Chapter 3. Small-Amplitude Water Wave Theory
3.1 Introduction
Assuming inviscid, incompressible fluid and irrotational flow, φ and exist, which
satisfy .
Ψ
2 0
2 =∇ Ψ=
∇ φ
3.2 Boundary Value Problem
3.3 Boundary Conditions
3.3.1 Kinematic boundary condition
The kinematic boundary condition is a condition that describes the water particle kinematics at a boundary (either fixed or moving). If we sit on a boundary (fixed or moving) and move with the boundary, we do not feel any change of the surface that constitutes the boundary. Mathematically, the rate of change of the surface must be zero, i.e., the total derivative of the surface is zero on the surface. Let the surface of the boundary be represented by F(x,y,z,t)=0. Then
0 on
0 =
∂ = + ∂
∂ + ∂
∂ + ∂
∂
= ∂ F
z w F y v F x u F t F Dt DF
2 2 2
,
; ⎟
⎠
⎜ ⎞
⎝
⎛
∂ + ∂
⎟⎟⎠
⎜⎜ ⎞
⎝
⎛
∂ + ∂
⎟⎠
⎜ ⎞
⎝
⎛
∂
= ∂
∂ ∇ +∂
∂ +∂
∂
= ∂
∇
∇
⋅
=
∇
⋅
∂ =
−∂
z F y
F x
F F z k
j F y i F x F F F n u F t u
F
0
/ on =
∇
∂
∂
=−
⋅ F
F t n F
u
3.3.2 BBC
Assume a fixed sea bed:
0 ) , ( )
, , , ( )
,
( ⇒ = + =
−
= h x y F x y z t z h x y z
The kinematic boundary condition on sea bed is ) , ( on
/ 0
y x h F z
t n F
u = =−
∇
∂
∂
= −
⋅
Recall that a⊥b if a⋅b=0. Therefore, u⋅n=0 means that u is parallel to the sea bed surface, i.e. no flow perpendicular to the bed.
Using
1
2 2
⎟⎟ +
⎠
⎜⎜ ⎞
⎝
⎛
∂ + ∂
⎟⎠
⎜ ⎞
⎝
⎛
∂
∂
∂ + +∂
∂
∂
∇ =
= ∇
y h x
h
k y j i h x h F
n F
we have
) , ( on
) , ( on
0 1
2 2
y x h y z
h y x h x z
y x h y z
v h x u h w
y h x
h
y w v h x u h n u
−
∂ =
∂
∂ + ∂
∂
∂
∂
=∂
∂
−∂
−
∂ =
− ∂
∂
− ∂
=
=
⎟⎟ +
⎠
⎜⎜ ⎞
⎝
⎛
∂ + ∂
⎟⎠
⎜ ⎞
⎝
⎛
∂
∂
∂ + + ∂
∂
∂
=
⋅
φ φ
φ
On a horizontal bottom, ∂h/∂x=∂h/∂y=0, so that ∂φ/∂z=0 on z=−h.
3.3.3 KFSBC On water surface,
0 ) , , ( )
, , , ( )
, ,
( ⇒ = − =
= x y t F x y z t z x y t
z η η
The kinematic boundary condition on free surface is
(
/) (
/)
1 on ( , , )/ /
2
2 z x y t
y x
t F
t n F
u η
η η
η =
+
∂
∂ +
∂
∂
∂
= ∂
∇
∂
∂
= −
⋅
Using
1
2 2
⎟⎟ +
⎠
⎜⎜ ⎞
⎝
⎛
∂ + ∂
⎟⎠
⎜ ⎞
⎝
⎛
∂
∂
∂ +
−∂
∂
−∂
∇ =
= ∇
+ +
=
y x
k y j xi F
n F
k w j v i u u
η η
η η
we have
) , , ( on
) , , ( on
t y x y z
y x x t z
t y x y z
x v t u
w
w t v y
u x
η η φ η φ η φ
η η η
η
η η
η
∂ =
∂
∂
−∂
∂
∂
∂
−∂
∂
=∂
∂
−∂
∂ = + ∂
∂ + ∂
∂
=∂
∂
=∂
∂ +
− ∂
∂
− ∂
Small-amplitude wave theory assumes L∼O(h)
H/L = wave steepness << 1 η ∼O(H)
t∼O(T) x∼O(L) z∼O(h)∼O(L)
u x
∂
= ∂φ ∼ ⎟
⎠
⎜ ⎞
⎝
⎛ T
O H
φ ∼ ⎟
⎠
⎜ ⎞
⎝
⎛ T O HL
( ) ( )
1 1 1) , , ( on
<<
⎟⎠
⎜ ⎞
⎝
⎟ ⎛
⎠
⎜ ⎞
⎝
⎛
⎟⎠
⎜ ⎞
⎝
⎟ ⎛
⎠
⎜ ⎞
⎝
⎟ ⎛
⎠
⎜ ⎞
⎝
⎟ ⎛
⎠
⎜ ⎞
⎝
⎛
∂ =
∂
∂
−∂
∂
∂
∂
−∂
∂
= ∂
∂
−∂
L H L
H
L H T H L H T H T H T H
t y x y z
y x x t
z η φ η φ η η
φ
) , , (
on z x y t
t
z η η
φ =
∂
=∂
∂
−∂
∴
Taylor series expansion about z =0 gives
⎟⎟⎠
⎜⎜ ⎞
⎝
⎟⎟ ⎛
⎠
⎜⎜ ⎞
⎝
⎟ ⎛
⎠
⎜ ⎞
⎝
⎟ ⎛
⎠
⎜ ⎞
⎝
⎛
=
⎟ +
⎠
⎜ ⎞
⎝
⎛
∂
−∂
∂
− ∂
∂ + ∂
⎟⎠
⎜ ⎞
⎝
⎛
∂
−∂
∂
−∂
∂ + ∂
⎟⎠
⎜ ⎞
⎝
⎛
∂
−∂
∂
−∂
⎟ =
⎠
⎜ ⎞
⎝
⎛
∂
−∂
∂
−∂
=
=
=
=
LT H LT H T
H T H
t z z
t z z t
z t
z z z z z
2 2
0 2
2 2
0 0
2 φ η L 0
η η
η φ η
φ η
φ
η
on =0
∂
=∂
∂
−∂ z
t z
η
φ (LKFSBC)
3.3.4 DFSBC
The Bernoulli equation on free surface is
( )
( ) on ( , , )2
1 2 2 2
t y x z t
C p g
w v
t u η η
ρ
φ + + + + η + = =
∂
− ∂
Using pη =0 (gauge pressure),
( )
( )
( )
1 1 ~( )
1) , , ( on
) 2 (
1
2 2
2 2
2 2 2
L O gT L
H T gH H T
HL
t y x z t
C g w v t u
⎟⎟⎠
⎜⎜ ⎞
⎝
<< ⎛
⎟⎠
⎜ ⎞
⎝
⎛
⎟⎟⎠
⎜⎜ ⎞
⎝
⎟ ⎛
⎠
⎜ ⎞
⎝
⎛
=
= + + +
∂ +
−∂φ η η
g C(t) on z (x,y,t)
t η η
φ + = =
∂
− ∂
∴
Taylor series expansion about z =0 gives
( )
( )
⎟⎟⎠
⎜⎜ ⎞
⎝
⎟ ⎛
⎠
⎜ ⎞
⎝
⎟⎟ ⎛
⎠
⎜⎜ ⎞
⎝
⎛
⎟⎟⎠
⎜⎜ ⎞
⎝
⎟⎟ ⎛
⎠
⎜⎜ ⎞
⎝
⎟ ⎛
⎠
⎜ ⎞
⎝
⎛
=
⎟ +
⎠
⎜ ⎞
⎝
⎛ +
∂
−∂
∂ + ∂
⎟⎠
⎜ ⎞
⎝
⎛ +
∂
−∂
⎟ =
⎠
⎜ ⎞
⎝
⎛ +
∂
−∂
=
=
=
L H L gT L H L
gT
L g H T
gH H T
HL
t C t g
g z g t
t z z z
2 2
2 2
2 2
0 0
1
) L ( φ η
η φ η
φ η
η
+ = ( ) on =0
∂
−∂ g C t z
t η
φ (LDFSBC)
3.3.5 LBC
Assuming 2-D (x−z) periodic wave,
) , ( ) , (
) , ( ) , (
T t x t x
t L x t x
+
= +
= φ φ
φ φ
3.3.6 Summary of 2-D periodic wave boundary value problem
GE ∇2φ =0 0≤x≤L, −h≤z≤η
BBC z h
z = =−
∂
− ∂φ 0 on
LKFSBC on =0
∂
=∂
∂
−∂ z
t z
η φ
LDFSBC + = ( ) on =0
∂
−∂ g C t z
t η
φ LBC
) , ( ) , (
) , ( ) , (
T t x t x
t L x t x
+
= +
= φ φ
φ φ
Note: The Laplace equation is linear, so the superposition of solutions is valid.
3.4 Solution to Boundary Value Problem
Using separation of variables, the velocity potential can be expressed as )
( ) ( ) ( ) , ,
(x z t =Χ x Ζ z Τt φ
Assume
t T
t) sinσ ; σ 2π
( = =
Τ
where σ = wave angular frequency. The above equation satisfies the periodicity condition in time, that is
t T
t T
t T
t T
t ) sin(σ σ ) sinσ cosσ cosσ sinσ sinσ
( + = + = + =
Τ
Now
t z x t
z
x σ
φ( , , )=Χ( )Ζ( )sin
Substitution into the Laplace equation gives
1 0 1
0 sin sin
2 2 2
2
2 2 2
2
Ζ= + Ζ
Χ Χ
Ζ = Χ + ΧΖ
dz d dx
d
dz t t d
dx
d σ σ
The first term is a function of x alone, whereas the second term is a function of zalone. Therefore, the two terms should be constants with opposite signs, that is,
2 2 2
2 2
2
1 1
dz k d
dx k d
Ζ = Ζ
− Χ= Χ
The possible solutions that satisfy these ODE’s are given in Table 3.1 of textbook.
Among these, the solution which satisfies the periodicity condition in x is
(
A kx B kx) (Ce De )
t
t z
x kz kz σ
φ( , , )= cos + sin + − sin
The periodicity condition in x gives
) sin cos cos
(sin )
sin sin cos
(cos
) ( sin )
( cos sin
cos
kL kx kL
kx B kL kx kL
kx A
L x k B L x k A kx B kx A
+ +
−
=
+ +
+
= +
For the above relation to be satisfied, the followings are needed:
0 sin and 1
coskL= kL=
which give the wave number,
k =2Lπ
Recalling that the superposition of solutions is valid for the Laplace equation, keep only
(
Ce De)
tkx A t z
x kz kz σ
φ( , , )= cos + − sin
Using BBC,
( )
kh kh kh
kz kz
De C
De Ce
h z t
kDe kCe
kx z A
2
0
on 0 sin cos
=
=
−
−
=
=
−
∂ =
−∂
−
− σ
φ
Therefore,
( )
( )
t z h k kx G
t e
e kx ADe
t De
e De kx A t z x
z h k z h k kh
kz kz
kh
σ
σ σ φ
sin ) ( cosh cos
sin cos
sin cos
) , , (
) ( ) ( 2
+
=
+
=
+
=
+
− +
−
Applying LDFSBC,
g t t C g kx
kh G
t C g t kh kx
G
z t
C t g
) cos (
cosh cos
) ( cos
cosh cos
0 on
) (
+
=
= +
−
=
=
∂ +
−∂
σ σ η
η σ σ
φ η
The spatial mean of η must be zero. Thus, C(t) must be zero. Expressing η as
t
H kx σ
η cos cos
= 2
we have
kh G gH
cosh 2σ
=
Finally,
t kh kx
z h t gH
z
x σ
φ σ cos sin
cosh ) cosh(
) 2 , ,
( = +
Applying LKFSBC,
tanh 2 2
sin 2 cos
sin cosh cos
sinh 2
0 on
kh H gkH
t H kx
t kh kx
kh gkH
t z z
σ σ
σ σ σ σ
η φ
=
−
=
−
∂ =
=∂
∂
− ∂
kh gktanh
2 =
∴σ (dispersion relationship)
The dispersion relationship gives relation among h, σ and k or the relation among h, T and L. The dispersion relationship can be solved for by Newton-Raphson method for given
k σ and . Approximate solutions are also available: h Eckart (1951):
g gk h
2
2 tanhσ
σ =
Hunt (1979):
textbook of
72 p.
in given
~ ,
; 1
)
( 1 6
2 6
1 2
2 d d
g y h y
d y y
kh
n n n
=σ +
+
=
∑
=
The dispersion relationship gives
gT kh L
L kh T g
2 tanh 2 tanh 2
2 2
π π π
=
⎟ =
⎠
⎜ ⎞
⎝
⎛
In deepwater (kh>π ), tanhkh≅1. Therefore, the deepwater wavelength is
π 2
2 0
L = gT
Then
π 2
tanh tanh
tanh
0 0
0 0
0
gT T C L
kh C
T kh L T C L
kh L
L
=
=
=
=
=
=
In summary,
t H kx
t x
t kh kx
z h t Hg
z x
σ η
σ σ φ
cos 2 cos
) , (
sin cosh cos
) cosh(
) 2 , , (
=
= +
which represents a standing wave.
Another standing wave associated with the sinkx term gives
t H kx
t x
t kh kx
z h t Hg
z x
σ η
σ σ φ
sin 2 sin
) , (
cos cosh sin
) cosh(
) 2 , , (
=
− +
=
Superposition of the two standing waves gives a progressive wave:
) 2 cos(
) sin sin cos
2 (cos )
, (
) cosh sin(
) ( cosh 2
) cos sin sin
cosh (cos ) cosh(
) 2 , , (
t H kx
t kx t
H kx t x
t kh kx
z h k H g
t kx t
kh kx z h t Hg
z x
σ
σ σ
η
σ σ
σ σ σ
φ
−
=
+
=
+ −
−
=
+ −
=
Progressive wave propagating to positive x-direction:
For progressive wave propagating to negative x-direction:
) sin(
)
sin(kx−σt ⇒ kx+σt
For the definitions of deepwater, shallow water, and intermediate depth water, see Fig.
3.12 of textbook. Also see Table 3.2 for asymptotic forms of hyperbolic functions.
Dispersion relationship in shallow water:
gh C
k gh C
h gk kh gk
=
⎟ =
⎠
⎜ ⎞
⎝
=⎛
≅
=
2 2
2
2 tanh
σ σ
The shallow water wave is non-dispersive, that is, all the waves with different frequencies propagate at the same speed.
Dispersion relationship in deepwater:
π
π π π σ
π π σ
2
2 2 2 2
2 tanh
0 0
2 2
0 2 2
gT T C L
gT T
g g
L k
gk kh gk
=
=
=
⎟⎠
⎜ ⎞
⎝
= ⎛
=
=
≅
=
3.5 Waves with Uniform Current
U0Assume: 1) Current direction and wave propagation direction is collinear, and 2) Current is uniform vertically as well as horizontally.
In the absence of waves,
x x U
U0 ⇒ =− 0
∂
−∂
= φ φ
For wave with current,
) cos(
) (
0x Acoshk h z kx t
U σ
φ =− + + −
which satisfies the periodicity condition and BBC. Then,
) sin(
) (
0 kAcoshk h z kx t
x U
u φ = + + −σ
∂
−∂
=
) (
sin ) ( cosh )
sin(
) ( cosh
2 0 2 2 2 2
2 0
2 U kAU k h z kx t k A k h z kx t
u = + + −σ + + −σ
Neglecting small nonlinear terms,
) sin(
) ( cosh
2 0
2 0
2 U kAU k h z kx t
u ≅ + + −σ
DFSBC is
( )
( ) ( )
( )2 1 2
1 2 1
0 2
2 0
2 2
2 2
t C g
w t u
g z w t u
g w t u
z z
z
⎥⎦ =
⎢⎣ ⎤
⎡ + + +
∂
−∂
∂ + ∂
⎥⎦⎤
⎢⎣⎡ + + +
∂
−∂
=
⎥⎦⎤
⎢⎣⎡ + + +
∂
−∂
=
=
=
φ η η
φ η φ η
η
Note that w2 is small compared to the linear terms but u2 is not. Then DFSBC gives
(
2 cosh sin( ))
( )2 ) 1 sin(
coshkh kx t U02 kAU0 kh kx t g C t
A − + + − + =
−σ σ σ η
or
) ( ) sin(
cosh 2 1
) ,
( 0
2
0 kU kh kx t C t
g A g t U
x ⎟ − +
⎠
⎜ ⎞
⎝⎛ − +
−
= σ
σ η σ
Since
∫
∫
= =− += T TC t dt
T g dt U
T 0
2 0
0 1 ( )
0 2
1 η
η
we have
g t U
C( ) constant 2
2
= 0
=
Now
) 2 sin(
) sin(
cosh 1
) ,
( 0 H kx t
t kx kU kh
g t A
x σ σ
σ
η σ ⎟ − = −
⎠
⎜ ⎞
⎝⎛ −
=
which gives
C kh U
gH kU kh
g A H
cosh 1
2 cosh
2 1 0 0
⎟⎠
⎜ ⎞
⎝⎛ −
=
⎟⎠
⎜ ⎞
⎝⎛ −
=
σ σ σ
Therefore,
) cosh cos(
) ( cosh 1
2 )
, , (
0
0 kx t
kh z h k
C U x gH
U t z
x σ
σ
φ + −
⎟⎠
⎜ ⎞
⎝⎛ − +
−
=
Applying KFSBC,
0
0 0
=
⎥⎦ +
⎢⎣ ⎤
⎡
∂ + ∂
∂
∂
∂
−∂
∂
∂
∂ + ∂
⎥⎦⎤
⎢⎣⎡
∂ +∂
∂
∂
∂
−∂
∂
= ∂
⎥⎦⎤
⎢⎣⎡
∂ +∂
∂
∂
∂
−∂
∂
∂
=
=
=
L
z z
z t x x z z t x x z
z x x t
φ η φ η η
φ η φ η φ
η φ η
η
0
0 on =
∂
−∂
∂ = + ∂
∂
∂ z
z U x
t
φ η
η
kh C gk
U
kh C gk
U C U
kh gk
kh C
U gk C
U
kh C
U gk U k
kh C
U k gk
U
t kx kh C
U t kgH
H kx k U t H kx
tanh 1
tanh 1
1 tanh
tanh 1
1
tanh 1
1
tanh 1
) cos(
tanh 1
2 ) 2 cos(
) 2 cos(
0 2 0 2
2 0 2
0 2
0
2 0 0
0 0
0 0
⎟=
⎠
⎜ ⎞
⎝⎛ −
=
⎟⎠
⎜ ⎞
⎝⎛ −
⎟⎠
⎜ ⎞
⎝⎛ −
=
⎟⎠
⎜ ⎞
⎝⎛ −
−
= +
−
⎟⎠
⎜ ⎞
⎝⎛ −
−
= +
−
⎟⎠
⎜ ⎞
⎝⎛ −
−
= +
−
−
⎟⎠
⎜ ⎞
⎝⎛ −
−
=
− +
−
−
σ σ σ
σ σ σ
σ σ
σ σ
σ σ
σ
Finally,
kh gk
k
U0 + tanh σ =
The second term on RHS indicates the angular frequency in no current, that is the angular frequency in moving frame of reference, while the total indicates the angular frequency in stationary frame of reference.
