Mechanical energy in an harmonic wave and in normal modes

[crick]

I think I miss something about energy of a mechanical wave.
In absence of dissipation the mechanical energy transported by an harmonic wave is constant.

$$E=\frac{1}{2} A^2 \omega^2 m$$

But, while studying normal modes on a rope, I find that the mechanical energy of a normal mode (still constant) is equal to

$$E=\frac{1}{4} A^2 \omega^2 m$$

Is the factor $\frac{1}{2}$ really present and why?

 

[nasu]

Are you sure the first formula is about a wave? What would be the meaning of m?
Isn't the energy of a harmonic oscillator?