where
The term in curly brackets multiplying represents a potential distortion of the density scale height term. Note however, that if with this distortion is completely removed and . The tracers and have a decisive effect in eq. (37) since setting either of them to zero reduces the order of the dispersion relation to quadratic implying the removal of two wave species. Consider first the effect of (with implied to avoid wave distortion ) and . The dispersion relation then reduces to:
which is identical to the inertia-gravity wave dispersion equation (36) and implies that setting completely removes the acoustic modes. Setting irrespective of gives the hydrostatic dispersion relation for inertia-gravity waves:
The search for wave modes in ignores the possibility of pure horizontal wave motion though clearly this can only be permitted if . If then eq.(24) gives . Combining the continuity equation (23) with eqs.(25) and (14) gives:
which has geostrophic solutions and, if , rotationally-modified acoustic wave solutions:
This case is known as the Lamb wave and is the only acoustic mode that survives the hydrostatic assumption
(). The high phase speed of acoustic modes ( ) is an inconvenience in numerical modelling since it imposes limitations on the size of time step in explicit time integrations scheme. The use of
semi-implicit time integration schemes in numerical weather prediction models successfully avoids this problem by effectively slowing down the speed of the acoustic waves. However the hydrostatic assumption is too severe for many problems in mesoscale dynamics and so the option n_2=n_3=0 option appears - in this context - to be the most satisfactory approximation to invoke to remove acoustic modes.