  We present an analysis of wave-mixing in recently developed Josephson
Travelling Wave Parametric Amplifiers (JTWPAs). Circuit simulations performed
using WRspice show the full behaviour of the JTWPA allowing propagation of all
tones. The Coupled Mode Equations (CMEs) containing only pump, signal, and
idler propagation are shown to be insufficient to completely capture complex
mixing behaviour in the JTWPA. Extension of the CMEs through additional state
vectors in the analytic solutions allows closer agreement with WRspice. We
consider an ordered framework for the systematic inclusion of extended
eigenmodes and make a qualitative comparison with WRspice at each step. The
agreement between the two methods validates both approaches and provides
insight into the operation of the JTWPA. We show that care should be taken when
using the CMEs and propose that WRspice should be used as a design tool for
non-linear superconducting circuits such as the JTWPA.
