The Cauchy problem for the wave equation $$\partial_t^2u=c^2\Delta_xu$$ is not too difficult to solve explicitly in $3$ space dimensions, by the method of spherical means. This yields a close formula for the fundamental solution.

It is much more difficult, if not impossible, to carry out the calculation directly in space dimension $2$. Actually, the explicit solution of the Cauchy problem and the fundamental solution are obtained by extending the initial data to ${\mathbb R}^3$ by $u_j(x_1,x_2)\mapsto v_j(x_1,x_2,x_3):=u_j(x_1,x_2)$ (here $j=0,1$ for the data of $u$ and $\partial_tu$ at initial time). This is called the descent method.