Stokes wave

In fluid dynamics, a Stokes wave is a non-linear and periodic surface wave on an inviscid fluid layer of constant mean depth.This type of modelling has its origins in the mid 19th century when Sir George Stokes – using a perturbation series approach, now known as the Stokes expansion – obtained approximate solutions for non-linear wave motion. u = ∇ Φ . {displaystyle mathbf {u} ={oldsymbol { abla }}Phi .}     (A ) ∇ 2 Φ = 0 {displaystyle abla ^{2}Phi =0}     (B ) ∂ η ∂ t + ∂ Φ ∂ x ∂ η ∂ x + ∂ Φ ∂ y ∂ η ∂ y = ∂ Φ ∂ z  at  z = η ( x , y , t ) . {displaystyle {frac {partial eta }{partial t}}+{frac {partial Phi }{partial x}},{frac {partial eta }{partial x}}+{frac {partial Phi }{partial y}},{frac {partial eta }{partial y}}={frac {partial Phi }{partial z}}qquad { ext{ at }}z=eta (x,y,t).}     (C) ∂ Φ ∂ t + 1 2 | u | 2 + g η = 0  at  z = η ( x , y , t ) , {displaystyle {frac {partial Phi }{partial t}}+{ frac {1}{2}},left|mathbf {u} ight|^{2}+g,eta =0qquad { ext{ at }}z=eta (x,y,t),}     (D) ⇒ ∂ 2 Φ ∂ t 2 + g ∂ Φ ∂ z + ∂ ∂ t ( | u | 2 ) + 1 2 u ⋅ ∇ ( | u | 2 ) = 0  at  z = η ( x , y , t ) . {displaystyle {color {Gray}{Rightarrow quad }}{frac {partial ^{2}Phi }{partial t^{2}}}+g,{frac {partial Phi }{partial z}}+{frac {partial }{partial t}}left(|mathbf {u} |^{2} ight)+{ frac {1}{2}},mathbf {u} cdot {oldsymbol { abla }}left(|mathbf {u} |^{2} ight)=0qquad { ext{ at }}z=eta (x,y,t).}     (E) ∂ Φ ∂ n = 1 1 + ( ∂ h ∂ x ) 2 + ( ∂ h ∂ y ) 2 { ∂ Φ ∂ z + ∂ h ∂ x ∂ Φ ∂ x + ∂ h ∂ y ∂ Φ ∂ y } = 0 ,  at  z = − h ( x , y ) , {displaystyle {frac {partial Phi }{partial n}}={frac {1}{sqrt {1+left({frac {partial h}{partial x}} ight)^{2}+left({frac {partial h}{partial y}} ight)^{2}}}},left{{frac {partial Phi }{partial z}}+{frac {partial h}{partial x}},{frac {partial Phi }{partial x}}+{frac {partial h}{partial y}},{frac {partial Phi }{partial y}} ight}=0,qquad { ext{ at }}z=-h(x,y),}     (F) [ ∂ 2 Φ ∂ t 2 + g ∂ Φ ∂ z ] 0 + η [ ∂ ∂ z ( ∂ 2 Φ ∂ t 2 + g ∂ Φ ∂ z ) ] 0 + [ ∂ ∂ t ( | u | 2 ) ] 0 + 1 2 η 2 [ ∂ 2 ∂ z 2 ( ∂ 2 Φ ∂ t 2 + g ∂ Φ ∂ z ) ] 0 + η [ ∂ 2 ∂ t ∂ z ( | u | 2 ) ] 0 + [ 1 2 u ⋅ ∇ ( | u | 2 ) ] 0 + ⋯ = 0 , {displaystyle {egin{aligned}&left_{0}+eta left_{0}+left_{0}\&quad +{ frac {1}{2}},eta ^{2}left_{0}+eta left_{0}+{iggl }_{0}\&quad +cdots =0,end{aligned}}}     (G) [ ∂ Φ ∂ t + g η ] 0 + η [ ∂ 2 Φ ∂ t ∂ z ] 0 + [ 1 2 | u | 2 ] 0 + 1 2 η 2 [ ∂ 3 Φ ∂ t ∂ z 2 ] 0 + η [ ∂ ∂ z ( 1 2 | u | 2 ) ] 0 + ⋯ = 0. {displaystyle {egin{aligned}&left_{0}+eta left_{0}+{iggl }_{0}\&quad +{ frac {1}{2}},eta ^{2}left_{0}+eta left_{0}+cdots =0.end{aligned}}}     (H) ∂ 2 Φ 1 ∂ t 2 + g ∂ Φ 1 ∂ z = 0 , {displaystyle {frac {partial ^{2}Phi _{1}}{partial t^{2}}}+g,{frac {partial Phi _{1}}{partial z}}=0,}     (J1 ) ∂ 2 Φ 2 ∂ t 2 + g ∂ Φ 2 ∂ z = − η 1 ∂ ∂ z ( ∂ 2 Φ 1 ∂ t 2 + g ∂ Φ 1 ∂ z ) − ∂ ∂ t ( | u 1 | 2 ) , {displaystyle {frac {partial ^{2}Phi _{2}}{partial t^{2}}}+g,{frac {partial Phi _{2}}{partial z}}=-eta _{1},{frac {partial }{partial z}}left({frac {partial ^{2}Phi _{1}}{partial t^{2}}}+g,{frac {partial Phi _{1}}{partial z}} ight)-{frac {partial }{partial t}}left(|mathbf {u} _{1}|^{2} ight),}     (J2 ) ∂ 2 Φ 3 ∂ t 2 + g ∂ Φ 3 ∂ z = − η 1 ∂ ∂ z ( ∂ 2 Φ 2 ∂ t 2 + g ∂ Φ 2 ∂ z ) − η 2 ∂ ∂ z ( ∂ 2 Φ 1 ∂ t 2 + g ∂ Φ 1 ∂ z ) − 2 ∂ ∂ t ( u 1 ⋅ u 2 ) − 1 2 η 1 2 ∂ 2 ∂ z 2 ( ∂ 2 Φ 1 ∂ t 2 + g ∂ Φ 1 ∂ z ) − η 1 ∂ 2 ∂ t ∂ z ( | u 1 | 2 ) − 1 2 u 1 ⋅ ∇ ( | u 1 | 2 ) . {displaystyle {egin{aligned}{frac {partial ^{2}Phi _{3}}{partial t^{2}}}+g,{frac {partial Phi _{3}}{partial z}}=&-eta _{1},{frac {partial }{partial z}}left({frac {partial ^{2}Phi _{2}}{partial t^{2}}}+g,{frac {partial Phi _{2}}{partial z}} ight)-eta _{2},{frac {partial }{partial z}}left({frac {partial ^{2}Phi _{1}}{partial t^{2}}}+g,{frac {partial Phi _{1}}{partial z}} ight)\&-2,{frac {partial }{partial t}}left(mathbf {u} _{1}cdot mathbf {u} _{2} ight)-{ frac {1}{2}},eta _{1}^{2},{frac {partial ^{2}}{partial z^{2}}}left({frac {partial ^{2}Phi _{1}}{partial t^{2}}}+g,{frac {partial Phi _{1}}{partial z}} ight)\&-eta _{1},{frac {partial ^{2}}{partial t,partial z}}left(|mathbf {u} _{1}|^{2} ight)-{ frac {1}{2}},mathbf {u} _{1}cdot {oldsymbol { abla }}left(|mathbf {u} _{1}|^{2} ight).end{aligned}}}     (J3 ) ∂ Φ 1 ∂ t + g η 1 = 0 , {displaystyle {frac {partial Phi _{1}}{partial t}}+g,eta _{1}=0,}     (K1 ) ∂ Φ 2 ∂ t + g η 2 = − η 1 ∂ 2 Φ 1 ∂ t ∂ z − 1 2 | u 1 | 2 , {displaystyle {frac {partial Phi _{2}}{partial t}}+g,eta _{2}=-eta _{1},{frac {partial ^{2}Phi _{1}}{partial t,partial z}}-{ frac {1}{2}},left|mathbf {u} _{1} ight|^{2},}     (K2 ) ∂ Φ 3 ∂ t + g η 3 = − η 1 ∂ 2 Φ 2 ∂ t ∂ z − η 2 ∂ 2 Φ 1 ∂ t ∂ z − u 1 ⋅ u 2 − 1 2 η 1 2 ∂ 3 Φ 1 ∂ t ∂ z 2 − η 1 ∂ ∂ z ( 1 2 | u 1 | 2 ) . {displaystyle {egin{aligned}{frac {partial Phi _{3}}{partial t}}+g,eta _{3}=&-eta _{1},{frac {partial ^{2}Phi _{2}}{partial t,partial z}}-eta _{2},{frac {partial ^{2}Phi _{1}}{partial t,partial z}}-mathbf {u} _{1}cdot mathbf {u} _{2}\&-{ frac {1}{2}},eta _{1}^{2},{frac {partial ^{3}Phi _{1}}{partial t,partial z^{2}}}-eta _{1},{frac {partial }{partial z}}left({ frac {1}{2}},left|mathbf {u} _{1} ight|^{2} ight).end{aligned}}}     (K3 ) u j = ∇ Φ j , ∇ 2 Φ j = 0 , ∂ Φ j ∂ n = 0  at  z = − h , } for all orders  j ∈ N + . {displaystyle left.{egin{array}{rcl}mathbf {u} _{j}&=&{oldsymbol { abla }}Phi _{j},\ abla ^{2}Phi _{j}&=&0,\displaystyle {frac {partial Phi _{j}}{partial n}}&=&0quad { ext{ at }}z=-h,end{array}} ight}qquad { ext{for all orders }}jin mathbb {N} ^{+}.}     (L ) In fluid dynamics, a Stokes wave is a non-linear and periodic surface wave on an inviscid fluid layer of constant mean depth.This type of modelling has its origins in the mid 19th century when Sir George Stokes – using a perturbation series approach, now known as the Stokes expansion – obtained approximate solutions for non-linear wave motion. Stokes' wave theory is of direct practical use for waves on intermediate and deep water. It is used in the design of coastal and offshore structures, in order to determine the wave kinematics (free surface elevation and flow velocities). The wave kinematics are subsequently needed in the design process to determine the wave loads on a structure. For long waves (as compared to depth) – and using only a few terms in the Stokes expansion – its applicability is limited to waves of small amplitude. In such shallow water, a cnoidal wave theory often provides better periodic-wave approximations. While, in the strict sense, Stokes wave refers to progressive periodic waves of permanent form, the term is also used in connection with standing waves and even for random waves. The examples below describe Stokes waves under the action of gravity (without surface tension effects) in case of pure wave motion, so without an ambient mean current. According to Stokes' third-order theory, the free surface elevation η, the velocity potential Φ, the phase speed (or celerity) c and the wave phase θ are, for a progressive surface gravity wave on deep water – i.e. the fluid layer has infinite depth: