  This study introduces novel, exact solutions to the scalar field
Signum-Gordon equation that feature a discontinuity near the light cone. These
solutions, applicable in higher spatial dimensions ($n > 1$), extend previous
limitations to one dimension. Our chosen ansatz leads to an ordinary equation
with exact solutions obtained for $n = 2$ and $3$ spatial dimensions. The shock
wave's energy trapped within the light cone is proportional to the wave's
$n$-dimensional volume and the field discontinuity at the wavefront. The
investigation delves further into the behavior of shock waves when their
driving force, represented by a delta function at the light cone, is disabled.
Disabling this delta function disrupts energy transfer, preventing the wave's
propagation as predicted by analytical calculations. We identify the region
within the light cones where the field remains unaffected. Two-dimensional ($n
= 2$) simulations reveal the formation of intriguing structures upon source
removal. These structures include a central, stable feature resembling an
oscillon and a surrounding ring that breaks down into smaller oscillations.
