We present a detailed analysis of the radio spectrum across the lobes of Cygnus A. These observations provide the first critical test of synchrotron spectral aging theory. The results are in good agreement with the jet model for powerful radio galaxies, involving particle acceleration at the hot spots and outflow into the radio lobes, with subsequent energy loss due to synchrotron radiation. The hot spot spectra are well represented by a spectral aging model involving continuous injection of relativistic particles. Both hot spots have spectral break frequencies around 10 GHz. We find an injection index of -0.5 for both hot spots, consistent with diffusive shock acceleration at a strong, nonrelativistic shock in a Newtonian fluid. The low-frequency hot spot emission spectrum falls below the injected power law. This effect is isolated to the hot spots, and is best explained by a low-energy cutoff in the particle distribution, as predicted by Bell in his original work on diffusive shock acceleration. We find that expansion losses may be significant going from the hot spots to the lobes, but that beyond a few arcseconds from the hot spots the dominant energy loss mechanism is synchrotron radiation. The break frequency distribution across each lobe shows a clear trend of decreasing break frequency with distance from the hot spot. The lowest break frequency in the source is 750 MHz, which implies a source age of 6 Myr in minimum-energy magnetic fields. The separation velocity assuming minimum-energy fields is 0.06c(H-0 = 75 km s-1 Mpc-1). This is much larger than the source advance speed required for ram-pressure confinement of the heads of the lobes, assuming a minimum-energy configuration for the particles and fields. A self-consistent model is possible in which the fields are a factor of 3 below minimum-energy values. This results in a source advance speed approximately equal to the separation velocity, almost-equal-to 0.01c, and a source age of 30 Myr. The lobe spectra fall off more steeply than allowed by the continuous injection model at high frequency, but less steeply than exponential. The lobe spectra are well represented by a spectral model involving "one-shot" injection and subsequent energy loss through synchrotron radiation, without continuous isotropization of the pitch-angle distribution. At 1".5 resolution, we find break frequencies much greater than 10 GHz beyond 2" from the hot spots. This is inconsistent with a simple hot spot model involving particle injection at a single point and radiative losses in an axisymmetric outflow from this point, and suggests spatially distributed particle acceleration in the vicinity of the hot spots and/or highly asymmetric outflow. We also find that the injection index across the lobes is -0.7. Such a value is difficult to reconcile with the observed hot spot injection indices of -0.5. The cause for this injection index discrepancy remains a mystery.
