# Study of Low-Frequency Hydroacoustic Waves' Behavior at the Shelf of Decreasing Depth
## 5 Analysis of the Results
The pressure of the hydroacoustic field with the frequency \(\omega=2\pi f\), generated by the source at a given point with the coordinates \((x,z)\) can be represented as the sum of the normal waves (modes) [13]:  
\[P(x,z,t)=\sum_{l}\left(\frac{2P_{0}R}{H}\left(\frac{2\pi}{\xi_{l}x}\right)^{1/ 2}\sin(\alpha_{l}z)\sin(\alpha_{l}z_{1})e^{i(\xi_{l}x+\frac{\pi}{l})}\right)e^ {-i\omega t}, \tag{6}\]  
where \(H\) is the depth of the sea, \(P\) is the pressure amplitude at the distance R from the emitter, and \(\xi_{l}\) and \(\alpha_{l}\) are horizontal and vertical wave numbers of the mode l.  
In this case, the group wave velocity can be written as \(U_{l}(f)=c\left(1-\frac{f_{l}^{2}}{f^{2}}\right)^{1/2}\), where \(f_{l}\) is the critical frequency of the mode number \(l\). At frequencies less than critical, the hydroacoustic wave becomes inhomogeneous and the energy completely transfers from hydroacoustic to seismo- acoustic. In case of the shallow sea, when the propagation condition is satisfied only for one mode (\(l=0\)), the critical frequency can be found by the formula \(f_{0}=\frac{\zeta}{4H\sqrt{1-n^{2}}}\), where \(n\) is the refractive index equal to the ratio of the speed of sound in water to the speed of sound in the sea bottom. Based on the formula for critical frequency, we can find the critical depth \(H_{cr}\) at which, at a given frequency of the hydroacoustic wave, all the energy will go to the bottom. Figure 8 shows graphs of dependence for the seismoacoustic and hydroacoustic wave energy densities on the depth of the points where the measurements were taken. The critical depth of the wave with a frequency of 22 Hz was 17.8 m. As we can see from the graphs, a significant part of the hydroacoustic wave energy density goes into the seismoacoustic wave energy density down to a depth of 36 m, which is approximately half the length of a wave with a frequency of 22 Hz (34 m). After the depth equal to that of the critical (17.8 m), it almost completely transforms into the energy density of the seismoacoustic wave. In the case of the experiment with a hydroacoustic emitter at a frequency of 33 Hz [3] with the same model of the seabed, the critical depth should be 11.8 m. The emitting stations described in [3] were located at significant distances from each other, which did not allow us to experimentally determine the critical depth for the frequency of 33 Hz. Figure 9 shows graphs of the energy density of the seismoacoustic wave recorded by the laser strainmeter and the difference in the energy density of the experiment and the cylindrical divergence in percent. As you can see, the graphs are almost identical, which can tell us that almost all the energy losses are along the path of the hydroacoustic wave, from the first measurement point to the last transfer into the elastic oscillations of the earth crust. Further, let us analyze the calculated data shown in Table 2. We perform the analysis accounting for the fact that the hydroacoustic energy is distributed along the shelf according to the cylindrical law and its absorption at such small distances is negligible. Suppose that the energy has come in full to the first measurement point, i.e., the losses amounted to 0%, both in the case of the experiment and in the case of cylindrical divergence. Then, we can calculate in terms of percent how much energy went into the bottom in the form of body waves and did not reach the laser strainmeter. As we can  
Figure 8: Graphs of the dependence of the seismoacoustic and hydroacoustic wave energy densities on the depth of the points where the measurements were taken. Figure 9: Graphs of the energy density of the seismoacoustic wave recorded by the laser strainmeter, and the difference in the energy density of the experiment and the cylindrical divergence in percent. see from the fifth column, at the distance of 335 m from the emission site, 83.8% of the energy given out by the emitter transferred into hydroacoustic energy. That is, about 16% went into the bottom in the form of body waves.