Let $Y$ be a smooth irreducible curve over an algebraically closed field $k$ and $f : X\rightarrow Y$ be a proper flat morphism of relative dimension 1 with smooth generic fiber and whose closed fibers have at worst ordinary double points as singularities. If $x\in X$ is a node in the fiber over $f(x)$, then the complete local ring at $x$ is isomorphic to $k[[x,y,t]]/(xy-ut^e)$ for some unit $u\in k[[t]]$.

Is this a rational singularity? Are there other singularity types that such singularities are examples of?

This is a very naive question, so references and general information would be appreciated!