How do all elecromagnetic waves travel at the same speed, if some have shorter wavelengths and therefore take a more circuitous route from A to B?

Question

Starting point: "electromagnetic waves of all frequencies travel through space at the same fixed speed - the speed of light". Hawking (The Universe in a Nutshell).

Which of the following interpretations of this is correct:

1. Waves of all frequencies get from A to B in the same time.

2. Waves of all frequencies move at the same speed, but those of higher frequencies take longer to get somewhere.

If #2 doesn't seem like a valid possibility, let me explain my reasoning. A plane flying at a given velocity X on a flat trajectory will get from Kansas to New York in less time than a plane flying at the same velocity X on a rollercoaster-esque trajectory.

If we define the speed of the plane in terms of how fast it is moving, the speed of the two planes is the same. If we define the speed of the plane just in terms of the time it took to get from the starting point to the finishing point, the speed of the two planes is different - the plane with the rollercoaster-style trajectory takes longer.

So my question is: what does "speed" mean in the context of electromagnetic waves? Is it the time to go from A to B? Or is it the actual speed of the wave, on its up-and-down curved trajectory?

If the latter, then I assume we conclude that some electromagnetic waves get to a destination faster than others (specifically, low-frequency waves get there quicker).

If the former, then surely a high-frequency wave must actually be traveling faster than the speed of light, in order to take its rollercoaster-y path and still get somewhere just as quick as a wave traveling at the speed of light and taking a flatter path through space.

Answer 1

$$c = \frac{\lambda}{T}$$

where $c$ is the speed, $\lambda$ the wavelength and $T$ the period.

As the wavelength increases (so does the period increases which is reflected in low frequency) therefore speed remains unchanged constant

Answer 2

You are supposing that light waves travel in jiggly paths like the graphs people draw for them.

But those jiggly graphs mean something else. It's a graph of force against distance. The graph assumes that the light is traveling in a straight line, and it creates a sideways force, first in one direction and then in another direction.

Does light really travel in a straight line? Yes, mostly. That's why shadows have sharp edges.

Well but when light refracts.... When light goes into glass at an angle, it gets bent. And the short wavelengths get bent more. When it comes out they have moved sideways more. That's why you see colors from a prism. So they don't travel the same path, some of them do go farther than others.

And light does refract around edges. That's why shadows don't have sharp edges. Partly the light source wasn't just a single point so different parts of the source go in slightly different directions. But partly it's that the light actually does bend around the edge of whatever makes the shadow.

Refraction and diffraction are complicated and kind of spooky. I'm not sure anybody completely understands them.

One time when I didn't know any physics I asked questions on a Usenet physics group. The group was kind of like this one except it was completely unmoderated, and there were a collection of smug physics majors and engineers who thought they understood everything, who trolled physics cranks until they went away. When I asked reasonable questions, they would give me answers that didn't explain anything, but that technically did make correct predictions, with an attitude of "I know and you don't, so I'm better than you".

There was an old British guy there who seemed to have a different understanding. The trolls left him alone and he left them alone. I started asking him questions, trying to find out what he knew, and he told me little hints that showed the standard stories didn't quite make sense. I wasn't sure I understood him, and I tried to figure it out.

Then he gave me links to three pictures of a distant galaxy. One of them was taken with radio waves, one with visible light, and one with x-rays. The arms of the galaxy were at different angles in the three pictures! The light must have moved at different speeds, and it was far enough away that the galaxy rotated in the time between! I couldn't tell how MANY times it rotated between those results....

I was spooked. It was proof that different wavelengths traveled at different speeds. What did it mean? I left alt.physics.

A couple of weeks later I realized that if intergalactic space was not sheer vacuum but contained a dilute gas, different light wavelengths would go through it at different speeds.