Page 17 - Spring 2009

The directional radiation pattern is a dipole pattern and the amplitude depends on (2πzs/λ) the proximity of the source to the pressure release surface. As the source approaches the surface, zs􏰀0 , it collides with its image and the result is zero, the characteristic of a doublet.
The mean-square pressure in the far field is
(9)
The farfield mean-square pressure decreases with the radial distance to the fourth power. This doublet characteristic is a consequence of the monopole beneath a pressure release sur- face. On the other hand the mathematical “point” dipole is derived by placing two monopoles of opposite sign separated by a distance 2Δzs and taking the limit as Δzs􏰀0 and 2posΔzs􏰀D the dipole source strength:
(10)
The subtle but pertinent issue is that a bubble below a pressure release surface has on average a dipole characteristic referred to here as a doublet; however as zs􏰀0 the radiated pressure goes to zero. On the other hand, the point dipole, such as a rain drop impact on the pressure release surface, radiates sound with dipole strength, D, and the following characteristic
(11)
the reactive term, 1/k2r2, becomes negligible at reasonable distance from the source.
The difference between the point dipole and the doublet is fundamental. For near surface sources one should expect
a dipole radiation pattern as shown in Fig. 2a. However as the depth of the source increases the pattern becomes more complex as shown in Figs. 2b and 2c. This effect can be observed with submerged sources such as large surface ship propellers that are generally at depths of less than a half of a wave length at shaft and blade-rate frequencies. Sound radi- ation from these propellers is also influenced by the hull in the forward direction and the wake in the aft direction. This results in a horizontal radiation pattern which is also co- sinusoidal.
In the frequency range between 100 Hz and 1 kHz, the image-interference pattern can be observed at considerable horizontal distances. These transmission characteristics are shown in Fig. 3. where the relative level, RL, is plotted versus range illustrating the near field, interference field and the far field, the Lloyd’s mirror range.
(12)
The discussion to this point has simply dealt with the case where the reflection coefficient was unity. If the sea state spectral density is written in terms of the roughness parame- ter h; the intensity can be shown proportional to μo≈-1 and the acoustic roughness R=2khsinθg, the Rayleigh parameter.
(13)
This formulation can be useful in determining the effect on the effective reflection coefficient μ. In the mid frequency range the increase in μ fills in the nulls of the interference pattern and reduces the magnitude of the peaks. The interesting feature is that for low sea states, Beaufort Number 3 (wind speeds between 3.4–5.4 m/s) the
ratio of the reflected to incident inten- sities is less than approximately 0.86 for grazing angles less than thirty degrees. Examples of at-sea measure- ments performed in the 1980’s at low sea states can be found in Carey (1997), see Figs. 4 and 5.
The utility of this image interfer- ence pattern is seen in the at-sea cali- bration of array hydrophone groups shown in Fig. 5. Common practice used in construction of seismic-type arrays was to combine multiple hydrophones connected in series and parallel to form groups with physical lengths on the order of a quarter wave length. Even though individual cali- brations were usually performed on each hydrophone, the calibration of the hundreds of array hydrophone groups with the long transmission cable was desirable. Such calibration geometry is shown in Fig. 4. This tech- nique requires stable and consistent motion of the vessel, good ship driv-
Fig. 3. The mid-frequency, 5 kHz, interference pattern is shown versus range for a reflection coefficient (μ= -0.9), a source depth of 6λ, and a receiver depth of 20 λ.
16 Acoustics Today, April 2009