The Jacobi elliptic function cn can be expanded into a Fourier series
(m) is known as the imaginary quarter period, while K(m) is also called the real quarter period of the Jacobi elliptic function. They are related through: (m) = K(1−m)
Since the interest here is in small wave height, corresponding with small parameter m ≪ 1, it is convenient to consider the Maclaurin series for the relevant parameters, to start with the complete elliptic integrals K and E:
Then the hyperbolic-cosine terms, appearing in the Fourier series, can be expanded for small m ≪ 1 as follows:
:  with the nome q given by  
The nome q has the following behaviour for small m:
Consequently, the amplitudes of the first terms in the Fourier series are:
So, for m ≪ 1 the Jacobi elliptic function has the first Fourier series terms:
The free surface η(x,t) of the cnoidal wave will be expressed in its Fourier series, for small values of the elliptic parameter m. First, note that the argument of the cn function is ξ/Δ, and that the wavelength λ = 2 Δ K(m), so:
Further, the mean free-surface elevation is zero. Therefore, the surface elevation of small amplitude waves is
Also the wavelength λ can be expanded into a Maclaurin series of the elliptic parameter m, differently for the KdV and the BBM equation, but this is not necessary for the present purpose.
For infinitesimal wave height, in the limit m → 0, the free-surface elevation becomes:
:  with  
So the wave amplitude is H, half the wave height. This is of the same form as studied in Airy wave theory, but note that cnoidal wave theory is only valid for long waves with their wavelength much longer than the average water depth.