Maybe the question can simply be answered by the observation that a "wave" like

$$\Psi(x,t)=\cos(x)+\cos(x+\omega\ t),$$

where the two cosines cancel at times
$$t_n=\frac{2\pi}{\omega}n\ \ \longrightarrow\ \ \Psi(x,t_n)=0,$$ 
still has nonvanishing kinetic energy, if it looks something like 
$$E=\sum_\mu(\partial_\mu \Psi)^2+\ ...$$

You would really have to construct an example.

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Since non-dissipative waves whose equations of motions can be formulated by a Lagrangian will have an energy associated to them, as you say, you'd have to find a sitation/theory without an energy quantity. So if the energy is defines as that which is constant because of time symmetriy and you don't have such a thing, then there is no question. 

Also don't make the mistake and talk about about two different waves with different energy. If you have a linear problem, the wave will be "one wave" in the energy expression, whereever its party may wander around.