  This work studies solutions of the scalar wave equation \[\Box_g\phi=0\] on a
fixed subextremal Reissner-Nordstr\"{o}m spacetime with non-vanishing charge
$q$ and mass $M$. In a recent paper, Luk and Oh established that generic smooth
and compactly supported initial data on a Cauchy hypersurface lead to solutions
which are singular in the $W^{1,2}_{loc}$ sense near the Cauchy horizon in the
black hole interior, and it follows easily that they are also singular in the
$W^{1,p}_{loc}$ sense for $p>2$. On the other hand, the work of Franzen shows
that such solutions are non-singular near the Cauchy horizon in the
$W^{1,1}_{loc}$ sense. Motivated by these results, we investigate the strength
of the singularity at the Cauchy horizon. We identify a sufficient condition on
the black hole interior (which holds for the full subextremal parameter range
$0<|q|<M$) ensuring $W^{1,p}_{loc}$ blow up near the Cauchy horizon of
solutions arising from generic smooth and compactly supported data for every
$1<p<2$. We moreover prove that provided the spacetime parameters satisfy
$\frac{2\sqrt e}{e+1}<\frac{|q|}{M}<1$, we in fact have $W^{1,p}_{loc}$ blow up
near the Cauchy horizon for such solutions for every $1<p<2$. This shows that
the singularity is even stronger than was implied by the work of Luk and Oh for
this restricted parameter range.
