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The lensing of gravitational waves (GWs) occurs when GWs experience localgravitational potential. In the weak lensing regime, it has been reported thata simple consistency relation holds between the variances of the magnificationand phase modulation. In this paper, we present two additional consistencyrelations between the averages and variances of the weakly lensed GW signals inwave optics. We demonstrate that these consistency relations are derived as theweak lensing limit of the full-order relations for the averages of theamplification factor and its absolute square. These full-order relations appearto originate from energy conservation and the Shapiro time delay, and they aredemonstrated to hold irrespective of the matter distribution.

New Consistency Relations between Averages and Variances of Weakly Lensed Signals of Gravitational Waves