Photoacoustic effect in a stratified atmosphere

The photoacoustic effect is usually studied in an isotropic medium on a laboratory scale. However, it is possible to use optical sources to launch pressure perturbations in the atmosphere consisting of both acoustic and gravity waves. Here photoacoustic theory is extended to incorporate the effects of a stratified atmosphere and a gravitational field on launching and propagation of pressure waves in the atmosphere. Properties of pressure waves corresponding to several optical excitation schemes are investigated. The acoustic component of the optically launched pressure waves is explored separately to delineate its properties from those found without the effects of a gravitational field.

Figure 1

Geometry of the theoretical model with the gravitational force antiparallel to the $z$ axis and the earth taken to be a flat perfect reflector.

Figure 2

(a) Phase and (b) group velocities as a function of ${\omega }_x=\omega sin\theta$ and ${\omega }_z=\omega cos\theta$. The phase velocity varies from zero to infinity, while its color plot is truncated at $v_p/c=3$ to make clear the details of the distribution. Within the blank areas at the left bottom corners, the velocities become imaginary, indicating evanescent wave behavior.

Figure 3

Pressure versus time for a Gaussian point source associated with the two source terms in Eq. (4). The source is positioned at $(0,0,100\mathrm{m})$ with $t_s=100$ s, $\sigma =1$ s, and $E_0=1$ kJ. The observation point is located 100 m below the source. Other physical parameters are taken as $g=9.8$ m/${\mathrm{s}}^2, \gamma =1.4, \beta =0.0034{\mathrm{K}}^{-1}, c=340$ m/s, and $C_p=1005$ J/kg/K. Note that (a) the first source term gives rise to an acoustically dominant wave with the waveform resembling the three-dimensional photoacoustic wave for a pulsed Gaussian point source [10], while (b) the second source term results in a long-tailed, low-frequency wave that is gravity wave dominant.

Figure 4

(a) Pressure distribution on the $yz$ plane with $x=20$ km and (b) the radiation pattern for a low-frequency monochromatic monopole radiator placed at the origin. The plot in (a) corresponds to the pressure distribution from a point source with its intensity oscillating at $f_0=2.6$ mHz. The other parameters are the same as in Fig. 3 except that $P_0=1$ MW and $t=100$ s. The plot in (b) shows that for point sources with ${\omega }_0<{\omega }_1$ the excited pressure wave will be evanescent within the cones.

Figure 5

Distribution of the photoacoustic pressure (a) on the $xy$ plane at $z=0$ and (b) along the $z$ axis for a modulated cw laser beam pointing upwardly. The laser intensity is modulated at $f_0=10$ Hz with $P_0=1$ kW and $t=1$ s. The other parameters are the same as in Fig. 3.

Figure 6

(a) Distribution of the photoacoustic pressure on the $xy$ plane and (b) pressure amplitude along the $z$ axis at $t=10$ s for a point source oscillating at $f_0=1$ Hz. The point source is placed at $(0,0,50\mathrm{km})$ with $P_0=1$ kW and the observation plane is 0.2 ct lower than the source. The other parameters are the same as given in Fig. 3. The plot in (b) shows that there exist both a local maximum and minimum for pressure in the case of $d=0.5h$, which evolve into a single saddle point for $d=h$.