One of the approaches they're using is generating lasers that follow a sinosoidal wave. Do any of you smarter than I have information as to why these sinosoidal lasers would be less affected by atmospheric conditions?

You appear to have misunderstood a part of the article. A free-electron laser forces electrons into a sinusoidal path in order to generate laser light, but the generated light is no more sinusoidal than any other laser light.

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