10. Numerical Modelling
335
Limited up to quadratic terms with respect to 7/ and the velocity potential expanded at the given élévation,
the following équations can be
derived. It is here assumed that cubic and higher order terms are negligible
and hence, posing weak nonlinearity.
drl , d
1 393u6
â + &I(d+”>U|>l = 6^a?dt
dx
9 dx
2 dtd2x
(10.15)
In the above équations, if the right-hand side terms are set to be zéro,
then, the resulting équations are called as “shallow water équations”.
Combining the above two équations with a linear approximation of ub
(horizontal flow velocity at z = -d) results the following:
d2^
d2^
d2
dT2~ ~d^ ~ dÇ2
= 0
(10.16)
The linear frequency dispersion characteristics of Boussinesq équations
without linear approximation of bottom particle velocity is,
C'2=gd
1 4. fc2d2
1
6
1 _i_ fe2d2
1 1"
2
(10.17)
where, C is the phase speed and k is the wave number.
For an approximated Boussinesq équation [Eq. (10.18)], the linear
frequency dispersion équation can be derived as,
2
/
k2d2
C2 = (7gH 1- —
(10.18)
Considering a relative error of 4% in the phase speed compared with linear wave theory estimate, Eq. (10.17) is valid for kd < zr/2 and Eq. (10.18)
is valid for kd < 2tt/7 for engineering applications. That is the former
équation is valid for wavelengths larger than four times the water depth
and the approximated form is valid for wavelengths larger than seven times
the water depth. That is, the latter is valid for very long waves. Depending on the applications, the simplified form of Boussinesq équation can
be adopted. However, the shallow water équations provide an estimate of
wave length with less than 4% accuracy is for the condition of wavelengths
larger than 13 times the water depth. The above clearly States the régime
of applications of various forms of Boussinesq approximation.
Any form of modification to Boussinesq équations, either in the form
of domain, i.e., varying bathymetry or higher order accuracy terms in the
sériés expansion or the incorporation of additional physics such as wave
335
Limited up to quadratic terms with respect to 7/ and the velocity potential expanded at the given élévation,
the following équations can be
derived. It is here assumed that cubic and higher order terms are negligible
and hence, posing weak nonlinearity.
drl , d
1 393u6
â + &I(d+”>U|>l = 6^a?dt
dx
9 dx
2 dtd2x
(10.15)
In the above équations, if the right-hand side terms are set to be zéro,
then, the resulting équations are called as “shallow water équations”.
Combining the above two équations with a linear approximation of ub
(horizontal flow velocity at z = -d) results the following:
d2^
d2^
d2
dT2~ ~d^ ~ dÇ2
= 0
(10.16)
The linear frequency dispersion characteristics of Boussinesq équations
without linear approximation of bottom particle velocity is,
C'2=gd
1 4. fc2d2
1
6
1 _i_ fe2d2
1 1"
2
(10.17)
where, C is the phase speed and k is the wave number.
For an approximated Boussinesq équation [Eq. (10.18)], the linear
frequency dispersion équation can be derived as,
2
/
k2d2
C2 = (7gH 1- —
(10.18)
Considering a relative error of 4% in the phase speed compared with linear wave theory estimate, Eq. (10.17) is valid for kd < zr/2 and Eq. (10.18)
is valid for kd < 2tt/7 for engineering applications. That is the former
équation is valid for wavelengths larger than four times the water depth
and the approximated form is valid for wavelengths larger than seven times
the water depth. That is, the latter is valid for very long waves. Depending on the applications, the simplified form of Boussinesq équation can
be adopted. However, the shallow water équations provide an estimate of
wave length with less than 4% accuracy is for the condition of wavelengths
larger than 13 times the water depth. The above clearly States the régime
of applications of various forms of Boussinesq approximation.
Any form of modification to Boussinesq équations, either in the form
of domain, i.e., varying bathymetry or higher order accuracy terms in the
sériés expansion or the incorporation of additional physics such as wave