We study properties of solutions $f$ of d'Alembert's functional equations on a topological group $G$. For nilpotent groups and for connected, solvable Lie groups $G$, we prove that $f$ has the form $f(x) = (\gamma(x) + \gamma (x^{-1}))/2$, $x \in G$, where $\gamma$ is a continuous homomorphism of $G$ into the multiplicative group $\mathbb{C}\setminus \{ 0\}$. We give conditions on $G$ and/or $f$ for equality in the inclusion $\{ u \in G \mid f(xu) = f(x)$ for all $x \in G\} \subseteq \{u \in G \mid f(u) =1\}$.