text: For very long nonlinear waves, with the parameter m close to one, m → 1, the Jacobi elliptic function cn can be approximated by
  :  with  
  Here sinh, cosh, tanh and sech are hyperbolic functions.  In the limit m = 1:
  with sech(z) = 1 / cosh(z).
  Further, for the same limit of m → 1, the complete elliptic integral of the first kind K(m) goes to infinity, while the complete elliptic integral of the second kind E(m) goes to one. This implies that the limiting values of the phase speed c and minimum elevelation η become:
  :  and  
  Consequently, in terms of the width parameter Δ, the solitary wave solution to both the KdV and BBM equation is:
  The width parameter, as found for the cnoidal waves and now in the limit m → 1, is different for the KdV and the BBM equation:
  But the phase speed of the solitary wave in both equations is the same, for a certain combination of height H and depth h.