  We consider linear and nonlinear hyperbolic SPDEs with mixed derivatives with
additive space-time Gaussian white noise of the form $Y_{xt}=F(Y) + \sigma
W_{xt}.$ Such equations, which transform to linear and nonlinear wave
equations, including Klein-Gordon, Liouville's and the sine-Gordon equation,
are related to what Zimmerman (1972) called a diffusion equation. An explicit
numerical scheme is employed in both deterministic and stochastic examples. The
scheme is checked for accuracy against known exact analytical solutions for
deterministic equations. In the stochastic case with $F=0$, solutions yield
sample paths for the Brownian sheet whose statistics match well exact values.
Generally the boundary conditions are chosen to be initial values $Y(x,0)$ and
boundary values $Y(0,t)$ on the quarter-plane or subsets thereof, which have
been shown to lead to existence and uniqueness of solutions. For the linear
case solutions are compared at various grid sizes and wave-like solutions were
found, with and without noise, for non-zero initial and boundary conditions.
Surprisingly, wave-like structures seemed to emerge with zero initial and
boundary conditions and purely noise source terms with no signal. Equations
considered with nonlinear $F$ included quadratic and cubic together with the
sine-Gordon equation. For the latter, wave-like structures were apparent with
$\sigma \le 0.25$ but they tended to be shattered at larger values of $\sigma$.
Previous work on stochastic sine-Gordon equations is briefly reviewed.
