In the Muskingum channel routing method, what specific physical process does the 'x' parameter represent or account for, which distinguishes it from simpler routing methods?

In the Muskingum channel routing method, the 'x' parameter specifically represents the weighting factor that accounts for the relative influence of inflow and outflow on the total storage within a channel reach during unsteady flow conditions. This effectively distinguishes between two primary forms of storage: prism storage and wedge storage. Prism storage is the volume of water that would exist in the channel if the flow were uniform and steady, primarily dependent on the outflow from the reach. Wedge storage is the additional, dynamic volume of water (positive or negative) that accumulates or depletes during the passage of a flood wave due to the difference between inflow and outflow. When inflow exceeds outflow on the rising limb of a hydrograph, a positive wedge forms; when outflow exceeds inflow on the falling limb, a negative wedge forms. The 'x' parameter quantifies the proportion of storage that is attributed to the inflow, thereby directly representing the contribution of this dynamic wedge storage. It ranges from 0 to 0.5. A value of x=0 means storage is solely a function of outflow (S = K O), representing pure prism storage or reservoir-type routing with maximum attenuation, where the channel behaves like a level-pool reservoir and wedge storage effects are ignored. A value of x=0.5 means storage is equally influenced by inflow and outflow (S = K (0.5 I + 0.5 O)), indicating a predominantly translatory wave with minimal attenuation. For most natural channels, 'x' typically falls between 0.1 and 0.3, signifying that both prism and wedge storage components are significant, but prism storage usually has a larger weighting. This capacity to explicitly account for and weight the dynamic wedge storage, alongside the more static prism storage, is what distinguishes Muskingum from simpler routing methods, which often implicitly assume x=0 (level-pool routing where storage depends only on outflow) or neglect the dynamic storage changes associated with varying water surface slopes and the difference between inflow and outflow.