It's what you responded to: It makes it very hard to hold a discussion if we don't pay read each other's posts carefully. Because we're both supposed to work through each methodology. I've given you enough information so that you can work through mine, you haven't given me enough to work through yours. I'm not asking for the detailed math description, I'm just asking for the approach you take. What factors do you plan to use to calculate the doppler shift at O'? The obvious thing to do is to transform the ion velocity to S', then use the transformed velocity to calculate the doppler shift. But that makes calculating the doppler shift redundant, because the ion velocity angle in S' is your end goal.