Normalized RMS pressure vs. position relative to the end of the pipe for a circular pipe at 870 Hz. Top: standing-wave (inside) fit; bottom: radiating-wave (outside) fit. The gray shading indicates points excluded from the fit and was determined using equation (2). LS (solid blue line) and empirical fits (dashed orange line) overlay closely with no visible difference, consistent with the nearly identical SSE values in Table 2.