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Harmonic maps give nonconstant stationary wave maps. For $n=2$, inverse stereographic projection givesIt satisfies $|Q|=1$ and $\Delta Q=-|\nabla Q{|}^{2}Q$, so $\varphi (t,x)=Q(x)$ is a smooth global wave map. Its wave map energy is $4\pi$, since $|\nabla Q{|}^2=8/(1+|x{|}^2{)}^2$. Translations and rescalings give further examples; these harmonic maps approach a constant at infinity.

A time-independent wave map is a harmonic map of its spatial domain into the target. For target $S^2$, its equation is $\Delta Q+|\nabla Q{|}^{2}Q=0$. In two dimensions inverse stereographic projection gives a smooth nonconstant example of finite wave map energy $4\pi$.

A wave map is a harmonic map with Lorentzian domain: a critical point of the action obtained by contracting the pullback target metric with the domain Lorentzian metric. For target sphere $S^2$ in Euclidean space and $\square =-{\partial }_t^2+\Delta$, its extrinsic equation is $\square \varphi =\varphi (|{\varphi }_t{|}^2-|\nabla \varphi {|}^2)$, subject to $|\varphi |=1$. The derivative contractions are null forms for wave equations. Wave map Cauchy data must include a tangent initial velocity.