Cnoidal wave

Non-dimensionalisation

All quantities can be made dimensionless using the gravitational acceleration g and water depth h:

η ~ = η h , {\displaystyle {\tilde {\eta }}={\frac {\eta }{h}},}   x ~ = x h {\displaystyle {\tilde {x}}={\frac {x}{h}}}   and   t ~ = g h t . {\displaystyle {\tilde {t}}={\sqrt {\frac {g}{h}}}\,t.}

The resulting non-dimensional form of the KdV equation is^[17]

∂ t ~ η ~ + ∂ x ~ η ~ + 3 2 η ~ ∂ x ~ η ~ + 1 6 ∂ x ~ 3 η ~ = 0 , {\displaystyle \partial _{\tilde {t}}{\tilde {\eta }}+\partial _{\tilde {x}}{\tilde {\eta }}+{\tfrac {3}{2}}\,{\tilde {\eta }}\,\partial _{\tilde {x}}{\tilde {\eta }}+{\tfrac {1}{6}}\,\partial _{\tilde {x}}^{3}{\tilde {\eta }}=0,}

In the remainder, the tildes will be dropped for ease of notation.

Relation to a standard form

The form

∂ t ^ ϕ + 6 ϕ   ∂ x ^ ϕ + ∂ x ^ 3 ϕ = 0 {\displaystyle \partial _{\hat {t}}\phi +6\,\phi \ \partial _{\hat {x}}\phi +\partial _{\hat {x}}^{3}\phi =0}

is obtained through the transformation

t ^ = 1 6 t , {\displaystyle {\hat {t}}={\tfrac {1}{6}}\,t,\,}   x ^ = x − t {\displaystyle {\hat {x}}=x-t\,}   and   ϕ = 3 2 η , {\displaystyle \phi ={\tfrac {3}{2}}\,\eta ,\,}

but this form will not be used any further in this derivation.

Fixed-form propagating waves

Periodic wave solutions, travelling with phase speed c, are sought. These permanent waves have to be of the following:

η = η ( ξ ) {\displaystyle \eta =\eta (\xi )\,}   with ξ {\displaystyle \xi } the wave phase:   ξ = x − c t . {\displaystyle \xi =x-c\,t.\,}

Consequently, the partial derivatives with respect to space and time become:

∂ x η = η ′ {\displaystyle \partial _{x}\eta =\eta '\,}   and   ∂ t η = − c η ′ , {\displaystyle \partial _{t}\eta =-c\,\eta ',\,}

where η′ denotes the ordinary derivative of η(ξ) with respect to the argument ξ.

Using these in the KdV equation, the following third-order ordinary differential equation is obtained:^[18]

1 6 η ‴ + ( 1 − c ) η ′ + 3 2 η η ′ = 0. {\displaystyle {\tfrac {1}{6}}\,\eta '''+\left(1-c\right)\,\eta '+{\tfrac {3}{2}}\,\eta \,\eta '=0.\,}

Integration to a first-order ordinary differential equation

This can be integrated once, to obtain:^[18]

1 6 η ″ + ( 1 − c ) η + 3 4 η 2 = 1 4 r , {\displaystyle {\tfrac {1}{6}}\eta ''+\left(1-c\right)\,\eta +{\tfrac {3}{4}}\,\eta ^{2}={\tfrac {1}{4}}r,\,}

with r an integration constant. After multiplying with 4 η′, and integrating once more^[18]

1 3 ( η ′ ) 2 + 2 ( 1 − c ) η 2 + η 3 = r η + s , {\displaystyle {\tfrac {1}{3}}\left(\eta '\right)^{2}+2\,\left(1-c\right)\,\eta ^{2}+\eta ^{3}=r\,\eta +s,\,}

with s another integration constant. This is written in the form

1 3 ( η ′ ) 2 = f ( η ) {\displaystyle {\tfrac {1}{3}}\left(\eta '\right)^{2}=f(\eta )\,}   with   f ( η ) = − η 3 + 2 ( c − 1 ) η 2 + r η + s . {\displaystyle f(\eta )=-\eta ^{3}+2\,\left(c-1\right)\,\eta ^{2}+r\,\eta +s.\,}

A

The cubic polynomial f(η) becomes negative for large positive values of η, and positive for large negative values of η. Since the surface elevation η is real valued, also the integration constants r and s are real. The polynomial f can be expressed in terms of its roots η₁, η₂ and η₃:^[7]

f ( η ) = − ( η − η 1 ) ( η − η 2 ) ( η − η 3 ) . {\displaystyle f(\eta )=-\left(\eta -\eta _{1}\right)\,\left(\eta -\eta _{2}\right)\,\left(\eta -\eta _{3}\right).}

B

Because f(η) is real valued, the three roots η₁, η₂ and η₃ are either all three real, or otherwise one is real and the remaining two are a pair of complex conjugates. In the latter case, with only one real-valued root, there is only one elevation η at which f(η) is zero. And consequently also only one elevation at which the surface slope η′ is zero. However, we are looking for wave like solutions, with two elevations—the wave crest and trough (physics)—where the surface slope is zero. The conclusion is that all three roots of f(η) have to be real valued.

Without loss of generality, it is assumed that the three real roots are ordered as:

η 1 ≥ η 2 ≥ η 3 . {\displaystyle \eta _{1}\geq \eta _{2}\geq \eta _{3}.\,}

Solution of the first-order ordinary-differential equation

Now, from equation (A) it can be seen that only real values for the slope exist if f(η) is positive. This corresponds with η₂ ≤ η≤ η₁, which therefore is the range between which the surface elevation oscillates, see also the graph of f(η). This condition is satisfied with the following representation of the elevation η(ξ):^[7]

η ( ξ ) = η 1 cos 2 ψ ( ξ ) + η 2 sin 2 ψ ( ξ ) , {\displaystyle \eta (\xi )=\eta _{1}\;\cos ^{2}\,\psi (\xi )+\eta _{2}\,\sin ^{2}\,\psi (\xi ),}

C

in agreement with the periodic character of the sought wave solutions and with ψ(ξ) the phase of the trigonometric functions sin and cos. From this form, the following descriptions of various terms in equations (A) and (B) can be obtained:

η − η 1 = − ( η 1 − η 2 ) sin 2 ψ ( ξ ) , η − η 2 = + ( η 1 − η 2 ) cos 2 ψ ( ξ ) , η − η 3 = ( η 1 − η 3 ) − ( η 1 − η 2 ) sin 2 ψ ( ξ ) , and η ′ = − 2 ( η 1 − η 2 ) sin ψ ( ξ ) cos ψ ( ξ ) ψ ′ ( ξ ) with ψ ′ ( ξ ) = d ψ ( ξ ) d ξ . {\displaystyle {\begin{aligned}\eta -\eta _{1}&=-\left(\eta _{1}-\eta _{2}\right)\;\sin ^{2}\,\psi (\xi ),\\\eta -\eta _{2}&=+\left(\eta _{1}-\eta _{2}\right)\;\cos ^{2}\,\psi (\xi ),\\\eta -\eta _{3}&=\left(\eta _{1}-\eta _{3}\right)-\left(\eta _{1}-\eta _{2}\right)\;\sin ^{2}\,\psi (\xi ),&&{\text{and}}\\\eta '&=-2\,\left(\eta _{1}-\eta _{2}\right)\;\sin \,\psi (\xi )\;\cos \,\psi (\xi )\;\;\psi '(\xi )&&{\text{with}}\quad \psi '(\xi )={\frac {{\text{d}}\psi (\xi )}{{\text{d}}\xi }}.\end{aligned}}}

Using these in equations (A) and (B), the following ordinary differential equation relating ψ and ξ is obtained, after some manipulations:^[7]

4 3 ( d ψ d ξ ) 2 = ( η 1 − η 3 ) − ( η 1 − η 2 ) sin 2 ψ ( ξ ) , {\displaystyle {\frac {4}{3}}\,\left({\frac {{\text{d}}\psi }{{\text{d}}\xi }}\right)^{2}=\left(\eta _{1}-\eta _{3}\right)-\left(\eta _{1}-\eta _{2}\right)\;\sin ^{2}\,\psi (\xi ),}

with the right hand side still positive, since η₁ − η₃ ≥ η₁ − η₂. Without loss of generality, we can assume that ψ(ξ) is a monotone function, since f(η) has no zeros in the interval η₂ < η < η₁. So the above ordinary differential equation can also be solved in terms of ξ(ψ) being a function of ψ:^[7]

1 Δ d ξ d ψ = ± 1 1 − m sin 2 ψ , {\displaystyle {\frac {1}{\Delta }}\,{\frac {{\text{d}}\xi }{{\text{d}}\psi }}=\pm \,{\frac {1}{\sqrt {1-m\sin ^{2}\,\psi }}},}

with:

Δ 2 = 4 3 1 η 1 − η 3 {\displaystyle \Delta ^{2}={\frac {4}{3}}\,{\frac {1}{\eta _{1}-\eta _{3}}}}   and   m = η 1 − η 2 η 1 − η 3 , {\displaystyle m={\frac {\eta _{1}-\eta _{2}}{\eta _{1}-\eta _{3}}},}

where m is the so-called elliptic parameter,^[19][20] satisfying 0 ≤ m ≤ 1 (because η₃ ≤ η₂ ≤ η₁). If ξ = 0 is chosen at the wave crest η(0) = η₁ integration gives^[7]

ξ = ± Δ ∫ 0 ψ d ψ ^ 1 − m sin 2 ⁡ ψ ^ = ± Δ F ( ψ | m ) , {\displaystyle \xi =\pm \,\Delta \,\int _{0}^{\psi }{\frac {{\text{d}}{\hat {\psi }}}{\sqrt {1-m\,\sin ^{2}{\hat {\psi }}}}}=\pm \,\Delta \,F(\psi |m),}

D

with F(ψ|m) the incomplete elliptic integral of the first kind. The Jacobi elliptic functions cn and sn are inverses of F(ψ|m) given by

cos ψ = cn ⁡ ( ξ Δ m ) {\displaystyle \cos \,\psi =\operatorname {cn} \left({\begin{array}{c|c}\displaystyle {\frac {\xi }{\Delta }}&m\end{array}}\right)}   and   sin ψ = sn ⁡ ( ξ Δ m ) . {\displaystyle \sin \,\psi =\operatorname {sn} \left({\begin{array}{c|c}\displaystyle {\frac {\xi }{\Delta }}&m\end{array}}\right).}

With the use of equation (C), the resulting cnoidal-wave solution of the KdV equation is found^[7]

η ( ξ ) = η 2 + ( η 1 − η 2 ) cn 2 ⁡ ( ξ Δ m ) . {\displaystyle \eta (\xi )=\eta _{2}+\left(\eta _{1}-\eta _{2}\right)\,\operatorname {cn} ^{2}\left({\begin{array}{c|c}\displaystyle {\frac {\xi }{\Delta }}&m\end{array}}\right).}

What remains, is to determine the parameters: η₁, η₂, Δ and m.

Relationships between the cnoidal-wave parameters

First, since η₁ is the crest elevation and η₂ is the trough elevation, it is convenient to introduce the wave height, defined as H = η₁ − η₂. Consequently, we find for m and for Δ:

m = H η 1 − η 3 {\displaystyle m={\frac {H}{\eta _{1}-\eta _{3}}}}   and   Δ 2 m = 4 3 H {\displaystyle {\frac {\Delta ^{2}}{m}}={\frac {4}{3\,H}}}   so   Δ = 4 3 m H . {\displaystyle \Delta ={\sqrt {{\frac {4}{3}}{\frac {m}{H}}}}.}

The cnoidal wave solution can be written as:

η ( ξ ) = η 2 + H cn 2 ⁡ ( ξ Δ m ) . {\displaystyle \eta (\xi )=\eta _{2}+H\,\operatorname {cn} ^{2}\left({\begin{array}{c|c}\displaystyle {\frac {\xi }{\Delta }}&m\end{array}}\right).}

Second, the trough is located at ψ = ⁠1/2⁠ π, so the distance between ξ = 0 and ξ = ⁠1/2⁠ λ is, with λ the wavelength, from equation (D):

1 2 λ = Δ F ( 1 2 π m ) = Δ K ( m ) , {\displaystyle {\tfrac {1}{2}}\,\lambda =\Delta \,F\left({\begin{array}{c|c}{\tfrac {1}{2}}\,\pi &m\end{array}}\right)=\Delta \,K(m),}   giving   λ = 2 Δ K ( m ) = 16 3 m H K ( m ) , {\displaystyle \lambda =2\,\Delta \,K(m)={\sqrt {{\frac {16}{3}}{\frac {m}{H}}}}\;K(m),}

where K(m) is the complete elliptic integral of the first kind. Third, since the wave oscillates around the mean water depth, the average value of η(ξ) has to be zero. So^[7]

0 = ∫ 0 λ η ( ξ ) d ξ = 2 ∫ 0 1 2 λ [ η 2 + ( η 1 − η 2 ) cn 2 ( ξ Δ m ) ] d ξ = 2 ∫ 0 1 2 π [ η 2 + ( η 1 − η 2 ) cos 2 ψ ] d ξ d ψ d ψ = 2 Δ ∫ 0 1 2 π η 1 − ( η 1 − η 2 ) sin 2 ψ 1 − m sin 2 ψ d ψ = 2 Δ ∫ 0 1 2 π η 1 − m ( η 1 − η 3 ) sin 2 ψ 1 − m sin 2 ψ d ψ = 2 Δ ∫ 0 1 2 π [ η 3 1 − m sin 2 ψ + ( η 1 − η 3 ) 1 − m sin 2 ψ ] d ψ = 2 Δ [ η 3 K ( m ) + ( η 1 − η 3 ) E ( m ) ] = 2 Δ [ η 3 K ( m ) + H m E ( m ) ] , {\displaystyle {\begin{aligned}0&=\int _{0}^{\lambda }\eta (\xi )\;{\text{d}}\xi =2\,\int _{0}^{{\tfrac {1}{2}}\lambda }\left[\eta _{2}+\left(\eta _{1}-\eta _{2}\right)\,\operatorname {cn} ^{2}\,\left({\begin{array}{c|c}\displaystyle {\frac {\xi }{\Delta }}&m\end{array}}\right)\right]\;{\text{d}}\xi \\&=2\,\int _{0}^{{\tfrac {1}{2}}\pi }{\Bigl [}\eta _{2}+\left(\eta _{1}-\eta _{2}\right)\,\cos ^{2}\,\psi {\Bigr ]}\,{\frac {{\text{d}}\xi }{{\text{d}}\psi }}\;{\text{d}}\psi =2\,\Delta \,\int _{0}^{{\tfrac {1}{2}}\pi }{\frac {\eta _{1}-\left(\eta _{1}-\eta _{2}\right)\,\sin ^{2}\,\psi }{\sqrt {1-m\,\sin ^{2}\,\psi }}}\;{\text{d}}\psi \\&=2\,\Delta \,\int _{0}^{{\tfrac {1}{2}}\pi }{\frac {\eta _{1}-m\,\left(\eta _{1}-\eta _{3}\right)\,\sin ^{2}\,\psi }{\sqrt {1-m\,\sin ^{2}\,\psi }}}\;{\text{d}}\psi =2\,\Delta \,\int _{0}^{{\tfrac {1}{2}}\pi }\left[{\frac {\eta _{3}}{\sqrt {1-m\,\sin ^{2}\,\psi }}}+\left(\eta _{1}-\eta _{3}\right)\,{\sqrt {1-m\,\sin ^{2}\,\psi }}\right]\;{\text{d}}\psi \\&=2\,\Delta \,{\Bigl [}\eta _{3}\,K(m)+\left(\eta _{1}-\eta _{3}\right)\,E(m){\Bigr ]}=2\,\Delta \,{\Bigl [}\eta _{3}\,K(m)+{\frac {H}{m}}\,E(m){\Bigr ]},\end{aligned}}}

where E(m) is the complete elliptic integral of the second kind. The following expressions for η₁, η₂ and η₃ as a function of the elliptic parameter m and wave height H result:^[7]

η 3 = − H m E ( m ) K ( m ) , {\displaystyle \eta _{3}=-\,{\frac {H}{m}}\,{\frac {E(m)}{K(m)}},}   η 1 = H m ( 1 − E ( m ) K ( m ) ) {\displaystyle \eta _{1}={\frac {H}{m}}\,\left(1-{\frac {E(m)}{K(m)}}\right)}   and   η 2 = H m ( 1 − m − E ( m ) K ( m ) ) . {\displaystyle \eta _{2}={\frac {H}{m}}\,\left(1-m-{\frac {E(m)}{K(m)}}\right).}

Fourth, from equations (A) and (B) a relationship can be established between the phase speed c and the roots η₁, η₂ and η₃:^[7]

c = 1 + 1 2 ( η 1 + η 2 + η 3 ) = 1 + H m ( 1 − 1 2 m − 3 2 E ( m ) K ( m ) ) . {\displaystyle c=1+{\tfrac {1}{2}}\,\left(\eta _{1}+\eta _{2}+\eta _{3}\right)=1+{\frac {H}{m}}\,\left(1-{\frac {1}{2}}\,m-{\frac {3}{2}}\,{\frac {E(m)}{K(m)}}\right).}

The relative phase-speed changes are depicted in the figure below. As can be seen, for m > 0.96 (so for 1 − m < 0.04) the phase speed increases with increasing wave height H. This corresponds with the longer and more nonlinear waves. The nonlinear change in the phase speed, for fixed m, is proportional to the wave height H. Note that the phase speed c is related to the wavelength λ and period τ as:

c = λ τ . {\displaystyle c={\frac {\lambda }{\tau }}.}

Résumé of the solution

All quantities here will be given in their dimensional forms, as valid for surface gravity waves before non-dimensionalisation.

Derivation

The derivation is analogous to the one for the KdV equation.^[22] The dimensionless BBM equation is, non-dimensionalised using mean water depth h and gravitational acceleration g:^[21]

∂ t η + ∂ x η + 3 2 η ∂ x η − 1 6 ∂ t ∂ x 2 η = 0. {\displaystyle \partial _{t}\eta +\partial _{x}\eta +{\tfrac {3}{2}}\,\eta \,\partial _{x}\eta -{\tfrac {1}{6}}\,\partial _{t}\,\partial _{x}^{2}\eta =0.}

This can be brought into the standard form

∂ t ^ φ + ∂ x ^ φ + φ ∂ x ^ φ − ∂ t ^ ∂ x ^ 2 φ = 0 {\displaystyle \partial _{\hat {t}}\varphi +\partial _{\hat {x}}\varphi +\varphi \,\partial _{\hat {x}}\varphi -\partial _{\hat {t}}\,\partial _{\hat {x}}^{2}\varphi =0\,}

through the transformation:

t ^ = 6 t , {\displaystyle {\hat {t}}={\sqrt {6}}\,t,}   x ^ = 6 x {\displaystyle {\hat {x}}={\sqrt {6}}\,x}   and   φ = 3 2 η , {\displaystyle \varphi ={\tfrac {3}{2}}\,\eta ,}

but this standard form will not be used here.

Analogue to the derivation of the cnoidal wave solution for the KdV equation, periodic wave solutions η(ξ), with ξ = x−ct are considered Then the BBM equation becomes a third-order ordinary differential equation, which can be integrated twice, to obtain:

1 3 c ( η ′ ) 2 = f ( η ) {\displaystyle {\tfrac {1}{3}}\,c\,\left(\eta '\right)^{2}=f(\eta )\,}   with   f ( η ) = − η 3 + 2 ( c − 1 ) η 2 + r η + s . {\displaystyle f(\eta )=-\eta ^{3}+2\,\left(c-1\right)\,\eta ^{2}+r\,\eta +s.\,}

Which only differs from the equation for the KdV equation through the factor c in front of (η′)² in the left hand side. Through a coordinate transformation β = ξ /  c {\displaystyle \scriptstyle {\sqrt {c}}} the factor c may be removed, resulting in the same first-order ordinary differential equation for both the KdV and BBM equation. However, here the form given in the preceding equation is used. This results in a different formulation for Δ as found for the KdV equation:

Δ = 4 3 m c H . {\displaystyle \Delta ={\sqrt {{\frac {4}{3}}{\frac {m\,c}{H}}}}.}

The relation of the wavelength λ, as a function of H and m, is affected by this change in Δ : {\displaystyle \Delta :}

λ = 16 3 m c H K ( m ) . {\displaystyle \lambda ={\sqrt {{\frac {16}{3}}\,{\frac {m\,c}{H}}}}\;K(m).}

For the rest, the derivation is analogous to the one for the KdV equation, and will not be repeated here.

Résumé

The results are presented in dimensional form, for water waves on a fluid layer of depth h.

The Jacobi elliptic function cn can be expanded into a Fourier series^[26]

cn ⁡ ( z | m ) = π m K ( m ) ∑ n = 0 ∞ sech ⁡ ( ( 2 n + 1 ) π K ′ ( m ) 2 K ( m ) ) cos ⁡ ( ( 2 n + 1 ) π z 2 K ( m ) ) . {\displaystyle \operatorname {cn} (z|m)={\frac {\pi }{{\sqrt {m}}\,K(m)}}\,\sum _{n=0}^{\infty }\,\operatorname {sech} \left((2n+1)\,{\frac {\pi \,K'(m)}{2\,K(m)}}\right)\;\cos \left((2n+1)\,{\frac {\pi \,z}{2\,K(m)}}\right).}

K′(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: K′(m) = K(1−m)^[27]

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:^[28][29]

K ( m ) = π 2 [ 1 + ( 1 2 ) 2 m + ( 1 ⋅ 3 2 ⋅ 4 ) 2 m 2 + ( 1 ⋅ 3 ⋅ 5 2 ⋅ 4 ⋅ 6 ) 2 m 3 + ⋯ ] , E ( m ) = π 2 [ 1 − ( 1 2 ) 2 m 1 − ( 1 ⋅ 3 2 ⋅ 4 ) 2 m 2 3 − ( 1 ⋅ 3 ⋅ 5 2 ⋅ 4 ⋅ 6 ) 2 m 3 5 − ⋯ ] . {\displaystyle {\begin{aligned}K(m)&={\frac {\pi }{2}}\,\left[1+\left({\frac {1}{2}}\right)^{2}\,m+\left({\frac {1\,\cdot \,3}{2\,\cdot \,4}}\right)^{2}\,m^{2}+\left({\frac {1\,\cdot \,3\,\cdot \,5}{2\,\cdot \,4\,\cdot \,6}}\right)^{2}\,m^{3}+\cdots \right],\\E(m)&={\frac {\pi }{2}}\,\left[1-\left({\frac {1}{2}}\right)^{2}\,{\frac {m}{1}}-\left({\frac {1\,\cdot \,3}{2\,\cdot \,4}}\right)^{2}\,{\frac {m^{2}}{3}}-\left({\frac {1\,\cdot \,3\,\cdot \,5}{2\,\cdot \,4\,\cdot \,6}}\right)^{2}\,{\frac {m^{3}}{5}}-\cdots \right].\end{aligned}}}

Then the hyperbolic-cosine terms, appearing in the Fourier series, can be expanded for small m ≪ 1 as follows:^[26]

sech ⁡ ( ( 2 n + 1 ) π K ′ ( m ) 2 K ( m ) ) = 2 q n + 1 2 1 + q 2 n + 1 {\displaystyle \operatorname {sech} \left((2n+1)\,{\frac {\pi \,K'(m)}{2\,K(m)}}\right)=2\,{\frac {q^{n+{\tfrac {1}{2}}}}{1+q^{2n+1}}}}   with the nome q given by   q = exp ⁡ ( − π K ′ ( m ) K ( m ) ) . {\displaystyle q=\exp \left(-\pi \,{\frac {K'(m)}{K(m)}}\right).}

The nome q has the following behaviour for small m:^[30]

q = m 16 + 8 ( m 16 ) 2 + 84 ( m 16 ) 3 + 992 ( m 16 ) 4 + ⋯ . {\displaystyle q={\frac {m}{16}}+8\,\left({\frac {m}{16}}\right)^{2}+84\,\left({\frac {m}{16}}\right)^{3}+992\,\left({\frac {m}{16}}\right)^{4}+\cdots .}

Consequently, the amplitudes of the first terms in the Fourier series are:

n = 0 {\displaystyle n=0\,}	:	π m K ( m ) sech ( π K ′ ( m ) 2 K ( m ) ) = 1 − 1 16 m − 9 16 m 2 + ⋯ , {\displaystyle {\frac {\pi }{{\sqrt {m}}\,K(m)}}\,\operatorname {sech} \,\left({\frac {\pi \,K'(m)}{2\,K(m)}}\right)=1-{\tfrac {1}{16}}\,m-{\tfrac {9}{16}}\,m^{2}+\cdots ,}
n = 1 {\displaystyle n=1\,}	:	π m K ( m ) sech ( 3 π K ′ ( m ) 2 K ( m ) ) = 1 16 m + 1 32 m 2 + ⋯ , {\displaystyle {\frac {\pi }{{\sqrt {m}}\,K(m)}}\,\operatorname {sech} \,\left({\frac {3\,\pi \,K'(m)}{2\,K(m)}}\right)={\tfrac {1}{16}}\,m+{\tfrac {1}{32}}\,m^{2}+\cdots ,}
n = 2 {\displaystyle n=2\,}	:	π m K ( m ) sech ( 5 π K ′ ( m ) 2 K ( m ) ) = 1 256 m 2 + ⋯ . {\displaystyle {\frac {\pi }{{\sqrt {m}}\,K(m)}}\,\operatorname {sech} \,\left({\frac {5\,\pi \,K'(m)}{2\,K(m)}}\right)={\tfrac {1}{256}}\,m^{2}+\cdots .}

So, for m ≪ 1 the Jacobi elliptic function has the first Fourier series terms:

cn ( z | m ) = ( 1 − 1 16 m − 9 16 m 2 + ⋯ ) cos α z + ( 1 16 m + 1 32 m 2 + ⋯ ) cos 2 α z + ( 1 256 m 2 + ⋯ ) cos 3 α z + ⋯ , {\displaystyle {\begin{aligned}\operatorname {cn} \,(z|m)&={\Bigl (}1-{\tfrac {1}{16}}\,m-{\tfrac {9}{16}}\,m^{2}+\cdots &&{\Bigr )}\;\cos \,\alpha \,z\;\\&+\;{\Bigl (}{\tfrac {1}{16}}\,m+{\tfrac {1}{32}}\,m^{2}+\cdots &&{\Bigr )}\;\cos \,2\,\alpha \,z\;\\&+\;{\Bigl (}{\tfrac {1}{256}}\,m^{2}+\cdots &&{\Bigr )}\;\cos \,3\,\alpha \,z\;+\;\cdots ,\end{aligned}}}   with   α ≡ π 2 K ( m ) . {\displaystyle \alpha \equiv {\frac {\pi }{2\,K(m)}}.}

And its square is

cn 2 ( z | m ) = ( 1 2 − 1 16 m − 1 32 m 2 + ⋯ ) + ( 1 2 − 3 512 m 2 + ⋯ ) cos 2 α z + ( 1 16 m + 1 32 m 2 + ⋯ ) cos 4 α z + ( 3 512 m 2 + ⋯ ) cos 6 α z + ⋯ . {\displaystyle {\begin{aligned}\operatorname {cn} ^{2}\,(z|m)&={\Bigl (}{\tfrac {1}{2}}-{\tfrac {1}{16}}\,m-{\tfrac {1}{32}}\,m^{2}+\cdots &&{\Bigr )}\\&+\;{\Bigl (}{\tfrac {1}{2}}-{\tfrac {3}{512}}\,m^{2}+\cdots &&{\Bigr )}\;\cos \,2\,\alpha \,z\;\\&+\;{\Bigl (}{\tfrac {1}{16}}\,m+{\tfrac {1}{32}}\,m^{2}+\cdots &&{\Bigr )}\;\cos \,4\,\alpha \,z\;\\&+\;{\Bigl (}{\tfrac {3}{512}}\,m^{2}+\cdots &&{\Bigr )}\;\cos \,6\,\alpha \,z\;+\;\cdots .\end{aligned}}}

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:

θ ≡ π ξ K ( m ) ξ Δ = 2 π ξ λ . {\displaystyle \theta \equiv {\frac {\pi \,\xi }{K(m)}}\,{\frac {\xi }{\Delta }}=2\,\pi \,{\frac {\xi }{\lambda }}.}  

Further, the mean free-surface elevation is zero. Therefore, the surface elevation of small amplitude waves is

η ( x , t ) = H ( 1 2 − 3 512 m 2 + ⋯ ) cos θ + H ( 1 16 m + 1 32 m 2 + ⋯ ) cos 2 θ + H ( 3 512 m 2 + ⋯ ) cos 3 θ + ⋯ . {\displaystyle \eta (x,t)=\;H\,{\Bigl (}{\tfrac {1}{2}}-{\tfrac {3}{512}}\,m^{2}+\cdots {\Bigr )}\,\cos \,\theta \;+\;H\,{\Bigl (}{\tfrac {1}{16}}\,m+{\tfrac {1}{32}}\,m^{2}+\cdots {\Bigr )}\,\cos \,2\theta \;+\;H\,{\Bigl (}{\tfrac {3}{512}}\,m^{2}+\cdots {\Bigr )}\,\cos \,3\theta \;+\;\cdots .}

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.

Note: The limiting behaviour for zero m—at infinitesimal wave height—can also be seen from:[31]

cn ⁡ ( z | m ) ≈ cos ⁡ ( z ) + 1 4 m ( z − sin ⁡ ( z ) cos ⁡ ( z ) ) sin ⁡ ( z ) + ⋯ , {\displaystyle \operatorname {cn} (z|m)\approx \cos(z)+{\tfrac {1}{4}}\,m\,{\bigl (}z-\sin(z)\,\cos(z){\bigr )}\,\sin(z)+\cdots ,}

but the higher-order term proportional to m in this approximation contains a secular term, due to the mismatch between the period of cn(z|m), which is 4 K(m), and the period 2π for the cosine cos(z). The above Fourier series for small m does not have this drawback, and is consistent with forms as found using the Lindstedt–Poincaré method in perturbation theory.

The phase speed of a cnoidal wave, both for the KdV and BBM equation, is given by:^[7][22]

c = g h [ 1 + H m h ( 1 − 1 2 m − 3 2 E ( m ) K ( m ) ) ] . {\displaystyle c={\sqrt {gh}}\,\left[1+{\frac {H}{m\,h}}\,\left(1-{\frac {1}{2}}\,m-{\frac {3}{2}}\,{\frac {E(m)}{K(m)}}\right)\right].}

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:^[7][22]

KdV :	λ = h 16 3 m h H K ( m ) , {\displaystyle \lambda =h\,{\sqrt {{\frac {16}{3}}\,{\frac {mh}{H}}}}\,K(m),}
BBM :	λ = h 16 3 m h H c g h K ( m ) , {\displaystyle \lambda =h\,{\sqrt {{\frac {16}{3}}\,{\frac {mh}{H}}\,{\frac {c}{\sqrt {g\,h}}}}}\,K(m),}

Introducing the relative wavenumber κh:

κ h = 2 π λ h , {\displaystyle \kappa \,h={\frac {2\,\pi }{\lambda }}\,h,}

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:

KdV :	c = g h [ 1 + ( κ h ) 2 4 3 π 2 K 2 ( m ) ( 1 − 1 2 m − 3 2 E ( m ) K ( m ) ) ] , {\displaystyle c={\sqrt {gh}}\,\left[1+(\kappa \,h)^{2}\,{\frac {4}{3\,\pi ^{2}}}\,K^{2}(m)\,\left(1-{\frac {1}{2}}\,m-{\frac {3}{2}}\,{\frac {E(m)}{K(m)}}\right)\right],}
BBM :	c = g h 1 1 − ( κ h ) 2 4 3 π 2 K 2 ( m ) ( 1 − 1 2 m − 3 2 E ( m ) K ( m ) ) . {\displaystyle c={\sqrt {gh}}\,{\frac {1}{\displaystyle 1-(\kappa \,h)^{2}\,{\frac {4}{3\,\pi ^{2}}}\,K^{2}(m)\,\left(1-{\frac {1}{2}}\,m-{\frac {3}{2}}\,{\frac {E(m)}{K(m)}}\right)}}.}

The limiting behaviour for small m can be analysed through the use of the Maclaurin series for K(m) and E(m),^[28] resulting in the following expression for the common factor in both formulas for c:

γ = 4 3 π 2 K 2 ( m ) ( 1 − 1 2 m − 3 2 E ( m ) K ( m ) ) = − 1 6 + 1 64 m 2 + 1 64 m 3 + ⋯ , {\displaystyle \gamma ={\frac {4}{3\,\pi ^{2}}}\,K^{2}(m)\,\left(1-{\frac {1}{2}}\,m-\,{\frac {3}{2}}\,{\frac {E(m)}{K(m)}}\right)=-{\tfrac {1}{6}}+{\tfrac {1}{64}}\,m^{2}+{\tfrac {1}{64}}\,m^{3}+\cdots ,}

so in the limit m → 0, the factor γ → −⁠1/6⁠. The limiting value of the phase speed for m ≪ 1 directly results.