1.2 Intro to Wave Theories

8/18/2019 1.2 Intro to Wave Theories

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Lessen Outcomes

To describe wave theories, their assumptions,

applications and limitations.


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Wave Properties


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Wave Profiles


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Introduction

• Numerous water wave theories applicable to different

environments dependent upon the specific environmental

parameters, e.g., water depth, wave height and wave period.

• All ocean wave theories assume that the waves are periodic

uniform, having a period T and height H.

• In developing a wave theory, a boundary value problem (BVP)

is developed from:

1. A number of differential equations that may be defined by continuity

equation, Bernoulli equation, stream function, potential function, etc.2. Certain boundary conditions.

• Complete boundary value problems difficult to be resolved

even in the simple case of uniform water depth.


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Waves at sea are very complex due to irregularity of wave shape.

Several theories exist to describe wave behaviour:

Wave Theories

Wave Theory Reference Water Condition

Linear wave theory Airy (1845)

Deep water

(d/Lo > 0.5)Stoke wave theory Stokes (1847)

Fenton (1985)

Cnoidal wave theory Korteweg & De Vries (1895)

Keulegan & Patterson (1940)

Svendsen (1974)Fenton (1979)

Transitional water

(0.16 < d/Lo > 0.5)

Solitary wave theory Boussinesq, 1872

McCowan (1981)

Grimshaw (1971)

Fenton (1972)

Shallow water

(d/Lo < 0.1)


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The simplest wave theory is the first-order, small-amplitude, or Airy

wave theory which will hereafter be called linear wave theory . The basis for the wave theory is the sinusoidal wave, and it

constitutes the 1st order of approximation of the Stokes’ theory.

Most commonly used wave theory due to less mathematically

complex.

Both crest and trough amplitudes must be equal. Most accurate for low amplitude waves in deep water (H


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When waves become large or travel toward shore into shallow

water, higher-order or non-linear wave theories are often

required to describe wave phenomena.

Non-Linear Wave Theory

• Stoke wave theory (Stokes, 1847)• Solitary wave theory (Boussinesq, 1872)

• Cnoidal wave theory (Korteweg & De Vries, 1895)

For the 1

st

order, still water level (SWL) is defined as the levelmidway between wave crest and trough. But for higher order

wave theories a rise of SWL from the origin may be expected.


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Stokes Wave Theory

• Mathematically complex as it takes into account the effects of wave

height & velocity.

• Higher order Stokes approximations can better describe the finite

amplitude waves, the kinematics and pressure prediction.

• The 5th-order Stokes finite-amplitude wave theory is widely used in

practical application both in deep- and shallow-water wave studies.


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Stokes’ 2nd Order Wave Profiles


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Stokes’ 3rd Order Wave Profiles


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Stokes’ 5th Order Wave Profiles


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Cnoidal Wave Theory

• Longer troughs and higher crests.

• Distortion of the wave shape is due to interference from the bottom.

• Applicable for shallow water.


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• An isolated crest moving in very shallow water which about to break.

• Applicable for modeling of tsunami waves.

• A solitary wave is neither oscillatory nor does it exhibit a trough.

• The solitary waveform lies entirely above the SWL.

Solitary Wave Theory


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1.2 Intro to Wave Theories