Difference between revisions of "Linear Plane Progressive Regular Waves"

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<u>Plane progressive linear waves</u>
+
{{Ocean Wave Interaction with Ships and Offshore Structures
 +
| chapter title = [[Linear Plane Progressive Regular Waves]]
 +
| next chapter = [[Wave Energy Density and Flux]]
 +
| previous chapter =  [[Linear and Second-Order Wave Theory]]
 +
}}
  
As illustrated in the first few slides, regular time-harmonic plane progressive waves are the fundamental building block in describing the propagation of surface wave disturbances in a deterministic & stochastic setting and in predicting the response of floating structures in a seastate.
+
{{incomplete pages}}
  
They satisfy the boundary-value problem:
+
== Introduction ==
  
<center><math> \begin{cases}
+
Regular time-harmonic linear plane progressive waves are the fundamental building block in describing the propagation of surface wave disturbances in a deterministic and stochastic setting and in predicting the response of floating structures in a seastate.
\frac{\partial^2\Phi_1}{\partial t^2} + g \frac{\partial\Phi_1}{\partial Z} = 0, \qquad Z=0 \\
 
\nabla^2\Phi_1 = 0, \qquad -H < Z < 0 \\
 
\frac{\partial\Phi_1}{\partial Z} = 0, \qquad Z=-H
 
\end{cases} </math></center>
 
  
The last condition imposes a zero normal flux condition across a sea floor of constant depth, <math>H</math>.
+
== Equations ==
  
The relation between <math> \omega </math> and <math> K </math> is known as the dispersion relation often written in the from:
+
We derive the solution for regular time-harmonic linear plane progressive waves
 +
for the first order potential in water of constant [[Finite Depth]].
 +
{{equations of motion time domain without body condition}}
 +
Note that we are assuming here no body is present so that <math>F</math> is the entire domain.
 +
It turns out that if the water is sufficiently deep the wave does not feel any effect from the sea floor and we can make the [[Infinite Depth]] assumption.
  
<center><math> \omega = f (K) </math></center>
 
  
<math> f(K) </math> depends on the physics of the wave propagation problem under study.
 
  
In the case of sound propagation:
+
== Expression in terms of velocity potential ==
  
<center><math> \omega = \mathbb{C} K \equiv f (K) </math></center>
+
{{frequency definition}}
  
Where <math> \mathbb{C} </math> is the speed of sound. The above relation suggests that all sound waves in a medium with isotropic properties propagate with the same phase speed regardless of their frequency of wave length. These are unknown to be non-dispersive waves.
 
  
Surface waves are dispersive since <math> f(k) </math> is a non linear function of <math> k </math>. It will be derived below.     
+
{{velocity potential in frequency domain}}
  
By definition, regular plane progressive waves are such that their free surface elevation is defined <u>apriori</u> as follows:
+
The equations therefore become
 
+
{{standard linear wave scattering equations without body condition}}
<center><math> \Im_1 (X,t) = A cos ( \omega t - K X) = A \mathbf{Re} \{ e^{-iKX+i\omega t} \} </math></center>
 
 
 
or schematically:       
 
 
 
Where A is an apriori known wave amplitude and the wave frequency and wave number <math> (\omega, K) </math> pair have the same definitions as in all wave propagation problems, namely:
 
 
 
<center><math> T = \frac{2\pi}{\omega} = \mbox{Wave Period} </math></center>
 
<center><math> \lambda = \frac{2\pi}{K} = \mbox{Wave Length} </math></center>
 
<center><math> C= \frac{omega}{K} = \mbox{Wave Phase Velocity} </math></center>
 
 
 
By virtue of the definition of the wave elevation of a plane progressive regular wave, we seek a compatible definition of the respective velocity potential. Let:
 
 
 
<center><math> \Phi_1 = A \mathbf{Re} \{ \phi (X,Z) e^{i\omega t} \} </math></center>
 
 
 
The problem reduces to the definition of <math> \phi(X,Z) </math> and the derivation of the appropriate dispersion relation between <math> \omega </math> and <math> K </math> so that the linear boundary value problem is satisfied.
 
 
 
Before proceeding with the algebra, certain underlying principles are always at work:
 
 
 
* Linear system theory states that when the input signal is <math> e^{i \omega t} </math> the output signal must also be harmonic and with the same frequency.
 
 
 
* We assume that a solution always exists, otherwise the statement of the physical and/or mathematical boundary value problem is flawed. If we can find a solution in most cases it is the solution. so simply try out solutions that may make sense from the physical point of view.
 
 
 
* If the boundary value problem is satisfied by a complex velocity potential then it is also satisfied by its real and imaginary parts.
 
 
 
In our case we will first derive the boundary value problem satisfied by the complex potential        and then we will try the plausible representation:
 
 
 
 
 
It follows upon substitution in the boundary value problem satisfied by    that        is subject to:
 
 
 
 
 
 
 
Allowing for:
 
 
 
 
 
 
 
It follows that            is subject to:
 
 
 
 
 
 
 
Verify by simple substitution that:
 
 
 
 
 
 
 
 
 
 
 
Satisfies the field equation            , the sea floor condition            and the free surface condition, only when
 
 
 
 
 
 
 
So by enforcing the free surface condition we have derived the dispersion relation for regular waves in finite depth.
 
 
 
Recall that:
 
  
 +
{{Separation of variable for a free surface first part}}
  
 +
We denote the
 +
positive imaginary solution of this equation by <math>k_{0} = \mathrm{i} k \,</math>
  
 
The resulting plane progressive wave velocity potential takes the form:
 
The resulting plane progressive wave velocity potential takes the form:
 +
<center><math> \phi(x,z,t) = A \ \mathrm{Re} \left\{ \frac{ig}{\omega} \frac{\cosh k (z+h)}{\cosh k h}
 +
e^{ \mathrm{i} k x - \mathrm{i} \omega t} \right\} </math></center>
 +
where <math>A</math> is the amplitude in displacement.
  
 +
This derivation is discussed in more mathematical depth in [[:Category:Eigenfunction Matching Method|Eigenfunction Matching Method]]
  
 +
== Propagating Wave ==
  
Verify that upon substitution:
+
[[Image:plane_wave.png|thumb|right|600px|Propagating Wave]]
  
 +
The free surface elevation is
 +
<center><math> \zeta (x,t) = A \mathrm{Re} \{ e^{\mathrm{i} kx - \mathrm{i} \omega t} \} \,</math></center>
 +
where A is an apriori known wave amplitude and the wave frequency and wave number <math> (\omega, k) \,</math>
 +
pair have the same definitions as in all wave propagation problems, namely:
  
 +
<center><math> T = \frac{2\pi}{\omega} = \mbox{Wave Period} </math></center>
 +
<center><math> \lambda = \frac{2\pi}{k} = \mbox{Wave Length} </math></center>
 +
<center><math> c= \frac{\omega}{k} = \mbox{Wave Phase Velocity} </math></center>
 +
The relation between <math> \omega \,</math> and <math> k \,</math> is known as the dispersion relation often written in the form
 +
<center><math> \omega = f (k) \,. </math></center>
 +
<math> f(k) \,</math> depends on the physics of the wave propagation problem under study,
 +
surface ocean waves are dispersive since <math> f(k)\, </math> is a non linear function of <math> k \,</math> as we will shortly show.
  
The corresponding flow velocity at some point            in the fluid domain or on            is simply given by:
+
== Flow Velocity and Pressure ==  
 
 
 
 
 
 
The linear hydrodynamic pressure due to the plane progressive wave, which must be added to the hydrostatic, is
 
 
 
 
 
 
 
From the lagrangian kinematic relation
 
 
 
 
 
 
 
We may obtain ordinary differential equations governing      . marking a paticular particle with the fluid at rest, so that          , we may write:
 
 
 
 
 
 
 
Where      is the prticle position at rest and     is its displacement due to the "arrival" of a plane progressive wave. Upon substitution in the equation of motion
 
 
 
 
 
Since        . Keeping therms of    on both sides, it follows that
 
 
 
 
 
 
 
This equation when forced by the velocity vector that corresponds to the plane progressive wave solution derived above, leads to a harmonic solution for the particle displaced trajectories                  which are circular [left as exercise].
 
 
 
If second-order effects are included, the particles under a plane progressive waves also undergo a steady-state drift known as the stokes drift. It can be easily modeled based on the approach described above by substituting second-order effects consistently into the right-hand side of the equation of motion (see M H).
 
 
 
Dispersion relation in deep & shallow waters
 
 
 
In finite depth:
 
 
 
 
 
 
 
 
 
This is a nonlinear algebraic equation for w = kh which has a unique solution as can be shown graphically
 
 
 
 
 
 
 
tanhw: Monotonically in creasing with an asymptotic value of 1.
 
 
 
      Hyperbola monotonically decreasing
 
 
 
Unique real root      can only be found numerically. Yet it always exists and the iterative methods that may be implemented always converge rapidly.
 
 
 
Give
 
 
 
 
 
So given the wave frequency
 
 
 
 
 
 
 
Indeed water j H
 
 
 
 
 
 
 
Deep water dispersion relation
 
 
 
Phase speed:
 
 
 
 
 
 
 
So the speed of the crest of a wave with period T=10 secs is approximately    or about 30 knots!
 
 
 
Often we need a quick estimate of the length of a deed water wave the period of which we can measure accurately with a stop watch. We proceed as follows:
 
 
 
                  By definition the phase speed is the ration of the wave length over the period, or the time it takes for a crest to travel that distance.
 
  
 +
The corresponding flow velocity at some point <math> \mathbf{x} = (x,z) </math> in the fluid domain or on <math> z=0, z=-h \,</math> is simply given by
 +
<center><math> \mathbf{v} = \nabla \phi </math></center>
 +
This equation leads to a harmonic solution for the particle trajectories which are ellipses (becoming circles as the depth becomes [[Infinite Depth|infinite]].
 +
If second-order effects are included, the particles under a plane progressive waves also undergo a steady-state drift known as the stokes drift.
  
 +
The linear hydrodynamic pressure due to the plane progressive wave, which must be added to the hydrostatic pressure, is
 +
<center><math> P = \rho \frac{\partial \phi}{\partial t} = \mathrm{Re} \left\{ \rho g A \frac{\cosh k (z+h)}{\cosh k h }
 +
e^{\mathrm{i} kx - \mathrm{i} \omega t} \right\} </math></center>
  
So the wave length of a deep water wave in m is approximately the square of its period is seconds plus half that amount. So a wave with period T=10 secs is about 150 m long or about 492 ft.
+
== Plane progressive waves in general directions ==
  
In the limit of shallow water; KH
+
[[Image:Plane_wave.jpg|thumb|right|400px|Plane wave]]
  
 +
For a linear plane progressive regular waves in a general direction the free surface elevation is given by
 +
<center><math> \zeta = \bar{A} \cos \left( \omega t - k x \cos \beta -k y \sin \beta +\chi \right) =
 +
\mathrm{Re} \left \{ A e^{- i k x \cos \theta - i k y \sin \theta + i \omega t} \right \}
 +
</math></center>
 +
where <math>A</math> is the complex amplitude which includes the phase factor <math>\chi</math>.
 +
<math>\theta</math> is the angle the wave makes to the <math>x</math> axis.
 +
and the velocity potential in finite depth is given by
 +
<center><math> \Phi = \mathrm{Re} \left \{ \frac{\mathrm{i}gA}{\omega} \frac{\cosh k (z+h)}{\cosh k h} \
 +
e^{ -\mathrm{i} k x \cos \beta - \mathrm{i} k y \sin \beta + \mathrm{i} \omega t} \right \} </math></center>
 +
where
 +
<center><math> \omega^2 = g k \tanh k h \,</math></center>
 +
from the [[Dispersion Relation for a Free Surface]] as before
  
 +
== [[Dispersion Relation for a Free Surface]] in [[Infinite Depth| Deep]] and [[:Category:Shallow Depth| Shallow Waters]] ==
  
Thus, according to linear theory shallow water waves become non dispersive as is the case with acoustic waves. Unfortunately, nonlinear effects become important in the limit of shallow water and must be carefully be taken into account. Solitons and wave breaking are some manifestations of nonlinearity. (See MEI).
+
In [[Finite Depth]]
 +
<center><math> \omega^2 = gk\tanh kh \,</math></center>
 +
which is a nonlinear algebraic equation for <math> \omega\,</math> as a function of <math>k\,</math> which has a unique positive real solution as can be shown graphically. It also has imaginary roots which are important in many application (see [[Dispersion Relation for a Free Surface]] and [[:Category:Eigenfunction Matching Method| Eigenfunction Matching Method]])
 +
The unique positive real root <math> k \,</math> can only be found numerically. Yet it always exists and the iterative methods that may be implemented always converge rapidly.
 +
In [[Infinite Depth|deep water]], <math> h \to \infty </math>
 +
and therefore
 +
<center><math> \tanh kh \to 1 </math></center>
 +
which in turn implies the deep water dispersion relation
 +
<center><math> \omega^2 = g k \,.</math></center>
 +
The phase speed is given by
 +
<center><math> c = \frac{\omega}{k} = \frac{\omega}{\omega^2/g} = \frac{g}{\omega}  = \frac{g}{2\pi/T} = \frac{gT}{2\pi} </math></center>
 +
So the speed of the crest of a wave with period <math> \,T = 10 \mathrm{secs} </math> is approximately <math> 15.6 \,\mathrm{m/s} </math> or about 30 knots!
  
The transition from deep to finite depth wave effects occurs for values of KH            . Check that:
+
Often we need a quick estimate of the wavelength of a water wave the period of which we can measure accurately with a stop watch. We proceed as follows:
  
 +
<center><math> c = \frac{\omega}{k} = \frac{\lambda}{T} \ \Longrightarrow \ \frac{\lambda}{T} = \frac{g}{\omega} = \frac{g}{2\pi/T} \ \Longrightarrow \ \lambda = \frac{gT^2}{2\pi} \simeq T^2 + \frac{1}{2} T^2 </math></center>
 +
(by definition the phase speed is the ration of the wave length over the period, or the time it takes for a crest to travel that distance). So the wave length of a deep water wave in m is approximately the square of its period is seconds plus half that amount. So a wave with period <math>T=10</math> secs is about <math>150 m</math> long.
  
 +
In the limit of [[:Category:Shallow Depth|Shallow Depth]] <math> kh \to 0</math>
 +
which in turn implies that
 +
<center><math> \tanh kh \simeq kh </math></center>
 +
It therefore follows that
 +
<center><math> \omega^2 = gk (kh) \ \Longrightarrow \ \frac{\omega^2}{k^2} = gh </math></center>
 +
or
 +
<center><math> \frac{\omega}{k} = c = \sqrt{gh} </math></center>
  
And for
+
Thus, according to linear theory shallow water waves become non dispersive as is the case with acoustic waves. Unfortunately, nonlinear effects become more important as waves propagate from deep to shallow water (because the wave amplidute rises). Solitons and wave breaking are some manifestations of nonlinearity.
  
 +
The transition from deep to finite depth wave effects occurs for values of <math> kh \le \pi </math>. This is because
 +
<center><math> \tanh \pi \simeq 1 </math></center>
 +
and for <math> kh = \pi \ \Longrightarrow \ \frac{2\pi h}{\lambda} = \pi \ \Longrightarrow \ \frac{h}{\lambda} = \frac{1}{2} </math>
 +
so for <math> \frac{h}{\lambda} > \frac{1}{2} </math> or <math> kh > \pi </math> we are effectively dealing with [[Infinite Depth]]. This means that for most of the world ocean and wave conditions the water depth may be approximate as infinite.
  
 +
-----
 +
This article is based on the MIT open course notes and the original article can be found
 +
[http://ocw.mit.edu/NR/rdonlyres/Mechanical-Engineering/2-24Spring-2002/C88824A3-CBFC-4857-A3DD-D463461C8B97/0/lecture3.pdf here]
  
So for                    we are effectively dealing with deep water.
+
[[Category:Linear Water-Wave Theory]]

Latest revision as of 10:28, 2 May 2010



Introduction

Regular time-harmonic linear plane progressive waves are the fundamental building block in describing the propagation of surface wave disturbances in a deterministic and stochastic setting and in predicting the response of floating structures in a seastate.

Equations

We derive the solution for regular time-harmonic linear plane progressive waves for the first order potential in water of constant Finite Depth. The equations of motion in the time domain are Laplace's equation through out the fluid

[math]\displaystyle{ \Delta\Phi\left( \mathbf{x,}t\right) =0,\ \ \mathbf{x}\in\Omega. }[/math]

At the bottom surface we have no flow

[math]\displaystyle{ \partial_{n}\Phi=0,\ \ z=-h. }[/math]

At the free surface we have the kinematic condition

[math]\displaystyle{ \partial_{t}\zeta=\partial_{n}\Phi,\ \ z=0,\ x\in \partial\Omega_{F}, }[/math]

and the dynamic condition (the linearized Bernoulli equation)

[math]\displaystyle{ \partial_{t}\Phi = -g\zeta ,\ \ z=0,\ x\in \partial\Omega_{F}. }[/math]

Note that we are assuming here no body is present so that [math]\displaystyle{ F }[/math] is the entire domain. It turns out that if the water is sufficiently deep the wave does not feel any effect from the sea floor and we can make the Infinite Depth assumption.


Expression in terms of velocity potential

We also assume that Frequency Domain Problem with frequency [math]\displaystyle{ \omega }[/math] and we assume that all variables are proportional to [math]\displaystyle{ \exp(-\mathrm{i}\omega t)\, }[/math]


The water motion is represented by a velocity potential which is denoted by [math]\displaystyle{ \phi\, }[/math] so that

[math]\displaystyle{ \Phi(\mathbf{x},t) = \mathrm{Re} \left\{\phi(\mathbf{x},\omega)e^{-\mathrm{i} \omega t}\right\}. }[/math]

The equations therefore become

[math]\displaystyle{ \begin{align} \Delta\phi &=0, &-h\lt z\lt 0,\,\,\mathbf{x} \in \Omega \\ \partial_z\phi &= 0, &z=-h, \\ \partial_z \phi &= \alpha \phi, &z=0,\,\,\mathbf{x} \in \partial \Omega_{\mathrm{F}}, \end{align} }[/math]


(note that the last expression can be obtained from combining the expressions:

[math]\displaystyle{ \begin{align} \partial_z \phi &= -\mathrm{i} \omega \zeta, &z=0,\,\,\mathbf{x} \in \partial \Omega_{\mathrm{F}}, \\ \mathrm{i} \omega \phi &= g\zeta, &z=0,\,\,\mathbf{x} \in \partial \Omega_{\mathrm{F}}, \end{align} }[/math]

where [math]\displaystyle{ \alpha = \omega^2/g \, }[/math])

The separation of variables equation for deriving free surface eigenfunctions is as follows:

[math]\displaystyle{ Z^{\prime\prime} + k^2 Z =0. }[/math]

subject to the boundary conditions

[math]\displaystyle{ Z^{\prime}(-h) = 0 }[/math]

and

[math]\displaystyle{ Z^{\prime}(0) = \alpha Z(0) }[/math]

We can then use the boundary condition at [math]\displaystyle{ z=-h \, }[/math] to write

[math]\displaystyle{ Z = \frac{\cos k(z+h)}{\cos kh} }[/math]

where we have chosen the value of the coefficent so we have unit value at [math]\displaystyle{ z=0 }[/math]. The boundary condition at the free surface ([math]\displaystyle{ z=0 \, }[/math]) gives rise to:

[math]\displaystyle{ k\tan\left( kh\right) =-\alpha \, }[/math]

which is the Dispersion Relation for a Free Surface

We denote the positive imaginary solution of this equation by [math]\displaystyle{ k_{0} = \mathrm{i} k \, }[/math]

The resulting plane progressive wave velocity potential takes the form:

[math]\displaystyle{ \phi(x,z,t) = A \ \mathrm{Re} \left\{ \frac{ig}{\omega} \frac{\cosh k (z+h)}{\cosh k h} e^{ \mathrm{i} k x - \mathrm{i} \omega t} \right\} }[/math]

where [math]\displaystyle{ A }[/math] is the amplitude in displacement.

This derivation is discussed in more mathematical depth in Eigenfunction Matching Method

Propagating Wave

Propagating Wave

The free surface elevation is

[math]\displaystyle{ \zeta (x,t) = A \mathrm{Re} \{ e^{\mathrm{i} kx - \mathrm{i} \omega t} \} \, }[/math]

where A is an apriori known wave amplitude and the wave frequency and wave number [math]\displaystyle{ (\omega, k) \, }[/math] pair have the same definitions as in all wave propagation problems, namely:

[math]\displaystyle{ T = \frac{2\pi}{\omega} = \mbox{Wave Period} }[/math]
[math]\displaystyle{ \lambda = \frac{2\pi}{k} = \mbox{Wave Length} }[/math]
[math]\displaystyle{ c= \frac{\omega}{k} = \mbox{Wave Phase Velocity} }[/math]

The relation between [math]\displaystyle{ \omega \, }[/math] and [math]\displaystyle{ k \, }[/math] is known as the dispersion relation often written in the form

[math]\displaystyle{ \omega = f (k) \,. }[/math]

[math]\displaystyle{ f(k) \, }[/math] depends on the physics of the wave propagation problem under study, surface ocean waves are dispersive since [math]\displaystyle{ f(k)\, }[/math] is a non linear function of [math]\displaystyle{ k \, }[/math] as we will shortly show.

Flow Velocity and Pressure

The corresponding flow velocity at some point [math]\displaystyle{ \mathbf{x} = (x,z) }[/math] in the fluid domain or on [math]\displaystyle{ z=0, z=-h \, }[/math] is simply given by

[math]\displaystyle{ \mathbf{v} = \nabla \phi }[/math]

This equation leads to a harmonic solution for the particle trajectories which are ellipses (becoming circles as the depth becomes infinite. If second-order effects are included, the particles under a plane progressive waves also undergo a steady-state drift known as the stokes drift.

The linear hydrodynamic pressure due to the plane progressive wave, which must be added to the hydrostatic pressure, is

[math]\displaystyle{ P = \rho \frac{\partial \phi}{\partial t} = \mathrm{Re} \left\{ \rho g A \frac{\cosh k (z+h)}{\cosh k h } e^{\mathrm{i} kx - \mathrm{i} \omega t} \right\} }[/math]

Plane progressive waves in general directions

Plane wave

For a linear plane progressive regular waves in a general direction the free surface elevation is given by

[math]\displaystyle{ \zeta = \bar{A} \cos \left( \omega t - k x \cos \beta -k y \sin \beta +\chi \right) = \mathrm{Re} \left \{ A e^{- i k x \cos \theta - i k y \sin \theta + i \omega t} \right \} }[/math]

where [math]\displaystyle{ A }[/math] is the complex amplitude which includes the phase factor [math]\displaystyle{ \chi }[/math]. [math]\displaystyle{ \theta }[/math] is the angle the wave makes to the [math]\displaystyle{ x }[/math] axis. and the velocity potential in finite depth is given by

[math]\displaystyle{ \Phi = \mathrm{Re} \left \{ \frac{\mathrm{i}gA}{\omega} \frac{\cosh k (z+h)}{\cosh k h} \ e^{ -\mathrm{i} k x \cos \beta - \mathrm{i} k y \sin \beta + \mathrm{i} \omega t} \right \} }[/math]

where

[math]\displaystyle{ \omega^2 = g k \tanh k h \, }[/math]

from the Dispersion Relation for a Free Surface as before

Dispersion Relation for a Free Surface in Deep and Shallow Waters

In Finite Depth

[math]\displaystyle{ \omega^2 = gk\tanh kh \, }[/math]

which is a nonlinear algebraic equation for [math]\displaystyle{ \omega\, }[/math] as a function of [math]\displaystyle{ k\, }[/math] which has a unique positive real solution as can be shown graphically. It also has imaginary roots which are important in many application (see Dispersion Relation for a Free Surface and Eigenfunction Matching Method) The unique positive real root [math]\displaystyle{ k \, }[/math] can only be found numerically. Yet it always exists and the iterative methods that may be implemented always converge rapidly. In deep water, [math]\displaystyle{ h \to \infty }[/math] and therefore

[math]\displaystyle{ \tanh kh \to 1 }[/math]

which in turn implies the deep water dispersion relation

[math]\displaystyle{ \omega^2 = g k \,. }[/math]

The phase speed is given by

[math]\displaystyle{ c = \frac{\omega}{k} = \frac{\omega}{\omega^2/g} = \frac{g}{\omega} = \frac{g}{2\pi/T} = \frac{gT}{2\pi} }[/math]

So the speed of the crest of a wave with period [math]\displaystyle{ \,T = 10 \mathrm{secs} }[/math] is approximately [math]\displaystyle{ 15.6 \,\mathrm{m/s} }[/math] or about 30 knots!

Often we need a quick estimate of the wavelength of a water wave the period of which we can measure accurately with a stop watch. We proceed as follows:

[math]\displaystyle{ c = \frac{\omega}{k} = \frac{\lambda}{T} \ \Longrightarrow \ \frac{\lambda}{T} = \frac{g}{\omega} = \frac{g}{2\pi/T} \ \Longrightarrow \ \lambda = \frac{gT^2}{2\pi} \simeq T^2 + \frac{1}{2} T^2 }[/math]

(by definition the phase speed is the ration of the wave length over the period, or the time it takes for a crest to travel that distance). So the wave length of a deep water wave in m is approximately the square of its period is seconds plus half that amount. So a wave with period [math]\displaystyle{ T=10 }[/math] secs is about [math]\displaystyle{ 150 m }[/math] long.

In the limit of Shallow Depth [math]\displaystyle{ kh \to 0 }[/math] which in turn implies that

[math]\displaystyle{ \tanh kh \simeq kh }[/math]

It therefore follows that

[math]\displaystyle{ \omega^2 = gk (kh) \ \Longrightarrow \ \frac{\omega^2}{k^2} = gh }[/math]

or

[math]\displaystyle{ \frac{\omega}{k} = c = \sqrt{gh} }[/math]

Thus, according to linear theory shallow water waves become non dispersive as is the case with acoustic waves. Unfortunately, nonlinear effects become more important as waves propagate from deep to shallow water (because the wave amplidute rises). Solitons and wave breaking are some manifestations of nonlinearity.

The transition from deep to finite depth wave effects occurs for values of [math]\displaystyle{ kh \le \pi }[/math]. This is because

[math]\displaystyle{ \tanh \pi \simeq 1 }[/math]

and for [math]\displaystyle{ kh = \pi \ \Longrightarrow \ \frac{2\pi h}{\lambda} = \pi \ \Longrightarrow \ \frac{h}{\lambda} = \frac{1}{2} }[/math] so for [math]\displaystyle{ \frac{h}{\lambda} \gt \frac{1}{2} }[/math] or [math]\displaystyle{ kh \gt \pi }[/math] we are effectively dealing with Infinite Depth. This means that for most of the world ocean and wave conditions the water depth may be approximate as infinite.


This article is based on the MIT open course notes and the original article can be found here