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We demonstrate agreement between measurements and ab initio calculations ofthe frequency shifts caused by distributed cavity phase variations in themicrowave cavity of a primary atomic fountain clock. Experimental verificationof the finite element models of the cavities gives the first quantitativeevaluation of this leading uncertainty and allows it to be reduced to delta nu/ nu = 8.4\times10^-17. Applying these experimental techniques to clocks withimproved microwave cavities will yield negligible distributed cavity phaseuncertainties, less than \pm1\times10^-17.

Evaluation of Doppler Shifts to Improve the Accuracy of Primary Atomic Fountain Clocks