The wave equation is a partial differential equation that describes the movement of waves. Here are just a few of the many descriptions and analyses available on web.
Heat-wave solves the linear wave equation on an isotropic 2D sheet. It does not model dispersion, where the speed of the wave depends on its frequency. Nor does it model non-linear effects where propagation speed varies with amplitude, which is a feature of large perturbations such as shock waves.
Heat-wave also only works with the scalar wave equation, as opposed to the vector wave equation. You need the vector wave equation to accurately model liquid and solid surface waves, and waves passing through different media at oblique angles. So even though the Heat-wave animation shows what looks like liquid surface waves, the Heat-wave simulation is an idealization that does not fully model this.
Heat-wave does model wave damping however, and thus can simulate highly damped or viscous liquid surface waves, such as those on molasses, fairly accurately.
In any case, the scalar wave equation can model model longitudinal waves such as sound waves in a gas, where the scalar field is modelling either density or pressure. It can also model transverse waves such as electromagnetic waves or sound waves inside a homogeneous solid, in which case the scalar field describes amplitude or displacement.