Linear Wave Theory

Linear wave theory is also known as:

• Airy theory;
• Sinusoidal wave theory;
•  Small amplitude wave theory.

Description:

• It has only single sinusoidal component;
• It is the simplest wave theory obtained by taking the wave height, H to me much smaller than both the wave length, λ and water depth, d. 
• Wave crest height, AC = Wave trough height, AT and is denoted the wave amplitude, A, hence H=2A.
• Linear wave theory is the first approximation to the wave motion. In order to achieve theoretical results closer to the reality requires to solve the equations to higher perturbations (non-linear system).

Wave Related Defination

Significant wave height, HS
The average height (trough to crest) of the highest one-third waves in the indicated time period, also denoted H1/3.

Peak Period, TP
It is the inverse of the frequency at which a wave energy spectrum has its maximum value.

Zero-up-crossing period, TZ
It is the average time interval between two successive up-crossing of the mean sea level.

Keulegan-Carpenter number, KC
KC=VT/D
Where,

• V = flow velocity; 
• T = period;
• D = pipeline diameter.

Keulegan-Carpenter number also known as Period Number. It generally describes two things:

1. Relative importance of drag force over inertia force (Small KC number, inertia dominates; large KC number, the drag force is important);
2. Wave ellipse size relative to the pipeline diameter (Small KC number, current dominated; large KC number, the wave is current-like)

Wave

Wave
Wave can be classified into:

1. Regular wave - a periodic wave with a single period.
2. Random wave (spectrum) - a superposition of a number of regular linear waves of differing heights and periods.

Regular Wave
There are two types of regular wave:

• Linear wave theory (Airy) - symmetric (Crest height = Trough height);
• Non-linear wave theories (Stoke's 5th order theory, Dean's stream function theory etc.) - asymmetric (Crest height > Trough height).

Random Wave
Random wave also known as spectrum. A real sea state is best described by a random wave model, since the ocean waves are irregular and random in shape, height, length and speed of propagation. Examples of random wave are JOHNSWAP, ISSC etc.