The phase speed of a cnoidal wave, both for the KdV and BBM equation, is given by:
In this formulation the phase speed is a function of wave height H and parameter m. However, for the determination of wave propagation for waves of infinitesimal height, it is necessary to determine the behaviour of the phase speed at constant wavelength λ in the limit that the parameter m approaches zero.  This can be done by using the equation for the wavelength, which is different for the KdV and BBM equation:
Introducing the relative wavenumber κh:
and using the above equations for the phase speed and wavelength, the factor H / m in the phase speed can be replaced by κh and m. The resulting phase speeds are:
The limiting behaviour for small m can be analysed through the use of the Maclaurin series for K(m) and E(m), resulting in the following expression for the common factor in both formulas for c:
so in the limit m → 0, the factor γ → −. The limiting value of the phase speed for m ≪ 1 directly results.
The phase speeds for infinitesimal wave height, according to the cnoidal wave theories for the KdV equation and BBM equation, are
with κ = 2π / λ the wavenumber and κh the relative wavenumber. These phase speeds are in full agreement with the result obtained by directly searching for sine-wave solutions of the linearised KdV and BBM equations. As is evident from these equations, the linearised BBM equation has a positive phase speed for all κh. On the other hand, the phase speed of the linearised KdV equation changes sign for short waves with κh > . This is in conflict with the derivation of the KdV equation as a one-way wave equation.