The potential energy density
with ρ the fluid density, is one of the infinite number of invariants of the KdV equation. This can be seen by multiplying the KdV equation with the surface elevation η(x,t); after repeated use of the chain rule the result is:
which is in conservation form, and is an invariant after integration over the interval of periodicity—the wavelength for a cnoidal wave. The potential energy is not an invariant of the BBM equation, but ρg [η +  h (&part; η)] is.
First the variance of the surface elevation in a cnoidal wave is computed. Note that η = −(1/λ) ∫ H cn(ξ/Δ|m) dx, cn(ξ/Δ|m)  = cos ψ(ξ) and λ = 2 Δ K(m), so
The potential energy, both for the KdV and the BBM equation, is subsequently found to be
The infinitesimal wave-height limit (m → 0) of the potential energy is E =  ρ g H, which is in agreement with Airy wave theory. The wave height is twice the amplitude, H = 2a, in the infinitesimal wave limit.