Influences of Cylindrical Carrier ’ s Size and Shape of Undersurface on the Directivity of Vector Sensor

When the acoustic vector sensor is installed on underwater carrier to detect and position targets, the measuring results will be influenced by the diffraction of the carrier. Due to the differences of the sizes and shapes of the carrier, the influences of the diffraction on acoustic vector sensor are also different. Using BEM, the influences of cylinder’s size and shape of undersurface on the directivity of acoustic vector sensor are calculated in this study. The results show that: the influences of diffraction on acoustic vector sensor are proportional to the length of cylinder’s radius and height; besides, the sharper the undersurface is, the less influences acoustic vector sensor subjects to diffraction of the carrier. The experiment that carried out in the lake proves the validity of the computing. The research is effective guidance for the application of acoustic vector sensor.


INTRODUCTION
Composed by omni directional sound pressure sensor and dipole directional particle velocity sensor, acoustic vector sensors picks up both pressure and vibration velocity information in acoustic field, which expends the signal processing space and provides new thought and method for solving underwater acoustic problems (Sun and Li, 2004).However, when it is installed on underwater carrier to detect and position targets, the diffraction of the carrier will distort the free space field, thus influencing the measuring results of vector sensor.To find out how the carrier's size and shape of undersurface affects the directivity of vector sensor and further select the reasonable carrier to reduce the diffraction, is very meaningful for the application of vector sensor.
In recent decades, the diffraction characteristics of objects in shape of sphere, cylinder and et al. have been studied by scholars home and abroad.The pressure and velocity of acoustic diffraction field by infinite rigid spheroid have been calculated using MATLAB PDE toolbox and the method of calculating the diffraction of shell structure in any shape is proposed (Kang, 2002).Systematic research on signal processing of acoustic vector sensor by a reflecting boundary has been carried out (Malcolm and Arye, 2000).The theory of resonance radiation of submerged elastic spherical and cylindrical shell have been studied (Tang and Fan, 2000;Liu et al., 2002).The mathematical expressions of pressure and velocity in diffraction field by prolate spheroid are deduced and the experimental results coincide well with the theoretical results (Ji et al., 2010).The model of the diffraction by elastic shell with filler inside using BEM+FEM is built and the calculating results show that the diffraction of the filler will change the characteristics of acoustic field (Zhuo et al., 2009).In this study, the influences of cylindrical carrier's sizes, shapes of undersurface on the directivity of acoustic vector sensor are studied and experiment was carried out to prove the validity of the calculation.

THEORY OF DIFFRACTION CALCULATION USING BEM
The wave propagating equation under single frequency can be guided to Helmholtz equation.As  1, the enclosed vibrating body V locates in infinite fluid mass whose density sound velocity are ߩ and ܿ respectively.ܲ is a point in the space and the Helmholtz equation in exterior field is expressed as (Шендеров et al., 1983): where, ݇ = ‫ܿ/ݓ‬ = The wave number of the incident wave ܳ = A point on surface S ݊ ሬ ሬԦ = The exterior normal direction of the vibrating body The integral equation of pressure at point ܲ in sound field can be expressed as (He and Zhao, 1992): where, ߙ = 4π, when ܲ ∈ ܸ ௨௧ (outside surface S); ߙ = 2π, when ܲ ∈ ܵ; ߙ = 0, when ܲ ∈ ܸ (inside surface S). ‫ܩ‬ is the three-dimensional Green function in free space and it satisfies the equation as follows: Utilizing the Euler equation, the pressure ‫‬ሺܳሻ and normal velocity ‫ݑ‬ሺܳሻ on surface S can be deduced as: By introducing Formula (4) into Formula (2), the pressure of point ܲ can be obtained as: From Formula (5), it can be indicated that if vibration velocity and pressure on surface ܵ are known parameters, then the pressure of any point in exterior field can be calculated.However, limited to the measuring conditions and the complicated shape of the surface, vibration velocity and pressure information usually can't be both obtained.According to BEM theory, surface S can be divided into many small elements named ∆ܵ .Then the integration form of Formula ( 5) is converted into the summing of each small element.Assume the pressure and velocity of any point in element ∆ܵ are expressed with the pressure ‫‬ and velocity ‫ݑ‬ of the nodes as follows: where, ܰ = The interpolation function of the element ሺߞ, ߟሻ = The local coordinate of the point in the element The transforming relationship between main and local coordinates is Jacobi Factor ‫ܬ‬ሺζ, ߟሻ, whose expression is: Assume that the number of the elements is M. By introducing Formula ( 6), (7) into Formula ( 5), the interpolation function can be converted into algebraic function as follows: If there are ܶ nodes on surface S and the column matrix of the pressure and velocity of the nodes are: By introducing Formula (10) into Formula (9), the matrix expression of Formula (9) can be obtained as follows: where, and ܰ is the interpolation matrix as follows ‫ܭ‬ is the transforming matrix between element nodes' degree of freedom and structure nodes' degree of freedom: When observing point ܲ is outside surface S, When observing point ܲ is on surface S, Utilizing Formula ( 17), the value of pressure and vibration velocity of any point on surface S can be obtained if either of them is known.By taking the value of pressure and vibration velocity of the surface into Formula (16), the sound field of any point outside the surface can be calculated.

SIMULATION AND ANALYSIS
Calculating model: As shown in Fig. 2, the radius of the cylinder is ‫ݎ‬ and the height is ℎ.Acoustic vector sensor locates ݈ away from the center of the carrier on the positive ‫ݔ‬ axis.The vertical and horizontal vibration velocity of the sensor points along the positive x, y axis.
The mass density and sound velocity of the cylinder are 7850 kg/m 3 and 5100 m/s respectively.The carrier is placed in quiescent, isotropic, homogeneous, infinite fluid water, whose mass density is 1000 kg/m 3 and sound velocity is 1500 m/s.When the incident plane wave is along the negative x axis, the incident angle is defined as 0°.The influences of diffraction by the carrier on the directivity of acoustic vector sensor will be calculated as the incident wave rotates a circle around the origin.

Carriers with different radiuses and heights:
The parameters of models with different radiuses and heights are shown in Table 1.Model 1 is the model for comparison, whose radius is 0.1 m and height is 0.4 m.The radius of model 2 is twice as large as model 1 and the height of model 3 is half as large as model 1.The distance from acoustic vector sensor and the center of the carrier is 0.4 m for all the models.
The influences of the diffraction by three models on the directivity of acoustic vector sensor are shown in Fig. 3.At 300 Hz, the amplitude of diffraction wave is small and the directivity of pressure is close to natural free field directivity.However, though the directivity of vertical and horizontal vibration velocity are still dipole, the amplitude of both channels has changed.Figure 3b illustrates the influences of the diffraction by three models on the directivity of vertical vibration velocity at different frequencies.Obviously the influences of diffraction by model 3 are the least and the influences of diffraction by model 1 are less, while the influences of diffraction by model 2 are the most of the three.With the increasing of incident frequency, the diffraction wave by the models grows larger and the directivity of all channels becomes worse.At 2500 Hz, the directivity of vertical vibration velocity by the diffraction of model 2 has seriously deviated from dipole directivity.
Combing with the parameters of Table 1, it can be inferred that the influences of diffraction by the models are proportional to the length of cylinder's radius and heights.Besides, the directivity of pressure and vibration velocity of the sensor by the diffraction of model 1 is better than that of model 2, which implies that influences from the radius are larger than that from the height.

Carriers with different shapes of undersurface:
The schematic diagram of models with different shapes of undersurfaces is shown in Fig. 4. The height and radius of all the models are h and r respectively.All three models are fixed with different shapes of undersurface whose heights are ℎ ଵ .Acoustic vector sensor still locates ݈ away from the center of the carrier and the parameters of three models are shown in Table 2.
The influences of the diffraction by three models on the directivity of acoustic vector sensor are shown in Fig. 5.The directivity of pressure and horizontal vibration velocity is less influenced by the diffraction, while the directivity of vertical vibration velocity is seriously distorted at different incident frequencies.This is because the intensity of diffraction wave concentrates in the x axis direction.From Fig. 5b, it can be inferred that the influences of diffraction by model 6 are the least and the influences of diffraction by model 5are less, while the influences of diffraction by model 4 are the most of all.
From Table 2 it can be inferred that the sharper the shape of the undersurface is, the less influences acoustic vector sensor subjects to the diffraction by the carrier.When the undersurface is sharp, the intensity of diffraction wave diffuses all directions and little is picked up by acoustic vector sensor, which makes its directivity close to natural free field directivity.

EXPERIMENTAL RESULTS AND ANALYSIS
The experiment of influences on the directivity of acoustic vector sensor by cylindrical carrier was carried   The experimental directivity of pressure and vibration velocity at 300,1200 Hz and 4600 Hz are given in Fig. 7.At 300 Hz, the theoretical results are almost the same as experimental results.At 1200 Hz, the differences between theoretical and experimental directivity of horizontal vibration velocity begin to grow, but the experimental directivity of pressure and vertical vibration velocity are basically in agreement with theoretical results.At 4600 Hz, the experimental directivity of pressure and horizontal vibration velocity coincides well with theoretical results, while the directivity of vertical vibration velocity deviates seriously from the theoretical directivity.The main reason is that the diffraction wave of the fixture becomes large at high frequency and it will affect the diffraction field of the carrier, thus influencing the measuring results.
Viewing the trend as a whole, the experimental results basically coincide well with the theoretical results, which verify the validity of the computing.

CONCLUSION
Using BEM, the directivity of pressure and vibration velocity of acoustic vector sensor by different carriers is calculated.The conclusions are as follows: • The influences of diffraction on acoustic vector sensor are proportional to the length of cylinder's radius and height.Besides, the influences from the radius are larger than that from the height.With frequency increasing, the directivity of pressure and vibration velocity becomes worse • The intensity of diffraction wave concentrates in the x axis direction, which makes the directivity of pressure and horizontal vibration velocity less influenced than vertical vibration velocity by the diffraction.When the shape of carrier's undersurface is sharp, the intensity of diffraction wave diffuses all directions and the directivity of acoustic vector sensor is close to natural free field directivity • Experiment was carried out and the results coincide well with the theoretical results at low frequencies, while at high frequency the diffraction of the fixture will affect the experimental results.

Fig. 2 :
Fig. 2: Schematic diagram of position relation between acoustic vector sensor and cylinder Fig. 3: Directivity of acoustic vector sensor by the diffraction of models in Table 1

Fig. 4 :
Fig. 4: Schematic diagram of models with different shapes of undersurface

Table 1 :
Parameters of carrier models with different radiuses and