Just take the Klein-Gordon equation for massive excitations and set the mass equal to zero. Since the Klein-Gordon equation is the Euler-Lagrange equation that's obtained from the Lagrangian for the scalar field, it suffices to set to zero the mass in the Lagrangian.

What's the problem with doing that?

The simple form of the Klein-Gordon equation for massless particles can be

(∂2/∂x2+∂2/∂y2+∂2/∂z2)u=0. The solution of this equation involves the result, physically verifiable, massless particles travel with constant physical velocity, with respect to a frame, assumed like a reference.

The Euler-Lagrange equation for massless particles has the simplest form: £=0. In an equivalent model in which a massless particle has an equivalent mass m, the potential energy of the particle, in a gravitational field, is mgz and, the kinetic energy is m(∂2x/∂t2+∂2y/∂t2+∂2z/∂t2)/2. The solution of the equation proves an elementary massless particle, in the equivalent model, moves with a parabolic trend with respect to the axis z of fall.

you can take the Euler-Lagrange equation for massless particles as an equivalent model to the Klein-Gordon equation for massless particles @

You can see my paper, i have written there.

Daniele

Missing the time term as in wave eqn?