See why a smooth wave can steepen into a shock

Consider a scalar right-moving approximation whose local characteristic speed is c₀+αu, where u is a small velocity-like perturbation and α is positive. Along each characteristic, u remains constant before characteristics intersect. Its initial value therefore fixes that characteristic’s speed. A descending portion of the initial profile places a faster high-amplitude region behind a slower low-amplitude one. Those characteristics approach one another, so a compressive slope becomes progressively steeper instead of translating rigidly.

For a weakly nonlinear isentropic right-going simple wave in a constant-γ ideal gas, the leading coefficient is α=(γ+1)/2 when u is particle velocity. This relation assumes no left-going disturbance, negligible dissipation and dispersion over the smooth evolution, and small amplitude relative to c₀. It is an asymptotic model, not a complete gas-dynamics equation for arbitrary shocks. Small nonlinear corrections can accumulate over many wavelengths, so small amplitude does not guarantee a linear approximation for arbitrarily long propagation.

Differentiate the characteristic position with respect to its initial label. The mapping from labels to positions first loses local invertibility when one plus αt times the initial slope reaches zero. If the initial derivative has a strictly negative minimum, the earliest positive crossing time is minus the reciprocal of α times that minimum. A purely increasing smooth profile has no such forward-time compression in this model, although boundaries or other dynamics could still matter in a different problem.

For a sinusoidal initial profile U sin(kξ), the most negative derivative is −Uk for positive U and k. The predicted first steepening time is therefore 1/(αUk). This is a gradient-catastrophe time: the smooth solution’s derivative grows without bound while its value remains finite. Continuing the characteristic drawing past intersection produces multiple values at one location, which cannot be interpreted as several simultaneous fluid velocities in this single-fluid scalar description. The failure identifies where a new mathematical and physical treatment is needed.

Write the scalar model in conservation form with flux c₀u+αu²/2. A discontinuity joining left and right states must move at the Rankine–Hugoniot speed obtained from that flux. For α positive, a compressive entropy-admissible shock has the larger state on the left, with incoming characteristics meeting the shock from both sides. A small positive viscosity selects this continuation in the scalar model. Full gas shocks additionally require mass, momentum, energy, and thermodynamic entropy accounting; this scalar calculation does not replace them.

The spectral lab can help reveal a separate issue: representing sharp spatial structure requires high spatial modes. Set coefficient decay to one and predict how increasing mode count from one to twelve changes the detail in its fixed finite sine sum. It does not evolve the nonlinear equation, cross characteristics, or select an entropy solution. Adding Fourier modes to a prescribed curve is not a physical mechanism for shock formation. Use the lab to inspect resolution, and use the characteristic calculation to establish steepening.