Fig. 3: Illustration of ligand wagging, core O–H stretch and O–O stretching motion of H9O\({}_{4}^{+}\) exemplary in one of the three arms of the cation.
From: The coupling of the hydrated proton to its first solvation shell

Note that in the ligand-wagging motion only the two hydrogen of the outer water molecules move as indicated by the arrow. The coordinate system on the left indicates the directions of components of the dipole moment surfaces (DMS).