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Astron. Astrophys. 354, 296-304 (2000)

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4. The energy propagation

For the system of the linearized Eqs. (1) we can write the conservation equation for total energy in the form

[EQUATION]

where the total energy density E (in units of [FORMULA]) is defined as

[EQUATION]

and the energy flux [FORMULA] as

[EQUATION]

In expression (18) for E the terms represent, respectively, kinetic energy, acoustic potential energy and gravitational potential energy. For the slowly varying wavetrain (11) we may consider energy and energy flux averaged over a period, denoted by [FORMULA] and [FORMULA], where the averaged energy flux may be written as

[EQUATION]

(Whitham 1974, Eq. 11.69). This (asymptotically correct) form of the energy flux shows the conservation of total energy in any volume in ordinary space whose boundaries move with the group velocity according to Eqs. (13-14). Eq. (17) can therefore be written as

[EQUATION]

One can also read Eq. (21) noting that a positive divergence of the group lines yields a decay of the energy density. Now, since [FORMULA] one can derive the temporal evolution equation for the amplitude [FORMULA],

[EQUATION]

(see Whitham 1974, Sect. 11.6), where n is the number of dimensions. Eq. (22) yields

[EQUATION]

Fig. 10 shows the temporal decay of the reduced pressure amplitude of the wake after the pulse has reached the positions of [FORMULA] on the vertical axis (top panel) and [FORMULA] on the horizontal axis (lower panel). The top panel compares the decay law of [FORMULA] with the actual behavior of the wake. The agreement is excellent even though the analytic result (23) is valid only for a slowly varying wavetrain. Note the (real) time delay of [FORMULA] in the horizontal direction with respect to the vertical, a consequence of the phase velocity of the wake that is smaller in the horizontal direction (see Fig. 3). Note also that the amplitude of the maxima in the horizontal direction is lower by a factor of 2 to 3. This follows from the properties of the energy propagation, as will be discussed below.

[FIGURE] Fig. 10. Pressure amplitude of the wake (in arbitrary units) vs time at [FORMULA] and [FORMULA] (top panel) and at [FORMULA] and [FORMULA]. The pulse has been removed; the amplitude decay of [FORMULA] is given as a comparison in the top panel.

At [FORMULA] the energy is spatially concentrated in the initial pulse, and as time elapses, the energy is dispersed into a wavetrain and the wavenumber distribution is spread out in ordinary space; both occur with the group velocity, as described by Eqs. (12) and (21). Thus the energy in any volume in wavenumber space remains fixed as the wavenumbers spread in physical space. Consequently

[EQUATION]

where [FORMULA] is the energy per unit volume in ordinary space, estimated at the point [FORMULA] and time t, and [FORMULA] is the energy per unit volume in the corresponding wavenumber space, where [FORMULA] is the wavenumber at position [FORMULA] and time t. Now, volumes transform as

[EQUATION]

where

[EQUATION]

is the Jacobian of the tranformation between wavenumbers and ordinary space; [FORMULA] depends on the initial distribution and the expansion or contraction of volumes in the mapping from wavenumber space to ordinary space, defined by the system (13-14). The energy density can therefore be written as

[EQUATION]

From the above discussion it is clear why the amplitude of the maxima following the pulse is much smaller in the horizontal than in the vertical direction. The behavior of the group velocity and the distribution of [FORMULA]values in space (see Figs. 1 and 2) show that in the horizontal direction a narrow range of k values near [FORMULA] is dispersed over a large portion of physical space, while the dispersion is much lower in the vertical direction; energy is therefore much more diluted in the horizontal than in the vertical direction and, as a consequence, the maxima have much lower amplitude.

This behavior becomes even more evident from the fractional energy contained in a cone of infinitesimal opening angle when its symmetry axis points in different directions. Fig. 11 shows this fractional energy at the time when the front has reached the height of [FORMULA], normalized to the total energy contained in a vertical cone, against distance measured from the front of a wave. Note that in the horizontal direction the energy rises from zero at the front to the asymptotic value in [FORMULA]. This implies that most of the wave energy is contained in a thin layer, of [FORMULA] width, following the pulse. But in the vertical direction about 80% of the energy is contained within a layer with a thickness of [FORMULA] extending in height from [FORMULA] to [FORMULA]. The wake therefore contains a significant fraction of the energy. Approximately 30% of the energy is in the pulse itself, i.e., within a thin layer of [FORMULA] at the head of the wave. Note also the sharp drop of the energy with polar angle in Fig. 11. Evidently the bulk of the energy goes into the vertical direction.

[FIGURE] Fig. 11. The fractional energy contained inside a cone vs distance from the front, for a cone whose axis is inclined at 0o (-), 45o ([FORMULA]), and 90o (- - -) with respect to the vertical.

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© European Southern Observatory (ESO) 2000

Online publication: January 31, 2000
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