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The combined longitudinal and torsional plastic waves in a thin-walled tube of rate-independent isotropic work-hardening material are used to illustrate the problems involved when the situation      c f = c 8 = c 2 {c_f} = {c_8} = {c_2}       occurs. Two examples are presented. In the first example, the stress paths in the      σ  ∼  τ  \sigma \sim \tau       plane for the fast and slow simple waves are examined in the region near the singular point      ( σ  ∗  , 0 ) \left ( {\sigma *,0} \right )       where      c f = c 8 = c 2 {c_f} = {c_8} = {c_2}      . For      η  ≥  1 2 \eta \ge \frac {1}{2}      , where      η  \eta       is a nondimensional material constant defined in the paper, there is  no  stress path passing through the singular point      ( σ  ∗  , 0 ) (\sigma *,0)       other than the      σ  \sigma      -axis itself. For      0 &gt; η  &gt; 1 2 0 &gt; \eta &gt; \frac {1}{2}      , there is a family of stress paths emanated from      ( σ  ∗  , 0 ) (\sigma *,0)       which span an angle of      tan −  1 ( 1 −  2 η  ) 1 / 2 {\tan ^{ - 1}}{\left ( {1 - 2\eta } \right )^{1/2}}       with the      σ  \sigma      -axis. In any case, the stress paths for the fast and slow simple waves are  not  orthogonal to each other at the singular point. In the second example, a study is made of the propagation of the plastic wave front into the tube which is initially prestressed at the stress state      ( σ  ∗  , 0 ) \left ( {\sigma *,0} \right )      . It is shown that the solution in the region next to a region of constant state is  not  necessarily a simple wave solution. In fact, an unloading can occur at the plastic wave front which changes its speed from      c 2 {c_2}       to      c 0 {c_0}       at the onset of the unloading.

On wave propagation problems in which 𝑐_{𝑓}=𝑐_{𝑠}=𝑐₂ occurs