Keywords

1 Introduction

Underwater acoustic velocity gradient sensors arise from the second-order approximation of the Taylor series expansion of the pressure field. They are a type of dyadic sensor, and they can measure scalar pressure, vector particle velocity, and tensor velocity gradient, etc. [1]. Underwater acoustic velocity gradient sensors have three types of directivities: monopole directivity, dipole directivity, and quadrupole directivities [2], and quadrupole directivities contain both longitudinal quadrupole directivity and lateral quadrupole directivity. An underwater acoustic velocity gradient sensor can produce a beamwidth of 65° compared to 105° for a vector sensor. Therefore, researchers focus on high-order directional sensors and consider putting them into arrays to improve the performance of the arrays [3, 4].

Underwater acoustic velocity gradient sensors can be combined with one pressure sensor and six biaxial vector sensors, and the structure for underwater acoustic velocity gradient sensors is given in this paper. With this type of design, particle velocity and velocity gradient will be measured approximately. Then the formulations of measurement errors of underwater acoustic velocity gradient sensors in theoretically ideal conditions are derived. Whereas, in practice, axis mismatches exist between pairs of vector sensors, and the formulations of measurement errors as axis mismatches can be obtained. Finally, the formulations are analyzed by figures in this paper, and the relationship among measurement errors, axis mismatches, azimuth angles of sources, and the frequency range of underwater acoustic velocity gradient sensors are discussed.

2 Taylor Series Expansion of an Acoustic Pressure Field

An nth-order Taylor series expansion of an acoustic pressure field contained within a finite region (x − x 0, y − y 0, z − z 0) about the point (x 0, y 0, z 0) can be expressed as

$$ p(\varvec{r}) = p(\varvec{r}_{0} ) + \sum\limits_{n = 1}^{\infty } {\frac{1}{n!}} \left[ {(\varvec{r} - \varvec{r}_{0} ){\nabla }} \right]^{n} p(\varvec{r}_{0} ) $$
(1)

where \( \varvec{r} = \left[ {x\;y\;z} \right] \) and \( \varvec{r}_{0} = \left[ {x_{0} \;y_{0} \;z_{0} } \right] \). The spatial gradient of pressure can be related to acoustic particle velocity via the linearized momentum equations

$$ \frac{{\partial p\left( {\varvec{r}_{0} } \right)}}{{\partial x_{0} }} = - j\omega \rho_{0} u_{x} ,\frac{{\partial p\left( {\varvec{r}_{0} } \right)}}{{\partial y_{0} }} = - j\omega \rho_{0} u_{y} ,\frac{{\partial p\left( {\varvec{r}_{0} } \right)}}{{\partial z_{0} }} = - j\omega \rho_{0} u_{z} $$
(2)

where \( \rho_{0} \) is the ambient density of the surrounding fluid, and u x , u y , and u z are the orthogonal components of the acoustic particle velocity vector. By substituting Eq. (2) into Eq. (1), taking the appropriate partial derivatives, and expanding the series to the second order (n = 2), Eq. (1) can be rewritten as

$$ p\left( \varvec{r} \right) \approx p\left( {\varvec{r}_{0} } \right) - j\omega \rho_{0} \left[ {\varvec{r} - \varvec{r}_{0} } \right]\left[ {\begin{array}{*{20}c} {u_{x} } \\ {u_{y} } \\ {u_{z} } \\ \end{array} } \right] - \frac{1}{2}j\omega \rho_{0} \left[ {\varvec{r} - \varvec{r}_{0} } \right]\left[ {\begin{array}{*{20}c} {\frac{{\partial u_{x} }}{{\partial x_{0} }}} & {\frac{{\partial u_{x} }}{{\partial y_{0} }}} & {\frac{{\partial u_{x} }}{{\partial z_{0} }}} \\ {\frac{{\partial u_{y} }}{{\partial x_{0} }}} & {\frac{{\partial u_{y} }}{{\partial y_{0} }}} & {\frac{{\partial u_{y} }}{{\partial z_{0} }}} \\ {\frac{{\partial u_{z} }}{{\partial x_{0} }}} & {\frac{{\partial u_{z} }}{{\partial y_{0} }}} & {\frac{{\partial u_{z} }}{{\partial z_{0} }}} \\ \end{array} } \right]\left[ {\varvec{r} - \varvec{r}_{0} } \right]^{\text{T}} $$
(3)

Notice that the pressure is a scalar quantity, equivalent to a tensor of rank zero; the velocity is a vector quantity, a first-rank tensor combined with three orthogonal components; the velocity gradient is a second-rank tensor combined with nine components. The propagating plane wave field is irrotational and, hence, the curl of the velocity vector is zero [5]

$$ \left\{ {\begin{array}{*{20}l} {\frac{{\partial u_{x} }}{{\partial y_{0} }} = \frac{{\partial u_{y} }}{{\partial x_{0} }}} \hfill \\ {\frac{{\partial u_{x} }}{{\partial z_{0} }} = \frac{{\partial u_{z} }}{{\partial x_{0} }}} \hfill \\ {\frac{{\partial u_{y} }}{{\partial z_{0} }} = \frac{{\partial u_{z} }}{{\partial y_{0} }}} \hfill \\ \end{array} } \right. $$
(4)

Then Eq. (3) can be simplified as

$$ p\left( \varvec{r} \right) \approx p\left( {\varvec{r}_{0} } \right) - {j}\omega \rho_{0} \left[ {\varvec{r} - \varvec{r}_{0} } \right]\left[ {\begin{array}{*{20}l} {u_{x} } \\ {u_{y} } \\ {u_{z} } \\ \end{array} } \right] - \frac{1}{2}{j}\omega \rho_{0} \left[ {\varvec{r} - \varvec{r}_{0} } \right]\left[ {\begin{array}{*{20}c} {\frac{{\partial u_{x} }}{{\partial x_{0} }}} & {\frac{{\partial u_{x} }}{{\partial y_{0} }}} & {\frac{{\partial u_{x} }}{{\partial z_{0} }}} \\ {\frac{{\partial u_{x} }}{{\partial y_{0} }}} & {\frac{{\partial u_{y} }}{{\partial y_{0} }}} & {\frac{{\partial u_{y} }}{{\partial z_{0} }}} \\ {\frac{{\partial u_{x} }}{{\partial z_{0} }}} & {\frac{{\partial u_{y} }}{{\partial z_{0} }}} & {\frac{{\partial u_{z} }}{{\partial z_{0} }}} \\ \end{array} } \right]\left[{\varvec{r} - \varvec{r}_{0}} \right]^{\text{T}} $$
(5)

Notice that the second-order terms above can be divided into two types: the pure partial derivatives of the velocity and the mixed partial derivatives of the velocity corresponding to the diagonal terms and off-diagonal, respectively. An acoustic velocity gradient sensor should measure pressure, particle velocity, and velocity gradient (ten components in plane wave field) at one point \( \varvec{r}_{0} \).

3 Structure of Velocity Gradient Sensors

A velocity gradient sensor is combined with one pressure sensor and three pairs of biaxial vector sensors as sensitive elements, as shown in Fig. 1. The pressure sensor is in the center of the velocity gradient sensor, three pairs of collinear vector sensors are laid on the coordinate axes (x, y, z), and each pair is oppositely positioned relative to the center. The predetermined distance between each pair of vector sensors is L, and the sensitive axes are shown by the arrows in Fig. 1.

Fig. 1
figure 1

The structure sketch of a velocity gradient sensor

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For this type of design, the acoustic pressure p can be measured directly, and the particle velocity components and velocity gradient components are obtained by principles of mean value and finite-difference approximation, respectively. For example

$$ \left\{ {\begin{array}{*{20}l} {u_{x} \approx \frac{{u_{2x} + u_{1x} }}{2}} \hfill \\ {\frac{{\partial u_{x} }}{\partial x} \approx \frac{{u_{2x} - u_{1x} }}{L}} \hfill \\ {\frac{{\partial u_{y} }}{\partial x} \approx \frac{{u_{2y} - u_{1y} }}{L}} \hfill \\ \end{array} } \right. $$
(6)

4 A Velocity Gradient Sensor in a Plane Wave Field

A velocity gradient sensor is simplified to a single axis and the plane wave field is simplified to two dimensions, as shown in Fig. 2. The analysis of the other two axes of the velocity gradient sensor would be analogous. A pair of biaxial vector sensors are laid in \( u_{1} \) and \( u_{2} \) positions respectively, and the sensor spacing is L. The azimuth angle between the x-axis and the propagation direction of the plane wave is \( \theta \). The sensitive axes of the vector sensors and the direction of the plane wave are shown by the arrows.

Fig. 2
figure 2

A velocity gradient sensor in a plane wave field

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The components of the velocity at the points \( u_{1} \) and \( u_{2} \) are

$$ \left\{ {\begin{array}{*{20}l} {u_{1x} = { \cos }\theta u_{0} {\text{e}}^{{j\left[ {\omega t - k\left( {x - \frac{L}{2}} \right)\,{ \cos }\theta - ky{ \sin }\theta } \right]}} } \hfill \\ {u_{1y} = { \sin }\theta u_{0} {\text{e}}^{{j\left[ {\omega t - k\left( {x - \frac{L}{2}} \right)\,{ \cos }\theta - ky{ \sin }\theta } \right]}} } \hfill \\ {u_{2x} = { \cos }\theta u_{0} {\text{e}}^{{j\left[ {\omega t - k\left( {x + \frac{L}{2}} \right)\,{ \cos }\theta - ky{ \sin }\theta } \right]}} } \hfill \\ {u_{2y} = { \sin }\theta u_{0} {\text{e}}^{{j\left[ {\omega t - k\left( {x + \frac{L}{2}} \right)\,{ \cos }\theta - ky{ \sin }\theta } \right]}} } \hfill \\ \end{array} } \right. $$
(7)

where u 0 is the amplitude of the plane wave, \( \omega \) is angular frequency (\( \omega = 2\uppi f \), f is the source frequency), and k is the wave number (k = ω/c, c = 1500 m/s). As shown in Fig. 2, the pure partial derivative of the velocity \( \frac{{\partial u_{x} }}{\partial x} \) can be written as

$$ \frac{{\partial u_{x} }}{\partial x} = - jk{ \cos }^{2} \theta u_{0} {\text{e}}^{{j\left[ {\omega t - kx{ \cos }\theta - ky{ \sin }\theta } \right]}} $$
(8)

Then the mixed partial derivative of the velocity \( \frac{{\partial u_{y} }}{\partial x} \) can be written as

$$ \frac{{\partial u_{y} }}{\partial x} = - {\text{j}}k{ \cos }\theta { \sin }\theta u_{0} {\text{e}}^{{j\left[ {\omega t - kx{ \cos }\theta - ky{ \sin }\theta } \right]}} $$
(9)

From Eqs. (6) to (9), the measurement errors of the particle velocity and velocity gradient can be expressed as

$$ \left\{ {\begin{array}{*{20}c} {\varepsilon_{{u_{x} }} = 20\lg \left| {\frac{{{{\left( {u_{1x} + u_{2x} } \right)} \mathord{\left/ {\vphantom {{\left( {u_{1x} + u_{2x} } \right)} 2}} \right. \kern-0pt} 2}}}{{u_{x} }}} \right| = 20\lg \left| {\cos \left( {\frac{{2{\uppi}fL\cos \theta }}{c}} \right)} \right|} \\ {\varepsilon_{{\frac{{\partial u_{x} }}{\partial x}}} = 20\lg \left| {\frac{{\left( {u_{2x} - u_{1x} } \right)}}{L}\frac{{\partial u_{x} }}{\partial x}} \right| = 20\lg \left| {\frac{{\sin \left( {{{2{\uppi}fL\cos \theta } \mathord{\left/ {\vphantom {{2{\uppi}fL\cos \theta } c}} \right. \kern-0pt} c}} \right)}}{{{{2{\uppi}fL\cos \theta } \mathord{\left/ {\vphantom {{2{\uppi}fL\cos \theta } c}} \right. \kern-0pt} c}}}} \right|} \\ {\varepsilon_{{\frac{{\partial u_{y} }}{\partial x}}} = 20\lg \left| {\frac{{\left( {u_{2y} - u_{1y} } \right)}}{L}\frac{{\partial u_{y} }}{\partial x}} \right| = 20\lg \left| {\frac{{\sin \left( {{{2{\uppi}fL\cos \theta } \mathord{\left/ {\vphantom {{2{\uppi}fL\cos \theta } c}} \right. \kern-0pt} c}} \right)}}{{{{2{\uppi}fL\cos \theta } \mathord{\left/ {\vphantom {{2{\uppi}fL\cos \theta } c}} \right. \kern-0pt} c}}}} \right|} \\ \end{array} } \right. $$
(10)

5 Axis Mismatches Between Pairs of Vector Sensors

Limited to the technic and elastic suspension of vector sensors, there will be an included angle \( (\theta_{m} ) \) which exists between the same sensitive axis of the vector sensors in practice, as shown in Fig. 3.

Fig. 3
figure 3

Axis mismatches between pairs of vector sensors

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Due to the included angle \( \theta_{m} \), the components \( (u_{1x} ,\;u_{1y} ) \) of the velocity of the No. 1 vector sensor will be changed and expressed as

$$ \left\{ {\begin{array}{*{20}c} {u_{1x} = {\cos}\left( {\theta - \theta_{m} } \right)u_{0} {\text{e}}^{{j\left[ {\omega t - k\left( {x - \frac{L}{2}} \right){\cos}\theta - ky{\sin}\theta } \right]}} } \\ {u_{1y} = {\sin}\left( {\theta - \theta_{m} } \right)u_{0} {\text{e}}^{{j\left[ {\omega t - k\left( {x - \frac{L}{2}} \right){\cos}\theta - ky{\sin}\theta } \right]}} } \\ \end{array} } \right. $$
(11)

Then the measurement errors of the particle velocity and velocity gradient are changed and expressed as

$$ \left\{ {\begin{array}{*{20}c} {\varepsilon_{{u_{x} }} = 20\lg \left| {\frac{{{{\left( {u_{2x} + u_{1x} } \right)} \mathord{\left/ {\vphantom {{\left( {u_{2x} + u_{1x} } \right)} 2}} \right. \kern-0pt} 2}}}{{u_{x} }}} \right| = 20\lg \left| {\frac{{{\cos}\left( {\theta - \theta_{m} } \right){\text{e}}^{{{j}\uppi\,f\frac{L}{c}\,{\cos}\,\theta }} + \cos \theta {\text{e}}^{{ - {j}\uppi\,f\frac{L}{c}\,{\cos}\,\theta }} }}{{2\,{\cos}\,\theta }}} \right|} \\ {\varepsilon_{{\frac{{\partial u_{x} }}{\partial x}}} = 20\lg \left| {\frac{{\left( {u_{2x} - u_{1x} } \right)}}{L}\frac{{\partial u_{x} }}{\partial x}} \right| = 20\lg \left| {\frac{{{\cos}\,\theta \,{\text{e}}^{{ - {j}\uppi\,f\frac{L}{c}\,{\cos}\,\theta }} - {\cos}\,\left( {\theta - \theta_{m} } \right){\text{e}}^{{{j}\uppi\,f\frac{L}{c}\,{\cos}\,\theta }} }}{{{{ - 2{j}\uppi\,fL\,{ \cos }^{2} \theta } \mathord{\left/ {\vphantom {{ - 2{j}\uppi\,fL\,{\cos}^{2} \theta } c}} \right. \kern-0pt} c}}}} \right|} \\ {\varepsilon_{{\frac{{\partial u_{y} }}{\partial x}}} = 20\lg \left| {\frac{{\left( {u_{2y} - u_{1y} } \right)}}{L}\frac{{\partial u_{y} }}{\partial x}} \right| = 20\lg \left| {\frac{{{sin}\,\theta \,{\text{e}}^{{ - {j}\uppi\,f\frac{L}{c}\,{\cos}\,\theta }} - {\sin}\,\left( {\theta - \theta_{m} } \right){\text{e}}^{{{j}\uppi\,f\frac{L}{c}\,{\cos}\,\theta }} }}{{{{ - 2{j}\uppi\,fL\,{\cos}\,\theta \,{\sin}\,\theta } \mathord{\left/ {\vphantom {{ - 2{j}\uppi\,fL\,{\cos}\,\theta \,{\sin}\,\theta } c}} \right. \kern-0pt} c}}}} \right|} \\ \end{array} } \right. $$
(12)

Assume that the included angle \( \theta_{m} = 5^{\circ } \), and Eq. (12) is drawn as contour maps and the 3-D surface plots of measurement errors \( \varepsilon \) versus fL and azimuth angle \( \theta \), as shown in Fig. 4.

Fig. 4
figure 4

Measurement errors while axis mismatches exist (θ m  = 5°). a, c, e Contour map of measurement errors of particle velocity component u x , velocity gradient components ∂u x /∂x and ∂u y /∂x, respectively; b, d, f 3-D surface plot of measurement errors of particle velocity component u x , velocity gradient components ∂u x /∂x and ∂u y /∂x, respectively

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Figure 4 shows that when the azimuth angle \( \theta \) is near ±π/2, the measurement errors of the particle velocity \( u_{x} \) and velocity gradient \( \frac{{\partial u_{x} }}{\partial x} \) are huge, and the measurement error of the velocity gradient \( \frac{{\partial u_{y} }}{\partial x} \) is huge when the azimuth angle \( \theta \) is near 0, ±π/2 and ±π.

Therefore, for a fixed permissible error, it cannot be satisfied on every azimuth angle \( \theta \). There should be a region to meet the permissible error, and the region should be a beamwidth. When permissible errors of the particle velocity and velocity gradient are within 1 dB in the region of the beamwidth, then for particle velocity \( u_{x} \), the range of fL is

$$ 0 \le fL \le 218 $$
(13)

For velocity gradient \( \frac{{\partial u_{x} }}{\partial x} \), the range of fL is

$$ 29 \le fL \le 380 $$
(14)

For velocity gradient \( \frac{{\partial u_{y} }}{\partial x} \), the range of fL is

$$ 149 \le fL \le 244 $$
(15)

From Eqs. (13) to (15), the intersection of fL is

$$ 149 \le fL \le 218 $$
(16)

Therefore, when spacing L is fixed, the frequency range of the acoustic velocity gradient sensors is fixed. Equations (13)–(15) show that for more quantities measured approximately, the smaller will be the intersection of fL, and moreover, the intersection of fL will be smaller or disappear as the included angle θ m becomes larger. Assume that the included angle θ m  = 1° and θ m  = 10°, respectively, and then compare to the measurement errors of different quantities, as shown in Fig. 5. For example, the permissible errors of the particle velocity and velocity gradients are within 1 dB, the region satisfies the permissible errors as the included angle θ m  = 1° are larger than the case that θ m  = 10°, and there is no intersection of fL as θ m  = 10°. Due to a constant L, axis mismatch (θ m ) will directly affect the frequency range of the acoustic velocity gradient sensors and moreover, the larger the included angle θ m , the larger the region over ranges the permissible errors.

Fig. 5
figure 5

Comparison of measurement errors in the cases of θ m  = 1° and θ m  = 10°. a, b, c Contour maps of measurement errors of particle velocity component u x , velocity gradient components ∂u x /∂x and ∂u y /∂x for the case θ m  = 1°; d, e, f contour maps of measurement errors of particle velocity component u x , velocity gradient components ∂u x /∂x and ∂u y /∂x for the case θ m  = 10°

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6 Conclusion

Velocity gradient sensors can be combined with one pressure sensor and six biaxial vector sensors, and they can measure ten quantities in a plane wave field, including pressure, three components of particle velocity and six components of velocity gradients. The formulations of measurement errors of acoustic velocity gradient sensors are obtained when axis mismatches exist, and the conclusions are summarized through the formulations as follows.

The performance of acoustic velocity gradient sensors will be strongly affected by axis mismatches between pairs of sensitive elements. Due to axis mismatches, the acoustic velocity gradient sensors have higher measurement errors of particle velocity and velocity gradient in some regions, such as the azimuth angle θ is near 0, ±π/2 and ±π. When axis mismatches become larger, the region of high measurement errors will increase. For the fixed permissible error and spacing L, the frequency range of the acoustic velocity gradient sensors is directly affected by axis mismatches between pairs of collinear vector sensors. The frequency range of acoustic velocity gradient sensors will be narrowed with increasing axis mismatches.