Oscillating charge, wave shift

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Hydr0matic
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Consider a charge oscillating along the Y axis. EM waves are emitted around it and hits an observer further down the X axis. The charge then gains speed along the X axis, moving towards the observer. He notes a blueshift in the measured EM wave, ie the Doppler effect.

My question is... Will any movement of the charge along the X axis, no matter how small, shift the wave measured by the observer?
 
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