4. Oblique propagation
The dispersion equations for are Eqs. (A19) and (A20), respectively, in terms of the Hill determinant. If we truncate the Hill determinant in Eq. (A19), retaining three rows centered around the row that contains , an approximate dispersion equation for the first three roots is obtained. For the motionless structured atmosphere, where , this approximate dispersion equation is
This solution is valid only if . If , then Eq. (43) should be used. When tends to zero, the approximate solution (44) reduces to the case of vertical propagation, Eq. (28). For horizontal propagation, in the low-frequency limit, relation (44) reduces to
In this limit the phase velocity turns out to be less than the sound speed. We recall that in the low-frequency limit the phase velocity of vertically propagating acoustic waves in the thermally structured atmosphere did not differ from the sound speed. For the moving atmosphere () the phase velocity of horizontally propagating waves () does not differ from the sound speed; this can be seen directly from Eq. (15), if one replaces by . At the same time the phase velocity of vertically propagating acoustic waves in a moving isothermal atmosphere in the low-frequency limit appears to be larger than the sound speed, see (32).
which follows from (43). The first mode is a waveguide mode with a low-frequency cut-off. The second mode is a classical wave mode in a 1D lattice (Ingard 1988); this mode has a high-frequency cut-off.
© European Southern Observatory (ESO) 1998
Online publication: March 10, 1998