Answer 1

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 $f(\mathbf r,t)=f(\mathbf r)e^{-i\omega t}$ 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 $f(\mathbf r)=e^{i\mathbf k\cdot \mathbf r}$, 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.

Answer 2

\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} -r\sin \left( \theta \right) \sin \left( \phi \right) \\ r\cos \left( \phi \right) \sin \left( \theta \right) \\ 0\end {array} \right] \,d\phi\\&+ \left[ \begin {array}{c} r\cos \left( \phi \right) \cos \left( \theta \right) \\ r\cos \left( \theta \right) \sin \left( \phi \right) \\ -r\sin \left( \theta \right) \end {array} \right] \,d\theta \end{align*}

thus $~\vec{dR}\ne (dr~,d\phi~,d\theta)$