Here is the math I attempted. This is a first attempt at calculating SR so it's a bit daunting and I'm not sure if I'm handling the variables correctly...
A train is traveling past an observer on a platform at velocity (v, equal to 0.1c). On the train a pulse of light is fired from a laser, in the direction of travel, at a mirrored target. The pulse leaves the laser and meets the target after (t) amount of time as seen by observers on the train. They also see that the light pulse has a wavelength (λ, equal to 1 meter) that is precisely equal to the distance between the laser and the target (L). The observer on the platform sees the distance between the laser and the target as (L') and the time from the emission of the light pulse to the time it meets the target is (t'). The question to be answered is if the observer on the platform also sees one and only one wavelength of light between the point of emission and the target. The wavelength as measured by the observer on the platform will be λ'.
As the train moves at velocity (v=0.1c), the observer on the platform sees the train travel a distance of (D) in the time (t') in takes the light pulse to go from the laser to the target. So, v=D/t' which gives D=vt'. The distance the pulse of light must travel (d) relative to the observer on the platform is the distance to the target L' plus the distance the train travels, because the target is moving away from the on coming pulse of light as seen by the observer on the platform during that time. So, d=L'+D. The speed of light is c=d/t' for the pulse of light as seen by the observer on the platform. This gives t'=d/c. I would like to point out that this t' is not the same t' which the time dilation equation of SR gives, instead it's just the time the observer measures. I'll address this later.
The distance the pulse travels as seen by the observer on the platform d=L'+D can be combined with D=vt' and t'=d/c to yield d=L'/(1-v/c). Using the length dilation equation of SR...
L'=0.99498743710662 meters
and then
d=1.105541596785133 meters
In this calculation we are checking to see if d is in fact identical to λ'. The Doppler equation for SR in the direction of travel is...
Which yields λ'= 1.105541596785133. So λ'=d, and we can see that both observers, on the platform and on the train, agree that exactly one cycle of the lights frequency was observed before meeting the target.
As mentioned earlier, t' is not t' as used in the time dilation equation of SR. I believe this is because there is a simultaneity shift that occurs at the laser relative to the target as see by the observer on the platform. I think that means that time is flowing differently in the direction of travel compared to the opposite direction. To test this, I decided to also calculate the time it would take for a pulse to return to the laser, reflected off of the mirrored target. By reversing the sign of v to -v, this yields...
d'=0.904534033733291
This should be the distance the light must travel relative to the observer on the platform on the return path.
And using t'=d/c from above, with a value for c=3x10^8m/s...
t''=0.301511344577764 x10^-8 seconds (the return time from target back to laser)
t'=0.368513865595044 x10^-8 seconds (the time originally observed from laser to target)
The total time round trip is t'+t''=0.670025210172808 x10^-8 seconds
Observers on the train should see the light pulse travel 2 meters total equaling...
0.666666666666667 x10^-8 seconds
The time dilation equation of SR...
Total time round trip as measured by the observer on the platform is
0.670025210172808 x10^-8 seconds
This is in agreement with the above calculations so I think it's correct. I'm not sure about my use of the equations and if there isn't a much simpler way of doing this. Thank you for looking at my post and pushing me to try it, any critical response is very appreciated.