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following Einstein

who thought about exactly that:

EM waves travel at 'c' in every reference frame. This is in distinction to sound or similar vibrations in an underlying medium.

As a result you can't 'travel along with an electromagnetic wave' to make it stationary other than experiencing the entirety of the Universe simultaneously. No matter how fast you go or where you travel, light waves will go by at 'c'.

Special relativity is the consequences of what happens when you follow that seriously. In 1905 it was revolutionary because it took that Maxwell's equations (then only discovered recently) were entirely correct as they were and it was Newton---the God of physics---whose results needed fixing, in contradiction to every scientist's experience until then.

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