which may be demonstrated by substituting for and and looking for its maximum value with respect to . It is easy to show that the term has a maximum for
and that this is equal to:
(using the fact that ). For this peaks at and using the same values for and as above, together with , gives a numerical value of around . For disturbances with vertical wavelength smaller than , this value will be very much smaller e.g. if it is .
The square-bracketted expressions of eqs.(33) and (34) which contain this term may, in view of its smallness, be expanded to two terms in a binomial series giving:
and
In the limit of small (short horizontal wavelengths; near vertical phase lines) we have and corresponding to ‘pure’ gravity and acoustic waves respectively.
Now let us return to the vertical structure equation (28) (with tracers) and examine the corresponding dispersion relation. This time the transformation
is required to remove the first derivative term. Following the same procedure as for the exact linear equation set, it can be shown that the dispersion relation is:
and