Doppler Effect

In order to carry out simulation research, the accuracy of the simulation model should be evaluated by comparing the consistency of the flyer velocity–time curve in the simulation and test. The velocity of flyer is the most important parameter used to evaluate the initiation ability of the detonation sequence. Difficulties with the velocity test include the high transient process and the miniaturization of the target to be tested [32 ]. The PDV system uses the Doppler effect to calculate the velocity of moving objects according to the difference in frequency between the reflected laser frequency and the reference laser frequency. Compared with other speed-measurement methods, it has the advantages of a high precision, a wide range, and simple composition. The principle of the PDV system is shown in Figure 4.
The principle of the PDV system was as follows: the laser emitted by the laser source was divided into two channels by the fiber coupler, one of which was used as the reference laser that was shined into the next fiber coupler; the other entered the circulator and then irradiated the flyer, and the frequency changed after being reflected by the flyer, which was called the signal laser. The signal laser was transmitted to the next fiber coupler after passing through the circulator, and a frequency difference occurred with the reference laser that was detected by the detector. Finally, the frequency-difference signal was recorded by the oscilloscope.
The frequency of the reference laser and signal laser met the following relationship:
where f₀ is the frequency of the reference laser in Hz, f_d is the frequency of the signal laser in Hz, C is the speed of light in km·s⁻¹, and v_f(t) is the velocity of the flyer changing with time in km·s⁻¹. The frequency difference can be expressed as:
The light speed, frequency, and wavelength of the reference laser satisfied the following relationship:
where λ₀ is the wavelength of the reference laser in nm.
When substituting Formulas (15) and (16) into Formula (14), the relationship between the flyer velocity v_f(t) and frequency difference Δf could be obtained: $$v_{\mathrm{f}}(t)=\frac{f_0}{2C}(f_{\mathrm{d}}-f_0)=\frac{{\lambda }_0}{2}\mathsf{\Delta }f$$
Therefore, the flyer velocity could be calculated by detecting the frequency difference of the laser caused by the flyer’s motion. Using a Fourier transform, the velocity–time curve of the flyer could be obtained by processing the frequency-difference signal Δf with the MATLAB program.
The test system was mainly composed of the test device of the flyer driven by a microcharge, an organic glass sheet, an optical fiber probe, the PDV, a laser amplifier, and a laser source (Figure 5).
In Section 2.3.1, it was introduced that the diameter of the lead azide was at the 1 mm level, so the diameter of the lead azide in this experiment was submillimeter (0.9 mm); when the pressing pressure was 188 MPa, the corresponding theoretical density of 3.83 g·cm⁻³ was used as the charge density of the lead azide [19 ], which was also close to that used in the literature [17 (link),18 (link)]. According to the preliminary simulation results, the flyer had a relatively high speed when the charge height was 1.2 mm, the thickness of the titanium flyer was 0.1 mm, and the aperture of the stainless steel acceleration chamber was 0.6 mm. Therefore, the flyer velocity–time curve under the above design parameters was obtained by the PDV test.
The test process was as follows:

Fix the optical fiber probe with a steel protective sleeve with the optical fiber probe clamp.

Fix the organic glass sheet on the path between the flyer and the probe to prevent the flyer from damaging the probe, and collect the flyer.

Assemble the flyer-type microdetonation sequence in the fixture, and fix the fixture position so that the optical fiber probe is aligned with the central axis of the acceleration chamber.

Check the continuity of the test circuit and oscilloscope settings, and fire after the inspection.

Read the original frequency difference signal from the oscilloscope, and obtain the flyer velocity–time curve through fast Fourier transform.

In the real-world underwater acoustic communication scenario, the influence of underwater acoustic channels on communication signals mainly includes two aspects: multi-path fading and OAN, as is shown in Figure 1.
(1) Multi-path fading
Multi-path propagation often exists in underwater acoustic communication. In a multi-path environment, the received signal can be represented as the superposition of a number of time-delayed and amplitude–attenuated versions of the transmitted signal. A typical underwater acoustic channel with multi-path fading can be seen as a filter whose impulse response function is $h\left(t,\tau \right)$ , in which $\tau$ is the delay time. The impulse response reflects the properties of the multi-path fading channel and can be expressed as
$$h(t,\tau )=\sum _{k=1}^K{a}_k\left(t\right)\delta (\tau -{\tau }_k\left(t\right))$$
where K is the total number of paths, $\delta \left(·\right)$ is the delta function, $a_k$ and ${\tau }_k$ are the attenuation, and the delay of the k-th path.
Since the source and receiver are usually not stationary and underwater acoustic reflection boundaries are unstable in most cases, multi-path fading is often accompanied by Doppler shift. We consider Doppler shift caused by relative motion in multi-path propagation. In a multi-path fading channel, each path has an independent Doppler shift factor $f_{dk}$ , which can be expressed as
$$f_{dk}=\frac{v_k}{c}{f}_c$$
where $f_c$ is the carrier frequency of the transmitted signal, and $v_k$ is the radial velocity of the source relative to the receiver of the k-th path.
Assume the transmitted signal is $s\left(t\right)$ , then the received signal $x\left(t\right)$ propagates through underwater acoustic channel can be expressed as
$$x\left(t\right)=\sum _{k=1}^K{a}_k\left(t\right)s(t-{\tau }_k\left(t\right))e^{j2\pi f_{dk}t}+n\left(t\right)$$
where $n\left(t\right)$ is the additive noise of the channel.
(2) Ocean ambient noise
Ocean ambient noise (OAN) is an additive interference in underwater acoustic channels. The composition of the OAN is very complicated and full of impulsive interference due to numerous noise sources, such as ship-radiated noise, industrial noise, wind noise, biological noise, etc. The reasons mentioned above make the OAN cannot be simulated accurately using white Gaussian noise. AMC method based on Gaussianity assumptions will suffer degradation in their performance to a low level. We use real-world OAN as the additive noise of underwater acoustic channels to enhance the robustness of the proposed AMC method in real-world underwater acoustic communication scenarios.