**3. Calculation**

We used a piezoelectric material PZT-5H (Lead Zirconate Titanate-5H, Baoding Hongsheng Acoustic-electric Equipment Co., Ltd., Baoding, Hebei 071000, China) in the construction of the cylindrical transducers. The physical parameters of the piezoelectric material PZT-5H are ε*T*33 = 300.9 × 10−<sup>10</sup> <sup>F</sup>/m2, *sE*11 = 16.5 × 10−<sup>12</sup> m<sup>2</sup>/N, *d*31 = −274 × 10−<sup>12</sup> m/V, and ρ*n* = 7.5 × 10<sup>3</sup> kg/m3. silicone oil was used as the coupling medium around the transducers, where the physical parameters are *vm* = 1424 m/s and ρ*m* = 856 kg/m<sup>3</sup> (Shandong Longhui Chemical Co., Ltd, Jinan, Shaandong 250131, China).

#### *3.1. Relationship between the Center Frequency versus the Radius for a Thin Cylindrical Transducer*

As described above, the transducer always works in an oscillatory mode. As a reference, we defined a free-mechanical-load transducer as a transducer in a vacuum, i.e. *R*ρ = 0, *<sup>m</sup>*ρ = 0, and *Rm* = 0. The center frequency of the piezoelectric PZT-5H thin cylindrical transducer as a function of average radius is presented in Figure 4, along with a case of the mechanical load, i.e., the transducer in the transformer oil, where *R*ρ - 0, *<sup>m</sup>*ρ - 0, and *Rm* - 0.

Figure 4 shows that: (i) the transducer's center frequency decreased with respect to its increased radius, with or without a mechanical load; (ii) the center frequency of the mechanical load was lower than that of the free mechanical load; and (iii) the mechanical load effect on the transducer's center frequency decreased with the increased radius.

**Figure 4.** The relationship between the transducer's center frequency versus its radius. The solid line is the center frequency for the case with a mechanical load (i.e. transducer was put in transformer oil) and the dashed line is the center frequency for the case of a free mechanical load (i.e. transducer was put in air or vacuum). The two star signs are the measured values for the case of the transducer with a mechanical load and *lt* = ρ0/8.

#### *3.2. Relationship between the Center Frequency versus a Forced Vibrational Frequency*

We built two piezoelectric PZT-5H shin-cylindrical transducers polarized in the radial direction, with an average radius ρ0 = 20.5 mm, height *H* = 6.0 mm, and wall thickness *lt* = ρ0/8. Figure 4 shows that the center frequency of the transducers with a mechanical load was 20.5180 kHz. We also determined that the center frequency of the free-mechanical-load transducer was at 22.5120 kHz.

For the case of the mechanical load, we selected the parameter *Rm* = 0.2*R*. When normalized by the transducer's free-load center frequency, for the transducer's forced harmonic vibrational motions at several frequencies, the electric–acoustic/acoustic–electric conversion properties were calculated and are presented in Figure 5.

At the center frequency of 22.5120 kHz for the free mechanical load, both the source and receiver transducers had a maximum transition amplitude, which was the largest for all cases. For the mechanical loaded transducers with harmonic forced vibrations, a lower forced vibration frequency corresponded to a lower center frequency with a larger maximum transition amplitude. With the increased vibration frequency going close to the free-mechanical-load resonance frequency *f*0, a mechanical-loaded transducer showed a maximum transition amplitude, but it was much smaller than that of the thin cylindrical transducer free vibration.

The observations above confirmed our understanding that the electric–acoustic/acoustic–electric conversion of the transducer was dependent not only on the physical and geometrical parameters of a transducer and the physical parameters of the medium around the transducer, but also the forced harmonic vibration frequency. The radiation resistance and radiation mass were parameters affecting the forced vibration frequency, which led to the variations of the electric–acoustic/acoustic–electric conversion properties of the transducer.

**Figure 5.** The impulse response and corresponding amplitude spectrum for a thin cylindrical transducer: (**a**) the impulse response and (**b**) the amplitude spectrum. The cyan line is the case of a free mechanical load. The other lines are for a mechanical load, where the magenta, blue, red, and black lines stand for the sinusoidal driving electric signals with frequency *fs* = 0.1 *f*0, 0.2 *f*0, 0.5 *f*0, and 1.5 *f*0, respectively, where *f*0 = 20.5180 kHz. The first numerical value in Figure 5b is the center frequency and the second one is the maximal value of the amplitude spectrum.

#### *3.3. Radiation Directivity of a Thin Cylindrical Transducer*

Different from the thin-shell spherical transducer, the radiation property of a thin cylindrical transducer has a determinative directivity, which can be described using [8]:

$$G(\theta) = \left[\frac{l\_0^2(\frac{2\pi\rho\_a}{\lambda}\cos\theta) + \cos^2\theta l\_1^2(\frac{2\pi\rho\_a}{\lambda}\cos\theta)}{l\_0^2(\frac{2\pi\rho\_a}{\lambda}) + l\_1^2(\frac{2\pi\rho\_a}{\lambda})}\right]^{1/2} \frac{\sin(\frac{\pi H}{\lambda}\sin\theta)}{\frac{\pi H}{\lambda}\sin\theta},\tag{28}$$

where *J*0 is the zeroth-order Bessel function; *J*1 is the first-order Bessel function; λ is the wavelength of the acoustic wave in coupling fluid, i.e., transformer oil; and θ is the angle between the propagation direction of radiated acoustic signal and the normal direction of the thin cylindrical transducer.

From Equation (28), the calculated radiation directivity of a thin cylindrical transducer is shown in Figure 6 It is shown that the acoustic energy radiated by the transducer was more centralized in the normal direction with the increased height of the thin cylindrical transducer.

**Figure 6.** The directivity of thin cylindrical transducer with varied height of the cylindrical transducer, where ρ0 = 20.50 mm. Red, black, green, and blue colors indicate the directivities of *H* = 40 mm, 60 mm, 80 mm, and 100 mm, respectively.

The effective radiating area of a cylindrical transducer, in the direction of θ, may be defined as the product of the area of the thin cylindrical transducer *Sa*(= <sup>2</sup>πρ*a<sup>H</sup>*) and the directivity *G*(θ):

$$ERA = S\_4 \cdot G(\theta) = 2\pi \rho\_a H \cdot G(\theta). \tag{29}$$

For a given cylindrical transducer with a fixed average radius, a normalized effective radiating area is labelled as *NERA*. The relationship of the normalized *NERA* versus *H* in several different directions are presented in Figure 7. The calculated results indicated that the transducer's effective radiating area did not increase monotonously with respect to its height or its radiation area. This was exactly the effect of the transducer's radiation directivity, which was a function of the radiation direction. With this understanding, by selecting a suitable height of the transducer, we may be able to achieve an increased effective radiation area of the cylindrical transducer for a set special direction.

**Figure 7.** The relationship of the normalized effective radiating area (*NERA*) versus height (*H*) for several different directions, where ρ0 = 20.5 mm. The green, blue, black, magenta, and red colors designate the cases of radiating direction θ = 20◦, θ = 30◦, θ = 40◦, θ = 50◦, and, θ = 60◦, respectively. The first numerical value in Figure 8 is the height, *H*, of the transducer and the second one is the maximal value of the normalized effective radiating area. The value of *NERA* is the value of *ERA* normalized using the maximum of *ERA* at θ = 20◦.

#### *3.4. Transient Response of a Cylindrical Transducer Excited by a Signal with Multi-Frequency Components*

As an example, let us consider a gated sinusoidal electric signal as the source of excitation with multi-frequency components as:

$$dL\_1(t) = \left[H(t) - H(t - t\_0)\right] lL\_0 \sin(\omega\_5 t),\tag{30}$$

where, *U*0, ω*s*, and *t*0 are the amplitude, angular frequency, and time window of the driving electric voltage signal, respectively; and *<sup>H</sup>*(·) is the Heaviside unit step function. A simple Fourier transform yields the source signal in the frequency domain as:

$$S\_1(\omega) = lL\_0 \Big| \omega\_s - \left(\omega\_s \cos \omega\_s t\_0 + j\omega \sin \omega\_s t\_0\right) \exp[-j\omega t\_0] / \left(\omega\_s^2 - \omega^2\right) \tag{31}$$

with a corresponding phase spectrum:

$$\phi(\omega) = \text{tg}^{-1}\left[\frac{\text{Im}(S\_1)}{\text{Re}(S\_1)}\right].\tag{32}$$

Now, we selected the source signal parameters *U*0 = 1 V, *t*0 = 4/ *fs*, and *fs* = <sup>ω</sup>*s*/2<sup>π</sup> = 20.5180 kHz, which was the mechanical-load center frequency of the transducer. The center frequency of this driving electric signal was then 20.1690 kHz, which was slightly smaller than the value of *fs*.

Figure 8 shows that the theoretical waveform of the driving electric voltage signal agreed well with the waveform synthesized using a discrete Fourier transform. In turn, it guaranteed the accuracy of our analysis in the following sections.

The gated sinusoidal driving electric signal was expanded on the basis of a series of sine-waves with different frequencies, amplitudes, and phases. Each of these sinusoidal components, as an individual excitation source, was applied to the parallel circuits of Figure 3. The output signal from each parallel circuit in the network (see Figure 3) was calculated and analyzed.

**Figure 8.** The waveforms of the gated sinusoidal driving electric signal with two cycles and *fs* = 20.5180 kHz, where *fs* = <sup>ω</sup>*s*/2<sup>π</sup>. (**a**) The amplitude spectrum, (**b**) the phase spectrum, and (**c**) the waveform. The solid line in (c) was from the theoretical calculation and the cycle line was the synthesized waveform from discretized amplitude and phase spectra.

The cumulative output of the waveform of the parallel circuits in part I of Figure 3 is presented in Figure 9a. The corresponding amplitude spectrum is presented in Figure 9b, which shows that the center frequency of the radiated acoustic signal was 20.2500 kHz. It was smaller than the load center frequency 20.5180 kHz, but larger than that of gated sinusoidal driving electric signal at 20.1690 kHz, which was the result of the joint action of the electric–acoustic conversion through the source transducer and the electric driving signal.

**Figure 9.** The waveform and amplitude spectrum of the acoustic signal radiated by the source transducer: (**a**) The normalized waveforms, which were the cumulative convolution of all frequency components in the network; and (**b**) the normalized amplitude spectrum of our transducer's transient response model.

The studied source transducer was cylindrical and had a radiation directivity. If we assumed that water is an ideal elastic medium, then the acoustic signal propagating inside the water could have a geometrical attenuation but not a viscous attenuation. Additionally, all frequency components of the acoustic signal would propagate at the same speed. The shapes of the waveform and frequency spectrum would not change. The amplitude would decrease with respect to the increased propagation distance only. Under these conditions, we calculated the signals at the electric terminals of the receiver transducer.

For a thin cylindrical transducer, the shape of its radiation directivity is the same as that of its receiving directivity; therefore, the acoustic impulse response in water can then be written as:

$$h\_2(t) = G\_1(\theta\_1) G\_2(\theta\_2) \delta(t - t\_1) / (1 + r),\tag{33}$$

where *h*21(*<sup>t</sup>*, <sup>ω</sup>1) = ... = *<sup>h</sup>*2*<sup>j</sup><sup>t</sup>*, ω*j*= ... *h*2*<sup>N</sup>*(*<sup>t</sup>*, <sup>ω</sup>*N*) = *h*2(*t*), *t*1 is the propagation time and *r* is the distance of the radiated signal in water, θ1 is the angle between of the propagation direction of the radiated acoustic wave and the normal direction of the side wall of the source-transducer, *<sup>G</sup>*1(<sup>θ</sup>1) is the corresponding radiation directivity at this propagation direction (θ1), θ2 is the angle between the direction of the acoustic wave propagating to the receiver transducer and the normal direction of the side-wall of the receiver-transducer, and *<sup>G</sup>*2(<sup>θ</sup>2) is the corresponding receiving directivity.

To determine the quality of a transducer for electric–acoustic/acoustic–electric conversion, the most important piece of information comes from the analysis of the output signal following electric–acoustic–electric transduction. To accomplish this operation, we set θ1 = θ2 = 0◦ and the distance from the source transducer to the receiver transducer was 1.0 m. Applying the gated sinusoidal driving electric signal to the source transducer, we calculated the cumulative output signals on the receiver transducer, as shown as solid lines in Figure 10.

Figure 10 shows the cumulative signals of the waveform and amplitude spectrum at the electric terminals of the receiver transducer. From the theoretical calculation, the center frequency of the output signal at the electrical terminals of the receiver transducer was 20.5000 kHz. This center frequency was slightly smaller than the transducer's load center frequency and slightly greater than the center frequency of the acoustic signal radiated by the source transducer. We also believe that this was the result of the joint action of a lower center frequency of the radiated acoustic signal and a higher center frequency of the transducer with a mechanical load.

**Figure 10.** Normalized electric signals at the receiver transducer (part III of Figure 4): (**a**) the waveform and (**b**) the amplitude spectrum. *S*3 stands for the amplitude spectrum corresponding to the electric-signals *U*3 at the receiver transducer. Solid lines come from the theoretical calculation and dotted lines come from the experimental measurement.

For an average radius of 20.5 mm, Figure 4 shows that the calculated center frequency of the mechanical-loaded transducer was 20.5800 kHz and the measured center frequencies of the two cylindrical transducers with the same size were 19.6780 kHz and 20.7520 kHz. The waveform and amplitude spectrum of the measured acoustic signal, i.e., the measured electric signal at the electric terminals of the receiver transducer, are shown as the dotted lines in Figure 10. Overall, the theoretical calculation had a good agreemen<sup>t</sup> with the experimental measurement.

#### *3.5. Influence of the Transducer's Radiating*/*Receiving Directivity on the Measured Acoustic Signal*

Now, let us consider a measurement system, where a cylindrical transducer is used as an acoustic source and four cylindrical transducers as a receiver array, as shown in Figure 11. The transition media of this system is water. The waveforms of the signals at the electric terminals of the four cylindrical transducers in the receiver array were calculated and are presented in Figure 12.

The results of the calculation show that the greater the deflection angle, the greater the influence on the signal at the electric terminals of each cylindrical transducer in the receiver array. This was especially true in the near-field area. The amplitude of the signal at the electric terminals decreased with increased angle θ*j*.

**Figure 11.** The acoustic-measurement system with a thin cylindrical transducer as a source and an array of four cylindrical transducers as the receivers, where *T* is the source transducer, *Ri* is each receiver transducer in the transducers line array, *ri* is the distance from the source transducer *T* to the receiver transducer *Ri* of the transducer line-array with {*i*} = {0, 1, 2, 3}, and θ*j* is the angle of *rj* with respect to *r*0 with {*j*} = {1, 2, 3}.

**Figure 12.** The signal waveforms at the electric-terminals of the receiver-transducer.

#### **4. Devices and Measurements**

To rationalize the proposed transient response and physical properties of the cylindrical transducers, we developed an experimental measurement system (see Figure S1 and Figure S2). In particular, this system consisted of a mechanical assembly, an electrical hardware module, and a system software module aimed at system control, measurement, and analysis. A structure flowchart is presented in Figure 13.

**Figure 13.** Schematic presentation of the experimental measurement system.

#### *4.1. Mechanical and Electrical Hardware*

The mechanical parts include steering engines, stepping motors, and sliding rails. The combined assembly with a microcontroller formed a positioning platform, which was used to slide and/or rotate the source/receiver transducers to the proper positions and directions in a silencing tank filled with water. The silencing tank was used to gauge the physical properties of the transducer, e.g., the electric–acoustic/acoustic–electric transient response, directivity, radiation power, and receiving sensitivity of the transducer.

The electric hardware was composed of a microcontroller, an electric-signal waveform generator, a power amplifier, a high-resolution and high-sampling-rate digitizer (up to 24 bit and 15 MHz), and a desktop/laptop for central system control.

## *4.2. System Software*

The system software was developed on the LabVIEW platform in Graphic Programming Language (G-code). Communications between the computer and the hardware were realized through USB serial ports. Powered by a graphic interface control panel, users have the options of selecting different driving signals, configuring the power amplifier, adjusting the position and rotational angle of the transducers, and attaining data acquisition of the measured acoustic signal.

The developed software consists of four functional modules, as shown in Figure 14. On a graphic computer interface panel, the VISA (Virtual Instrument Software Architecture) library from the LabVIEW platform was employed to achieve the control of the four modules through serial port communications. In the function/selection boxes in the human–computer interface modules, users input the parameters and commands for special operations and acoustic measurements.

**Figure 14.** Structure flowchart of the modules for an electric signal source, power amplifier, date display/storage, and slippage/rotation.

#### 4.2.1. Electric Signal Module

In controlling the widgets, knobs, and switches on the software interface panel, this module configures and modulates the types of signal and frequency, amplitude, and the cycle of an electric source signal. The options for the electric source signal are the gated sinusoidal wave, square wave, triangular wave, and sawtooth wave.

#### 4.2.2. Power Amplification Module

This module configures the amplification gain of a power amplifier, which amplifies the electric signal created by the electric signal waveform generator. The amplified electric signal is used to excite the source transducer to radiate the acoustic signal. This is achieved through a rotary button on the interface control panel. For an output voltage signal with a frequency belt from 0.15 MHz to 1.5 MHz, the maximum peak-to-peak value can reach 220 V. Meanwhile, for an output voltage signal with a frequency range from 10 kHz to 150 kHz, the maximum peak-to-peak value can be up to 1600 V. For an electric signal source with general impedance, a suitable input impedance is 50 Ω. For an electric signal source with high impedance, an appropriate input impedance is 50 k Ω.
