Smoothness of the solution.

  Mathematically the difference between the wave equation and the Laplace equation is only a minus sign in the second term, but this makes a great difference. While the wave equation propagates singularities, the Laplace equation, like the heat equation, smooths them out. For the example, while at the lower boundary of the plate the temperature has a jump from 100 to zero degrees, in the interior this singularity disappears. Two temperature profiles, one at the lower boundary and one a bit inside the plate, are shown in figure 13. Figure 14 shows the shape of the isotherms in the plate near the jump in the boundary condition. At the jump, the temperature is multiple-valued, but away from the boundary, the isotherms fan out into a smooth solution. Clearly, solutions of elliptic equations such as the Laplace equation look very different from solutions of hyperbolic equations such as the wave equation.

  

Figure 13: Smoothing of a singularity.

  

Figure 14: Smoothing of a singularity.