CE5307 Wave Hydrodynamics

Linear Wave Theory

– Formulation and Solution

Lecturer: Low Ying Min

email: ceelowym@nus.edu.sg

Course Outline

Part I Wave theory (Low YM)

Fundamentals of Fluid mechanics

Linear wave theory
Introduction to second-order wave theory
Random waves and their properties

Part II Wave loads (Allan Magee)

Wave loads on slender structures
Wave loads on large structures
 References

Dean, R.G. & Dalrymple R.A. (1991). Water wave

mechanics for engineers and scientists.

Sarpkaya, T. & Isaacson. (1981). Mechanics of wave

forces on offshore structures.

Faltinsen O.M. (1990). Sea loads on ships and offshore

structures.

Background
 In general, wave is an up and down, or back and forth
motion with a certain period.

 Common types of waves include: sound waves, light

waves, radio waves, and ocean waves

 Ocean waves are most commonly generated by wind in the

deep ocean, although it may also be caused by other
factors, for example tsunamis.

 Gravity is the restoring force for ocean waves

 Wave modelling is applied in two main areas:

- Offshore engineering
- Coastal engineering
 Videos

 Big wave near the beach

https://www.youtube.com/watch?v=ZGuYj5RM7u0

 Ships travelling in storm

https://www.youtube.com/watch?v=HW6uiRWkgcM

 Lighthouse with very big wave during stormy weather

https://www.youtube.com/watch?v=m7RSryuJAwE

Water waves
 Known as gravity waves
 Actual characteristics
- Random
- Nonlinear
- Multidirectional
- Energy losses from breaking, etc
 Simple mathematical model
- Sinusoidal
- Linear
- Unidirectional (plane waves)
 Overview of wave theories
Wave theories

Deterministic Probabilistic
(regular) (irregular/random)


Analytical Numerical time

e.g. Dean’s stream function theory

Our present
focus Short waves Long waves
(wind generated) (e.g. tsunamis)

Linear Nonlinear
(sinusoidal)
e.g. Stokes 2nd to 5th order theories
Cnoidal wave theory

Basic definitions
z (upwards +ve !!) L Wavelength
Wave speed, c=L/T
T Wave period
 (x,t) H a a Wave amplitude
0
x
d Water depth
Still water level
d  Surface elevation
L
Seabed, z = –d H Waveheight
x Horiz axis
z Vertical axis
 x, t   a coskx  t 
k Wave number
2 2
k   Wave freq (rad/s)
L T
 Basic definitions

L
 x, t   a coskx  t 
T
Total phase angle

Varying in space & time

t
y y y = b(x – c)
y = bx

x x c x

-- Wave form after time t

Some Fundaments of Fluid Mechanics

We need to have basic background on fluid mechanics before we can
proceed further….but nothing too scary!

Lagrangian Description
 Considers the motion of individual fluid particles
 Seeks to define their trajectory within the flow field
 Highly complex, difficult to obtain solutions
 Can be informative in some cases

Eulerian Description (our focus)

 Considers fluid properties at a specific location fixed in
space. Does not consider history of fluid particles, only
interested in properties at one point
 Much simpler, solutions can be more easily obtained
 More frequently used.
 Mass continuity
Consider 2-dimensional flow (i.e. v = 0)
For incompressible fluid,
Flow in – flow out = 0, i.e.

Continuity eqn:

Irrotational flow
A flow in which the fluid elements rotate is said to have
vorticity. Consider 2D flow

1
z A B
C After a time , AB will
have rotated by an angle

A B
u
Angular velocity of AB
w ω
x
Convention: anti-clockwise +ve
 Irrotational flow
Likewise, after a time , AC will have rotated
by an angle
C
–ve because clockwise
2
rotation

Angular velocity of AC
A
ω

The rate of rotation of the fluid particle, , is the average of 1 and 2

1 1
2 2

Irrotational flow
Important assumptions that are widely made are:
 Ideal fluid: If the viscous effect is negligible in a fluid flow,
the fluid is referred to as the ideal fluid. In an ideal fluid the
vorticity cannot be created nor destroyed.
 Irrotational flow: If the fluid flow is initially irrotational, it
will remain so, i.e.,  = 0. This type of flow is called the
irrotational flow.

For an irrotational flow

1
 0
2
 Velocity potential
Working with the velocity vector [u v w] can be complicated

Velocity potential  (x, z, t) is a scalar, and is simpler for

describing flow field. It is defined as

(also applies to 3D which we

, will not consider)

 only exists for irrotational flow.

Vector calculus form,
Example when  does not exist
which we will not use
Direction of flow
Friction at u = 
boundary layer Vortices
Wake
Lift Turbulent flow If velocity potential
(Higher velocities exists, we call the
Drag = reduced pressure)
flow potential flow

Velocity potential
Proof that  exists if and only if the flow is irrotational.

Starting with Setting this equal to some function 

we obtain
Integrating w.r.t. x gives
 , ,

Hence, ,
∴ Check:
Subst into
Integrating w.r.t. z gives

∴
Hence,  = 0 i.e. irrotational
 Laplace equation

2D continuity eqn: + ,

If vel potential exists, then

Incompressibility +
+ irrotationality gives
Laplace eqn

Laplace eqn in
Laplace eqn in 2D Vector calculus form
2 = 0

Non-examinable
Stream function
An alternative description is the stream function (x, z, t),
defined as

Stream functions only exist

for 2D flow, or 3D flow with
axisymmetry

Streamlines – lines with a constant

value of the stream function – for
the incompressible potential flow
around a circular cylinder in a
uniform onflow.

https://en.wikipedia.org/wiki/Stream_function
 Navier-Stokes’s equation

 The Navier-Stokes equations are the most complete fluid

mechanics version of Newton’s law, allowing for shear
stresses on the fluid
 Navier-Stokes are very complex, and analytical solutions
are very rare
 A common assumption is to assume that the flow is
inviscid, leading to Euler’s equations

Euler’s equation
p = pressure

z-direction similar, but there is

gravity, hence
Applying Newton’s law F = ma
in the x-direction
d
d d
d

1 d 1 d
d d
 Euler’s equation
Consider the acceleration, . Since u = u(x, z, t)

d Chain rule for partial differentiation; e.g.

https://calculus.subwiki.org/wiki/Chain_rule
d _for_partial_differentiation

d
d Euler’s eqns
unsteady convective 1
acceleration acceleration

Unsteady Bernoulli’s equation

If we assume irrotational and hence  exists, then by
integrating Euler’s eqns, we get

Unsteady Bernoulli’s eqn

ρ const

unsteady term

Provides useful relationship between pressure field and kinematics

We can just say Bernoulli’s equation for short

Details of derivation see Dean & Dalrymple, pp. 33-34

 Linear wave theory
Also known as:
 Small amplitude wave theory
 Airy wave theory

Key assumptions (must know!) Note: a/L is

 Fluid is incompressible known as the
 Fluid is irrotational wave steepness
 Small amplitude: a << L, a << d

Other assumptions (for background knowledge)

 Seabed is horizontal, i.e. depth is constant
 Waveform is periodic in space and time
 Waves are planar, i.e. long-crested (two-dimensional)
 Coriolis effect due to earth’s rotation is neglected

Boundary Value Problem

There are infinite solutions to the differential eqn:

We select the solution rejecting those that are not

compatible with the boundary conditions.

What are the boundary conditions for our problem?

 Boundary conditions

Periodic boundary conditions

In time (x, t) = (x, t + T)
In space (x, t) = (x + L, t)

Seabed
Vertical velocity must be zero,

i.e. at z = –d

Boundary conditions
Kinematic free surface boundary condition (KFSBC)
A fluid particle at the free surface stays there
i.e. the velocity of the fluid normal to the surface
equals the velocity of the surface along that normal

x t = t + t For a given x, let z =  represent a point P

z =  +  on the surface at time t.
 At time t = t + t, the surface is given by
 t=t z =  + 

z= The velocity of the surface along a normal

P through P is given by
w
cos cos

But the velocity of the fluid along this
u
normal is cos sin
 Boundary conditions

at z=

tan

at z = 

For a << L, and  are small, hence

at z = 0 or at z = 0

Boundary conditions
Dynamic free surface boundary condition (DFSBC)
Pressure at free surface is constant (i.e. atmospheric)
Unsteady form of Bernoulli’s eqn
ρ const

Applying p = const at z = ,
1
ρ  0
2
For small amplitude waves, neglect 2nd-order terms u2 and w2,
and apply B.C. at z = 0

 0 at z = 0
 Boundary conditions
Summary

Governing differential eqn: 0

Seabed: 0 at z = –d
Free surface:
can be combined to give
Kinematic 0 at z = 0
0 at z = 0
Dynamic  0 at z = 0

Periodicity in space and time: (x, t) = (x, t + T)

(x, t) = (x + L, t)

Solution of the PDE

Governing eqn: 0
Solve PDE by separation of variables
Solution is of the form (x, z, t) = F(z) G(x) H(t)
Considering the periodicity of x and t,
 =F(z)[C3cos(kx –t) + C4sin(kx –t)]
From DFSBC:

1
 sin cos

We want the wave crest to be at t =0, x = 0 (we are free to define convention)
Hence, C3 = 0
Further reading: Dean & Dalrymple, pp. 53-62
 =F(z)C4sin(kx –t) or Sarpkaya & Isaacson, pp. 154-155
 Solution of the PDE
It can be shown that F(z) has the form
cosh sinh (non-periodic)

Using 0 at z = –d

sinh cosh sin 0 at z = –d

∴ sinh cosh 0
∴ = tanh
Hence, ϕ cosh tanh sinh sin

cosh cosh sinh sinh

sin
cosh

sin where C = C1C4

Solution of the PDE

To find C, let us first relate  and  via

 (DFSBC)

cosh
cos cos
cosh kd

Hence, wave amplitude, 

cosh
sin
cosh kd

Subst above into the B.C. 0

cosh sinh
sin sin 0 at 0
cosh kd cosh kd
sinh
0
cosh kd
 Dispersion relationship
Hence,

tanh
This is the linear dispersion relationship, which relates , k and d

Unfortunately, the eqn is implicit in k,

 solve by trial and error

Using the dispersion relationship, we can obtain another expression for :

a cosh k  z  d 
 sin kx  t 
k sinh kd

Equations for linear wave theory

a cosh k  z  d 
Velocity potential  sin kx  t  Recall that:
k sinh kd
e x  e x
sinh x 
 a cosh k z  d  2
Horiz velocity u  coskx  t 
x sinh kd e x  e x
cosh x 
 a sinh k z  d  2
Vertical velocity w  sin kx  t  sinh x
z sinh kd tanh x 
cosh x
u a 2 cosh k  z  d 
Horiz acceleration u   sin kx  t 
t sinh kd
u a 2 sinh k z  d 
Vertical acceleration w   coskx  t 
t sinh kd

Dispersion relationship  2  gk tanh kd

(implicit in k) , k and d are
not independent!
 Deepwater approximations

 Valid if kd >  (i.e. d > L/2)

 Exact as d  

Example: 0


a 1
sin kx  t  1 kd
2 e 
kz  kd
 e  kz  kd 
k 2 e  e  kd 
a kz
 e sin kx  t  0
k

Deepwater approximations
a kz
Velocity potential  e sin kx  t 
k

Horiz velocity u  ae kz coskx  t 

Vertical velocity w  ae kz sin kx  t 

Horiz acceleration u  a 2 e kz sin kx  t 

Vertical acceleration w  a 2e kz coskx  t 

Dispersion relationship  2  gk since tanh kd  1

No longer
implicit
 Wave trajectories (deepwater)
Horizontal displacement, x   udt

  ae kz coskx  t dt

  ae kz sin kx  t 

Vertical displacement, z   wdt

  ae kz sin kx  t dt

 ae kz coskx  t 

 
 x 2  z 2  ae kz
2
Eqn of circle

Radius of orbit = aekz

Wave trajectories (deepwater)

z
Radius

Rate of decay = ekz < 1 (z is –ve!!)

 2  L  
At z = –L/2, e kz  exp      e  0.043
 L  2 
https://www.youtube.com/watch?v=7yPTa8qi5X8
 Wave trajectories (shallow water)
a cosh k z  d 
Horizontal displacement, x   udt   coskx  t dt
sinh kd
a cosh k z  d 
 sin kx  t    Rx sin 
sinh kd
a sinh k z  d 
Vertical displacement, z   wdt   sin kx  t 
sinh kd
a sinh k z  d 
 coskx  t   R y cos
sinh kd
2 2
 x   z 
       1 Eqn of ellipse
 Rx   Rz 

a cosh k z  d  a sinh k z  d 
Rx  Rz 
sinh kd sinh kd
Major radius Minor radius

Rx  Ry always

Wave trajectories (shallow water)

Very shallow Moderately shallow

10 10
1 1 1
20 20 2

Both Rx and Ry decay with water depth