### ASA 130th Meeting - St. Louis, MO - 1995 Nov 27 .. Dec 01

## 3aPAb5. Omega-to-the-one-third term in dispersion relation for acoustic
pulse propagation through turbulence.

**Allan D. Pierce
**

**
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*Boston Univ., Dept. of Aerospace and Mech. Eng., 110 Cummington St.,
Boston, MA 02215
*

*
*
The author's earlier formulation of pulse propagation through turbulence
required a somewhat ad hoc separation of the effects of large scale and small
scale turbulence with the selection of a cut-off turbulent wave number k[inf c]
that separates the two regimes. A neater-cleaner formulation proceeds with the
premise that the frequency dispersion of pulses is caused by that part of the
turbulence spectrum which lies in the inertial range originally predicted by
Kolmogoroff. The acoustic propagating wave's dispersion relation has the
acoustic wave number being of the form k=((omega)/c)+F((omega)), where c is a
spatially averaged sound speed and where, for mechanical turbulence, the extra
term F((omega)) must depend on only the angular frequency (omega), the sound
speed c, and the turbulent energy dissipation (epsilon) per unit fluid mass and
per unit time. If the turbulence is weak, then the quantity F((omega)) has to be
of second order in the portions of the turbulent fluid velocity in the intertial
range, so, following Kolmogoroff's reasoning, it must vary with (epsilon) as
(epsilon)[sup 2/3]. Simple dimensional analysis then reveals that F((omega)) is
K(epsilon)[sup 2/3]c[sup -7/3](omega)[sup 1/3], the latter factor being as
announced in the title of this abstract, and K being a universal dimensionless
complex constant. A similar result holds for thermal turbulence. The analysis
showing that the separating-out of the effects of turbulence in the inertial
regime is in fact possible yields K=-0.37e[sup i(pi)/3]. The dispersion is
typically small, but has an accumulative effect that leads to a sizable pulse
distortion over large propagation distances. [Work supported by NASA Langley
Research Center.]