LINEAR WAVE THEORY
Linear wave theory can be defined as first-order, small amplitude gravity wave with a sinusoidal shape. This theory has been developed by Airy in 1845. It is easy to apply and give reasonable approximation of wave characteristic for a wide range of wave parameters. However, in some situations, waves are better described by higher order theories or also referred as finite-amplitude wave theories. Although there are limitations to its application, linear theory is still useful provided assumptions made in developing this theory are not violated.
To develop linear wave theory, several assumptions need to be made. First assumption need to be made is the fluid is homogenous and incompressible, hence the density is constant. …show more content…
The relative water depths help to determine whether waves are disperse or non-disperse and whether the wave celerity, length and height are influenced by water depth. Wave steepness is a measure of how large a wave is relative to its height and whether the linear wave assumption is valid. Large values of the wave steepness suggest that the small amplitude assumption may be questionable. A third dimensionless parameter is used to replace either the wave steepness or relative water depth. This new parameter is defined as the relative wave …show more content…
Wave perturbation theory have assumed the wave slope ka is small where k is the wave number and a the amplitude of the wave. The perturbation solution, developed as a power series in terms of ε =ka, is expected to converge as more and more terms are considered in the expansion. Convergence does not occur for steep waves unless a different perturbation parameter from that of Stokes is chosen. The perturbation expansion for velocity potential Φ can be written as below;
Φ= ε Φ_1+ ε^2 Φ_2+⋯⋯
in which ε=ka is the perturbation expansion parameter. In this expansion, Φ_1 is the first order theory (linear theory), Φ_2 is the second order theory and so on.
The Stokes expansion method is formally valid under the conditions that H⁄d≪ (kd)^2 for kd<c, the wave would become unstable and break. Stokes found that a wave having a crest angle less than 〖120〗^° would break. Michell (1893) found that in deep water the theoretical limit for wave steepness is
(H_0/λ_0 )_max=0.142
Miche (1944) gives the limiting steepness for waves travelling in depths less than λ_0⁄2 without a change in form as;
(H/λ)_max= 0.142
Linear wave theory can be defined as first-order, small amplitude gravity wave with a sinusoidal shape. This theory has been developed by Airy in 1845. It is easy to apply and give reasonable approximation of wave characteristic for a wide range of wave parameters. However, in some situations, waves are better described by higher order theories or also referred as finite-amplitude wave theories. Although there are limitations to its application, linear theory is still useful provided assumptions made in developing this theory are not violated.
To develop linear wave theory, several assumptions need to be made. First assumption need to be made is the fluid is homogenous and incompressible, hence the density is constant. …show more content…
The relative water depths help to determine whether waves are disperse or non-disperse and whether the wave celerity, length and height are influenced by water depth. Wave steepness is a measure of how large a wave is relative to its height and whether the linear wave assumption is valid. Large values of the wave steepness suggest that the small amplitude assumption may be questionable. A third dimensionless parameter is used to replace either the wave steepness or relative water depth. This new parameter is defined as the relative wave …show more content…
Wave perturbation theory have assumed the wave slope ka is small where k is the wave number and a the amplitude of the wave. The perturbation solution, developed as a power series in terms of ε =ka, is expected to converge as more and more terms are considered in the expansion. Convergence does not occur for steep waves unless a different perturbation parameter from that of Stokes is chosen. The perturbation expansion for velocity potential Φ can be written as below;
Φ= ε Φ_1+ ε^2 Φ_2+⋯⋯
in which ε=ka is the perturbation expansion parameter. In this expansion, Φ_1 is the first order theory (linear theory), Φ_2 is the second order theory and so on.
The Stokes expansion method is formally valid under the conditions that H⁄d≪ (kd)^2 for kd<c, the wave would become unstable and break. Stokes found that a wave having a crest angle less than 〖120〗^° would break. Michell (1893) found that in deep water the theoretical limit for wave steepness is
(H_0/λ_0 )_max=0.142
Miche (1944) gives the limiting steepness for waves travelling in depths less than λ_0⁄2 without a change in form as;
(H/λ)_max= 0.142