The author starts with a Brownian motion X on \({\mathbb{R}}^ d \)with exponential killing rate 1/2 and considers the Gaussian (free) field \(\Phi =\{\Phi f;f\in H\}\), where H is the Dirichlet space associated with X. Moreover, he considers a continuous function \(\rho\) on \({\mathbb{R}}^ d\) with values in an interval I. Then \((\Phi_ t;t\in I)\) given by the subfields \(\Phi_ t=\Phi (\rho^{-1}(t))\) is a Gaussian process and (\(\Phi,\rho)\) is called a Gaussian wave. The Markov property of \(\Phi\) implies that \((\Phi_ t,t\in I)\) is a non-homogeneous Markov process. The author evaluates its generator. In particular, he gives explicit calculations in the case \(\rho (x,s)=s(x\in {\mathbb{R}}^{d-1},s\in {\mathbb{R}})\).