relativistic waves

a free massless scalar field \(\varphi\) is described by the Lagrange density \(\mathcal{L} = \partial_\mu\varphi\partial^\mu\varphi\), leading to an equation of motion \(\partial_\mu\partial^\mu\varphi=0\), which is solved by plane waves travelling at the speed of light \(c\). can you explain in what way electromagnetic (or gravitational) waves differ from scalar waves in this respect? where do their wave properties and propagation come from?