The linear stability of two-dimensional surface gravity waves on fluid of finite depth is investigated for superharmonic disturbances. For this problem, Zufiria & Saffman (Stud. Appl. Maths vol. 74, 1986, p. 259) suggested that an exchange of stability occurs when the total wave energy becomes stationary as a function of wave speed for fixed ‘Bernoulli constant’. In defining the potential energy of the above total wave energy, the surface displacement was measured from the quiescent surface with the same ‘Bernoulli constant’. We have re-examined this problem both analytically and numerically, and found that the above ‘Bernoulli constant’ must be replaced by ‘mean surface height’ for the statement to be valid.