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J.R. Holton in his An Introduction to Dynamic Meteorology (p.199) gives the linearized version of the governing equations for a two-dimensional internal gravity wave system as:
The solution for w' in this system of equations was found by Holton to be of the form w'=Re[Weiφ] where W is a complex constant and φ=kx+mz-νt. The system of equations is homogeneous so the solutions for u', p' and θ' must be of the same functional form as that of w'; i.e.,
where U, P and Θ are complex constants.
When the forms, Ueiφ, Peiφ and Θeiφ are substituted into the equation system the result is a set of linear complex equations in U, P and Θ; i.e.,
This system of linear homogeneous equation has only the trivial solution unless the determinant of the coefficient matrix is zero. The coefficient matrix is
| 0 | (uk-ν) | k/ρ0 | 0 | |
| i | 0 | (im/ρ0) | -g/θ | |
| m | k | 0 | 0 | |
| (dθ/dz) | 0 | 0 | i(uk-ν) | |
The evaluation of a fourth order determinant seems formidable but there are enough zero elements to make the evaluation quite easy. The value of the determinant D is given by
If the determinant is to be zero it must be that
where K2=(k2+m2).
Let g(d ln(θ)/dz) be denoted as N2. Then the allowable frequencies ν are given by
When the determinant condition is satisfied the solutions for U, P and Θ can be expressed in terms of W.
From the third equation it is found that
In summary
Taking into account the dispersion relation these reduce to:
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