An analytical method based on the wave theory is proposed to calculate the pressure at the interfaces of coated plate subjected to underwater weak shock wave. The method is carried out to give analytical results by summing up the pressure increment, which can be calculated analytically, in time sequence. The results are in very good agreement with the finite element (FE) predictions for the coating case and Taylor’s results for the noncoating case, which validate the method that is suitable for underwater weak shock problem. On the other hand, Taylor’s results for the coating case are invalid, which indicates a potential application field for the method. The extension of the analytical method to qlayer systems and dissipation case is also outlined.
Coated plates are widely used in the panels of naval vessels in order to make better protection for shock input such as underwater explosion (UNDEX). Therefore, systematic understanding of wave propagation mechanism in multilayered mediums is of great significance in guiding us to a better design for the coatings.
A lot of researches have been performed to study the response of the plane plates that interacted with underwater shock wave. Taylor [
Analytical solutions of transient wave propagation in multilayered mediums have a series of results in the literature. For onedimensional problem of plane wave propagation in the direction normal to the layered medium, Sun et al. [
Because of the difficulties mentioned above, the problems were always solved numerically in time domain. Desceliers et al. [
From the above introduction, no analytical method can consider coated plates subjected to underwater shock wave. The primary objective of this paper is to propose an analytical method to calculate the pressure at the interfaces of multilayered mediums, which is suitable to solve the UNDEX problems. FE analysis is applied to validate the method.
The remainder of this paper is organized as follows. Section
The underwater pressure wave will generate a series of reflection and transmission waves when the wave propagates in the multilayered mediums. As the number of reflection and transmission waves increases, the problem becomes very complicated. Rather than considering the specific reflection and transmission propagation process in time sequence, the method determines the pressure history by accumulating the pressure increments at the interfaces at a specific time, which can be calculated analytically, in time sequence.
In this paper, wave propagation process in the water, coating, and steel plate 3medium system is assumed to be a onedimensional problem as shown in Figure
pressure wave propagates as longitudinal wave in the medium;
each medium is homogeneous with fixed linear impedance;
rigid body motion is negligible;
the varying of the wave propagation time in each medium due to the medium deformation is negligible.
Simplified model.
The formulas derived below are all based on the simplified model shown in Figure
This implies that for any function
Define that the propagation time of wave propagation in the coating for one time is
Cases for wave propagation in the multilayered mediums: (a) the wave propagating in medium 2 for one time and medium 3 for one time, (b) the wave transmitting 0 time into medium 3, (c) the wave transmitting 1 time from medium 2 into medium 3, and (d) the wave transmitting 2 times from medium 2 into medium 3.
At a specific time, there must be a pressure wave transmitting from the coating into the water. Assuming that the time that the pressure wave has propagated,
The pressure increment at a specific time
After the initial time (
If
The wave that transmits one time from medium 2 into medium 3 is presented as
The wave that transmits two times from medium 2 into medium 3 is presented as
More generally, the wave that transmits
To sum up, the pressure increment at boundary I at the specific time
The basic idea to calculate the pressure increment at boundary II is the same as that at boundary I. It becomes more complicated here, since the direction of the wave coming to boundary II for the last time can be different. That wave comes to boundary II can be from medium 2 or medium 3 so that the case should be separated into two parts.
The specific time of the pressure increment at boundary II is a little different from that at boundary I which is expressed as
Assuming that
First, the wave that arrives at boundary II from medium 2 at the specific time
If
If
Then, the wave that arrives at boundary II from medium 3 at the specific time
If
If
In conclusion, the wave that arrives at boundary II from medium 2 at the specific time
The procedure to calculate the pressure increment at boundary III is similar to that has been introduced to calculate the pressure increment at boundary I.
The specific time is defined by
Assuming that
To sum up, the pressure increment at boundary III at the specific time
The method to calculate the pressure increment at the specific time has been introduced above. After the specific time and the corresponding pressure increment
If the input is the exponentially decaying wave, then
Assuming that
In this section, the solutions with dissipation are derived.
Considering the damping term, the wave equation is modified as
Assuming that the equation has the harmonic wave solution with the form
It can be proved that the wave equation (
The difference between the solution of the wave equation with and without dissipation lies that the decaying term which is frequency dependent is added and the wave velocity turns to be frequency dependent in the solution with dissipation. It implies that despite the varying wave shapes for an arbitrary wave input, the wave shape keeps unchanged for a harmonic input except for the decay of the wave crest. In order to obtain the solutions of the wave propagation in the multilayered mediums, the relation between the acoustic pressure
Therefore, the solutions of the wave propagation in the multilayered mediums with dissipation under harmonic input are easily obtained if the wave velocity, the decay of wave crest, and the specific acoustic impedance are modified by (
For the exponentially decaying wave, the cosine Fourier expansion is presented as
In the complex exponential form, the Fourier expansion of the exponential decay wave becomes
Then, the analytical solutions of the wave propagation in the mediums with dissipation will be derived in the way similar to those without dissipation, except that the solutions with dissipation are the superposition of the Fourier components.
The extension of the analytical method to the
In this section, calculation examples is presented, and comparison between analytical results and FE predictions is conducted to validate the analytical method. The analytical results are achieved by using MATLAB, and FE analysis is performed with FE analysis software ABAQUAS. Medium properties are presented in Table
Physical and geometrical parameters of the mediums.
Medium  Material properties  

Density 
Wave velocity 
Specific acoustic impedance 
Thickness 

(1) Water  998  1,483  1,480,034  — 
(2) Coating  1,200  1,570  1,884,000  0.05 
(3) Steel plate  7,800  5,050  39,390,000  0.024 
The model is subjected to an exponentially decaying wave input with a decay time of 0.6 ms and a peak pressure of 1 Pa. (Here 1 Pa is used for simplicity. However, it is enough for weak underwater shock wave. The reasons will be given in Section
Onedimensional system subjected to exponentially decaying wave in ABAQUS.
Analytical calculation examples are presented in this subsection to give a general idea of analytical results when the plate is coated.
Using the medium properties presented in Table
Analytical results with (a) rigid boundary, (b) airbacked boundary, (c) waterbacked boundary: —: watercoating surface,  : coatingsteel surface, and …: boundary III.
To validate the analytical method, comparisons between the analytical results and FE predictions with the varying boundaries at the different interfaces are conducted. The medium parameters are taken from Table
Comparisons between the analytical results and FE predictions with rigid boundary at (a) the watercoating surface, (b) the coatingsteel surface, and (c) boundary III: —: analytical results and  : FE predictions.
Comparisons between the analytical results and FE predictions with airbacked boundary at (a) the watercoating surface, (b) the coatingsteel surface, and (c) boundary III: —: analytical results and  : FE predictions.
Comparisons between analytical results and FE predictions with waterbacked boundary at (a) the watercoating surface, (b) the coatingsteel surface, and (c) boundary III: —: analytical results and  : FE predictions.
Comparisons between analytical results and FE predictions with airbacked boundary at the watercoating surface, where the thickness of the steel plate changes to 1mm: —: analytical results and  : FE predictions.
Comparisons between analytical results and FE predictions with waterbacked boundary at the watercoating surface, where the wave velocity of medium changes to 570 m/s: —: analytical results and  ; FE predictions.
As seen in Figures
Figure
Figure
To sum up, one can conclude that the comparisons validate the analytical method, since the analytical results agree reasonably well with the FE predictions for all boundaries and interfaces for different medium properties. FE analysis can hardly give perfect predictions under some extreme conditions, while analytical method has no such problem.
The analytical method takes multiple reflections and transmissions into account. However, someone may ask whether the approximate method that only considers several reflections and transmissions at the beginning is enough to obtain reasonable results. In this section, comparison between the analytical results and the approximate predictions at the watercoating surface is presented.
Comparisons at the watercoating surface with rigid, airbacked, and waterbacked boundary are investigated in Figure
The approximate predictions that only consider the reflection at boundary I underestimate the results in comparison to the analytical results.
The approximate predictions that consider the reflection at boundary II only one time agree well with the analytical results with rigid boundary, but the approximate predictions cannot reveal the oscillation effect of the pressures caused by multiple reflections and transmissions. The approximate predictions that consider the reflection at boundary II only one time are in conflict with the analytical results with airbacked boundary and waterbacked boundary. The decaying rate of the case that considers multiple reflections and transmissions is much larger than the case that considers the reflection at boundary II only one time.
Comparisons between analytical results and approximate predictions at the watercoating surface with (a) rigid boundary, (b) airbacked boundary, and (c) waterbacked boundary: —: only consider the reflection at boundary I, …: consider the reflection at boundary II only one time, and  : consider that the wave arrives at boundary I for
The specific examples show that the approximate method that only considers several reflections and transmissions with neglecting the effect of subsequent ones is incorrect. To obtain the correct result, multiple reflections and transmissions should be considered.
The analytical method is applied to solve the underwater weak shock problem as Taylor [
To apply the method to underwater weak shock problem, the restrictions of the method must be checked first. The shock wave applied to the coating and the steel plate must be weak enough to guarantee that the medium property is still linear, and the deformation of the mediums is negligible. Since the time span of the incident exponentially decaying wave is always short, the rigid motion can also be ignored.
In this section, the comparisons between Taylor’s results and the analytical results are presented. Two cases will be investigated here. Firstly, the noncoating case where the material of medium 2 and medium 3 that is steel for both of them is conducted. Secondly, the coating case, where the material of medium 2 and medium 3 used is listed in Table
Since the analytical results and Taylor’s results both behave linearly with respect to the incident peak pressure, the results can be normalized by dividing the incident peak pressure. Therefore, if the analytical results agree well with Taylor’s results for the noncoating case, one can say the method can be applied to weak underwater shock problem, which is independent of the peak pressure employed. This explains the choice of 1 Pa in Section
Figure
Comparisons between analytical results and Taylor’s results at the watercoating surface with airbacked boundary, where the material of medium 2 and medium 3 is both steel: —: analytical results and  : Taylor’s results.
Though Taylor’s results are no longer valid for the coating case, revised results based on Taylor’s method are still obtained by only modifying the mass when the plate is coated. Using the material of medium 2 and medium 3 listed in Table
Comparisons between analytical results and Taylor’s results at the watercoating surface with airbacked boundary, where the material of medium 2 and medium 3 used is listed in Table
An analytical method to calculate the pressure at the interfaces of coated plate subjected to underwater shock wave is obtained in this paper based on the wave theory. The analytical results are in good agreement with the FE predictions for the coating case and Taylor’s results for the noncoating case, which indicates that the method can be used for solving underwater weak shock problem. The failure predictions of Taylor’s method for the coating case show a potential application field of the analytical method. The approximate predictions that only consider several reflections and transmissions are also compared with analytical results. The comparisons show that the results cannot be obtained by simple approximation and multiple reflections and transmissions should be taken into account.
The analytical method is not restricted to two layers and could be executed for
Wave velocity (m s^{−1})
Bulk modulus of medium (N m^{−2})
Pressure increment (Pa)
Arbitrary functions (Pa)
Thickness of medium (m)
Impulse (Pa s)
Pressure (Pa)
Reflection coefficient
Specific acoustic impedance (kg s m^{−2})
Transmission coefficient
Time (s)
Time length (s)
Energy flow per unit area (J m^{−2})
Acoustic absorption coefficient (m^{−1})
Damping of medium (N s m^{−4})
Decay time (s)
Density of medium (kg m^{−3})
Particle displacement (m).
Quantity related to the times of transmission.
Amplitude
Normalized
Input
Specific
Quantity related to damping
Quantity related to boundary
Quantity related to medium
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors gratefully acknowledge the support for this work by the National Natural Science Foundation of China (NSFC) under Grant no. 11172173 and no. 11272215.