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Astron. Astrophys. 332, 314-324 (1998)

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4. Oblique propagation

In the case of inclined wave propagation there are solutions of Eq. (22) with period [FORMULA] ([FORMULA]) and [FORMULA] ([FORMULA])

[EQUATION]

The dispersion equations for [FORMULA] 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 [FORMULA], an approximate dispersion equation for the first three roots is obtained. For the motionless structured atmosphere, where [FORMULA], this approximate dispersion equation is

[EQUATION]

where the squared full wavenumber is [FORMULA]. For [FORMULA] (43) has a root that corresponds to [FORMULA], up to a term of order [FORMULA]. In this case the distortion due to the atmospheric structure yields

[EQUATION]

This solution is valid only if [FORMULA]. If [FORMULA], then Eq. (43) should be used. When [FORMULA] 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

[EQUATION]

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 ([FORMULA]) the phase velocity of horizontally propagating waves ([FORMULA]) does not differ from the sound speed; this can be seen directly from Eq. (15), if one replaces [FORMULA] by [FORMULA]. At the same time the phase velocity of vertically propagating acoustic waves in a moving isothermal atmosphere in the low-frequency limit [FORMULA] appears to be larger than the sound speed, see (32).

In the case of oblique propagation each vibrational wave mode splits into two modes. For [FORMULA] and [FORMULA] their frequencies are given by

[EQUATION]

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.

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© European Southern Observatory (ESO) 1998

Online publication: March 10, 1998
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