You provided the usual derivation for plane waves. But you didn't give a proof that it doesn't hold in general. And it is pretty trivial that it doesn't hold for an arbitrary superposition of plane waves.

@freecharly Explicit examples are in the linked thread, there's no real need to repeat them here so long as one is clear on what the results do and do not say.

Dear Emilio - You have not given a proof of your claim that "the wave number $\mathbf k$ and the electric and magnetic fields $\mathbf E$ and $\mathbf B$ are not perpendicular to each other", with the exception of plane waves. The (linked) counter-examples are mostly waves determined by metal boundary conditions, superpositions of plane wave solutions or solutions of the inhomogeneous wave equation. I also wonder where it is shown that $\vec E$ and $\vec B$ and $\vec k$ are not mutually perpendicular in spherical wave solutions of the homogeneous electromagnetic wave equation.

@ZeroTheHero - This seems to describe a TE guided wave propagating between two parallel metal plates in z-direction which has a B component in z-direction. It is well known that such waves can be described as the superposition of two plane TEM waves, that their $E$ and $B$ field are not perpendicular, and that they have $B$ components in propagation direction. This has already been stated by Emilio. But these waves are not free space waves, but waves determined by the metal boundary conditions. I don't doubt at all that such superposition solutions of Maxwell's equations exist.

@ZeroTheHero - But I would like to know whether only plane free space waves have mutually perpendicular $\vec E$, $\vec B$ and propagation direction. Especially also, whether it doesn't hold, as stated, for the free spherical solutions of the homogeneous equations.

@freecharly that's a pretty weaselly target you're setting up, with a helping of spoon-feed-me-the-info-you've-just-linked on top, but sure. This paper has explicit expressions for a vector gaussian beam in free space, including clear longitudinal components; here is a more condensed look at those same fields. That's about as physical as it gets, i.e. the output of a laser pointer with maybe some focusing thrown in. If you want additional examples, then you'll need to be clear about exactly what aspects you're looking for.

@ZeroTheHero -Thanks for the interesting link.

@EmilioPisanty - Thank you for the link to the articles. I have to read them in detail but they seem to refer to rather complicated situations. I am thinking of the much simpler question whether there is a free space spherical analogue to a plane wave where the EM fields and local propagation are still mutually perpendicular. Simply put, are there spherically propagating solutions of the homogeneous EM wave equations for $\vec E$ and $\vec B$ in polar coordinates where this still holds.

@EmilioPisanty - It seems that, in contrast to a purely scalar wave equation, there seem to exist no EM wave solutions that are completely spherically symmetric. There has always to be at least a $\theta$ dependence, which in itself is already rather intriguing. Whether the $\theta$ dependent solutions can still have the property of $\vec E$, $\vec B$, $\vec S$ (or $\vec k$) to be mutually perpendicular is not clear to me. In this context, I also wonder if an atom can emit a photon as a spherically symmetric EM wave with identical probabilities to detects anywhere on a sphere.

@freecharly "They seem to prefer rather complicated situations" is a bizarre thing to say - counter-examples are counter-examples, period, and complexity plays no role in that structure. As to why you seem to think that spherical waves are simpler go-to benchmarks than gaussian beams (which do see everyday use in pretty much any laser you care to name), I'm completely mystified.

If you want the details of spherical waves, Jackson 3rd ed. §9.7 is the place to look (as already referred to, multiple times), but the possible existence of orthogonal-vector solutions in a spherical-wave context is completely irrelevant. What matters are solutions where $E·B \neq 0 \neq E·k$, and those are readily in evidence if you actually bother to look. Your final question veers waaay off topic, but see here for more.

@EmilioPisanty _ You are missing my point. I am not looking for more counter-examples to a general orthogonality misconception, which I never adhered to, I am looking for a possible counter-example to the idea that said orthogonality only occurs in plane waves.

@EmilioPisanty - Emilio, am familiar with the mentioned Jackson chapter. Thank you for pointing it out and for the interesting link to the "final question".

@freecharly I really don't know what you're shooting for, but this isn't the place to do it. The claim is not that the orthogonality is exclusive to plane waves (which is clearly a meaningless claim ─ null EM fields with a vanishing invariant $E·B=0$ are obviously much wider than just plane waves; see here for an example), but that this orthogonality is not a generic property (and only that it is not a generic property) as commonly held by many people, as a result of insufficiently clear materials.

If you've never met the misconception in the wild, I congratulate you! Those of use that have encountered it kindly ask that you help contain it by writing text that cannot be misinterpreted by an inattentive reader. That's the only thing that's going on here.

@EmilioPisanty - Emilio, I really don't want to bother you, but I have the impression that you are a theoretically rather knowledgeable contributor on this site. Therefore the question whether there are some solution of the EM wave equations that are the spherical analogue to transverse plane waves. In Jackson, the considered Helmholtz equations are for the variables $\vec r·\vec E$ and $\vec r·\vec H$ which makes possible transverse EM wave solutions to trivial zero solutions.

I have the following thought experiment. Take two parallel metal plates with a standing transverse EM wave in between. You bend the plates to two concentric spherical shells with the standing transverse wave mode inside. Is this possible and is it still a transverse wave? Then you suddenly eliminate the metal shells and let the EM wave propagate outwards. Is it still a transverse EM wave?

@freecharly so... lemme get this straight. You post an answer that can be considered misleading under some perspectives. That fact is pointed out to you with a clear proposed fix. You get incredibly defensive and intransigent in comments under your answer. You go on the offensive and raise a fuss over already-presented evidence under other people's answers. You then veer off-topic. And then you request personalized tutoring / discussion?

I didn't want to bring this up, but there are appropriate ways to respond to flaws in one's posts.

@freecharly That's not an uninteresting question (though at the level of detail at which you've kept your description of the deformation it is not at a level I would consider answerable - you'd need to be much more specific about what the bending implies and how the standing wave is maintained in the process). You should ask it on main ─ why are you restricting your field of potential answerers to just me?

Or in other words, why does "I answered a question before you did and pointed out a potential flaw in your presentation" suddenly make me available to you for discussion of anything you want?

@EmilioPisanty - My questions are driven by pure interest in physics not by any any supposed or perceived personal intransigence. I don't want to continue on this, but as you bring it up, my answer really assumed that the question was related to a plane wave only. I didn't think at all that the question could refer to general EM waves. I don't take your criticism of my answer at all personal, I only disagreed with the assessment that it claimed a general orthogonality assertion.

@EmilioPisanty - I thought that you might also discuss the general topic as you are pretty expert in the field. If you don't want to discuss it, it's also OK with me. Maybe its a good suggestion to eventually pose a question on the main.

@freecharly Post it on main.

It will force you to actually work through the question to make sure it makes sense, and to have a clear sense of what type of answer you're expecting in terms of scope and structure, before you actually open the floor to discussion.

.... which, frankly, was a big shortcoming of your comments above, in the sense that you oscillate between wanting free-space solutions to wanting spherical-wave solutions to being wholly unclear what you were expecting.

I was thinking of free-space spherical wave solutions, i.e., solution of the homogeneous spherical EM wave equations, or Helmholtz equation if you want.

Without additional boundary conditions in space.

Jackson excludes possible TEM waves by considering only solutions where not both E and H are orthogonal to r.

@freecharly I won't get into a detailed breakdown of how your comments were ineffective at communicating whatever it is you actually wanted to say.

I don't see the point of this discussion.

If you actually have something that can benefit specifically from my perspective, please clarify it now. Otherwise, I have a train to catch soon.

The question is simply whether there are spherically propagating TEM waves or not.

@freecharly ask it on main.

TEM means that there are only electric and magnetic field orthogonal to the propagation direction

I will do it. Thank you...