# What Really Caused the ROKS Cheonan Warship Sinking?

Authors: Hwang Su Kim; Mauro Caresta
Journal: Advances in Acoustics and Vibration
 (2014)

---

## Abstract

This paper is concerned with the sinking of the Korean naval warship (ROKS Cheonan) and the reported spectra of the seismic signals recorded at the time of the incident. The spectra of seismic signals show prominently amplitude peaks at around 8.5 Hz and its harmonics. These frequencies were explained with the vibrations of a water column due to an underwater explosion. This explanation is highly doubtful and concerns about its validity have already been raised in the scientific community. In this work an alternative explanation is presented: it is shown that the recorded seismic spectra are consistent with the natural frequencies of vibrations of a large submarine with a length of around 113 m. This finding raises the possibility that the ROKS Cheonan sunk because of the collision with a large submarine rather than the explosion of a torpedo or an underwater mine.


---

## Body

## 1. Introduction

Several years have passed since the incident of the Korean naval warship (ROKS Cheonan) sinking that occurred on the 26th of March 2010 [1, 2]. The Cheonan warship was split largely into two parts as shown in Figure 1 and sank off Korea’s west coast near Baekryeong Island in the Yellow Sea. The seismic signals were recorded at the time of the incident in many stations including Baekryeong Island. The analysis of the seismic signals followed soon after the incident as reported in [3]. Based on the official government report of a summary of its investigation on 20 May 2010, the warship was sunk by a North Korean torpedo fired by a midget submarine (JIG’s report [1] printed in September 2010). Nevertheless controversy over the scientific legitimacy of the report has been aroused among scientists [4–8]. The doubts raised by each of the reference are summarized in Appendix A.(a) ROKS warship; (b) top view. Sketch of the part damaged during the incident.
(a)
(b)This work was motivated from the report of [3, 9] showing the spectra of the seismic signals generated from the Cheonan incident: the spectral Figure 7 in [3] shows very interesting spectral peaks at around 8.5 Hz and its harmonics. The author in [3] has explained these characteristic frequencies as those of the hydroacoustic reverberation waves generated by an underwater explosion. This explanation is based on the theory in the report of [10] that underwater explosion generates reverberation waves in the column of seawater with characteristic odd harmonic series of a fundamental frequency by f
1
=
c
/
4
H, where c is the wave speed in water (~1.5 km/s) and H is the water column depth (for f
1
=
8
.
5 Hz, H
=
4
4 m). However contrary to this prediction, Figure 7 in [3] clearly shows a series including 2
f
0
≈
17
.
7 Hz and 4
f
0
≈
3
4
.
4 Hz, which is not an odd harmonic series. Thus it is very questionable for the validity of the analysis result in [3]. Recently in another paper based on the same underwater explosion theory as in [9] where the explosive source was indicated as a South Korean land control mine rather than a North Korean torpedo this problem was recognized and the authors stated that“these unusual unexplained spectral peaks at 17 Hz and its second harmonic frequency at 34 Hz can be attributed to the nonlinear deformation of the hull structure by shock waves along with overlapping readings from later arriving T-phases.”This statement in our opinion is a speculation and no proof or simulations to support it were shown in [9]; besides the corresponding amplitude matching of the recorded spectra had not been attempted in [3, 9].We think that it is worthwhile to search an alternative theory to explain more properly the spectra including amplitudes of the seismic signals. For instance, it will be shown in this work that a possible explanation for the sinking of the ROKS Cheonan warship is a collision with a large submarine. The reason for this explanation relies on the fact the frequencies observed in the spectra of the seismic data are consistent with the natural frequencies of vibrations of a large submarine, as it will be shown later in the paper. The collision of the ship with the submarine would generate a large force transmitted to the submarine hull with the consequent structure born sound radiated in the sea-water. The sound waves then transmit into the earth crust as seismic waves and can be recorded at seismic detecting stations. The purpose of this paper is to explore this possibility to gain insight into the reality of the cause of the Cheonan sinking. For this aim both a simplified and more accurate model of a submarine hull will be presented in the following sections. A simplified model will be used first to find the main dimension of the submarine hull; a more realistic model will be then used to properly match the seismic signature.


## 2. Theoretical Formulation and Calculation

### 2.1. Natural Vibrations of a Submarine Hull

A model of a submarine is presented according to the approach of [11, 12], where the submarine was modeled as having a main cylindrical hull with internal bulkheads and ring stiffeners. The cylindrical shell is closed by truncated conical shells, which are closed by circular plates at each end as shown in Figure 2(a). Hereafter we call this the realistic model of a submarine; however we also consider a simplified version, as shown in Figure 2(b), where the hull is modeled as a cylindrical shell with ring stiffeners closed by circular plates at ends of the shell. This is because the effect of the end cones on the structural response is minimal [11] and the purpose of using the simplified model is to get a closed form solution for the main parameters of the hull and get a first matching with the recorded seismic frequencies. The detailed model as in [12] will be used later to calculate a more realistic structural and acoustic response of the submarine.Figure 2
(a) Schematic diagram of the more realistic model of a submarine for the free-free cylindrical shell with bulkheads, stiffeners, and closed by plates following conical end caps. (b) Cylindrical coordinate system (z, θ, and r) and the corresponding displacements (u, ν, and w) for a thin walled shell. The figure indicates the simplified model of a submarine for the cylindrical shell, stiffeners, and closed by plates at ends of the shell.At first, let us consider a cylindrical shell of lengthL with shear diaphragm boundary conditions to model the main part of the hull (Figure 2(b)). The equations of motion and the solutions and the boundary condition equation are given in Appendix B. The solution of the free vibration of the cylindrical shell satisfying the Flügge differential equations of motion can be written as given by Caresta et al. [12, 13]:
(1)
u
n
z
,
θ
,
t
=
U
n
cos
⁡
n
θ
cos
⁡
k
z
e
-
j
w
t
,
v
n
z
,
θ
,
t
=
V
n
sin
n
θ
sin
k
z
e
-
j
w
t
,
w
n
z
,
θ
,
t
=
W
n
cos
⁡
n
θ
sin
k
z
e
-
j
w
t
.
Here u
(
n
), v
(
n
), and w
(
n
) are the orthogonal components of shell displacements in the z-axial, the θ-circumferential, and the r-radial directions, respectively. n is the circumferential mode number (n
=
0, 1, 2…), j is the imaginary unit, ω is the angular frequency, and k is the axial wave number. The shear diaphragm boundary conditions are given by
(2)
v
n
z
,
θ
,
t
=
0
,
w
n
z
,
θ
,
t
=
0
,
N
z
=
M
z
=
0
,
z
=
0
,
L
,
where N
z is the membrane force and the M
z is the bending moment. With these boundary conditions, the axial wave number k is given by
(3)
k
m
=
m
π
L
,
m
=
0,1
,
2
…
.
Substituting (1) and (3) into the Flügge equations gives the eigenvalue equation in the matrix form
(4)
L
11
-
Ω
2
L
12
L
13
L
21
L
22
-
Ω
2
L
23
L
31
L
32
L
33
-
Ω
2
U
(
n
)
V
(
n
)
W
(
n
)
=
0
.
In (4)   Ω is the dimensionless frequency parameter given by Ω
=
ω
γ
a
/
c
L. The elements of the matrix are given by
(5a)
L
11
=
k
a
2
+
(
1
-
υ
)
2
n
2
(
1
+
β
2
)
(5b)
L
12
=
-
(
1
+
υ
)
2
n
k
a
(5c)
L
13
=
-
υ
k
a
-
β
2
k
a
3
-
1
-
υ
2
n
2
k
a
(5d)
L
21
=
L
12
(5e)
L
22
=
(
1
-
υ
)
2
k
a
2
(
1
+
3
β
2
)
+
n
2
(
1
+
μ
)
(5f)
L
23
=
n
(
1
+
μ
+
χ
)
-
χ
n
3
+
β
2
(
3
-
υ
)
2
n
k
a
2
(5g)
L
31
=
L
13
(5h)
L
32
=
L
23
(5i)
L
33
=
(
1
+
μ
+
2
χ
)
-
2
χ
n
2
+
β
2
k
a
2
+
n
2
2
+
1
-
2
n
2
,
where
(5j)
Ω
2
=
γ
a
2
c
L
2
ω
2
,
γ
=
1
+
A
h
b
+
m
eq
ρ
h
,
c
L
2
=
E
ρ
(
1
-
υ
2
)
,
β
2
=
h
2
12
a
2
.
μ
=
(
1
-
υ
2
)
A
h
b
,
χ
=
(
1
-
υ
2
)
A
z
e
h
b
a
.
In the previous equations, a is the mean radius of the cylindrical shell, h is the shell thickness, A is the cross-section area of the ring stiffener, b is the stiffener spacing, and z
e is the distance between the shell mid surface and the center of a ring. E, ρ, and υ are Yong’s modulus, density, and Poisson’s ratio of the shell, respectively. c
L is the longitudinal wave speed in the shell and m
eq is the equivalent distributed mass on the shell to take into account the onboard equipment and ballast tanks. For the each set of (n, m) the matrix equation of (4) gives three real eigenvalues of Ω
2 and the corresponding eigenvectors (U
(
n
), V
(
n
), and W
(
n
)) under the usual normalization condition U
(
n
)
2
+
V
(
n
)
2
+
W
(
n
)
2
=
1.


### 2.2. Axisymmetric Modes of Vibrations

Ifn
=
0, the wave modes are independent of θ and if we do not consider the torsional modes, we can set v
(
n
)
(
z
,
θ
,
t
)
=
0. The matrix equation (4) simplifies in a 2
×
2 matrix equation:
(6a)
L
11
-
Ω
2
L
12
L
21
L
22
-
Ω
2
U
(
n
)
W
(
n
)
=
0
,
where
(6b)
L
11
=
k
a
2
,
L
12
=
-
υ
k
a
-
β
2
k
a
3
(6c)
L
21
=
L
12
,
L
22
=
(
1
+
μ
+
2
χ
)
+
β
2
{
k
a
4
+
1
}
.
Under the condition, (
k
a
)
<
1, the off-diagonal elements become small values and then the solutions of (6a) approximately give for the nonzero of (
U
n
,
V
(
n
)
)
(7)
L
11
-
Ω
2
≈
0
,
L
22
-
Ω
2
≈
0
.
The first equation (7) with (3) and the first relation in (5j) and L
11
=
(
k
a
)
2in (6b) give
(8)
f
m
(
0
)
≈
m
2
L
c
L
γ
,
where
m
=
1,2
…
.
It is very important to note that (8) is a harmonic series with a fundamental frequency of f
1
(
0
), which can be used to match the characteristic frequencies of the seismic signals reported in [3, 9]. The validity of (8) in the approximation can be seen up to the wave number m
=
8 from the solution of (6a). The first 4 frequencies and the corresponding normalized eigenvectors of (6a) are listed in Table 1. Using (8) and (5j), we can obtain analytically the length L of a submarine giving f
1
(
0
)
=
8
.
5 Hz as to match the recorded seismic signature. The submarine hull is assumed to be made of steel with physical properties of ρ
≈
7800 kg/m
3, E
≈
2.1
×
10
11 N/m
2, and υ
≈
0.3; c
L
=
5
.
450 km/s. It is widely known that the typical dimension of submarines has the ratio of 2
a
/
L
= ~0.1 and h
/
a
=  ~0.01. In γ
=
(
1
+
A
/
h
b
+
m
eq
/
ρ
h
), A
=
0.08 m
×
0.15
m and b
=
0.5
m can be used as in [12]. The stiffeners parameters A and b give minor contribution in γ so it is not necessary to know accurate values. m
eq is supposed to be less than the value estimated by the relation m
eq
=
ρ
f
V
s
/
S
s where V
s and S
s are the volume and the surface area of submarine, respectively. ρ
f is the density of seawater, about 1025 kg/m3. If V
s
=
π
a
2
L and S
s
=
2
π
a
L, m
eq
=
ρ
f
a
/
2. Inserting these values in (8) gives L
=
113
m. This value is the typical length of a large submarine. In summary we have L
=
113
m, a
=
5.65
m (radius of the cylindrical model), h
=
0.0565
m (thickness of the hull), and m
eq
=
2896 kg/m2 giving √
γ
=
2.84. This model of submarine according to (8) has the natural frequencies as a harmonic series of 8.5 Hz. In this result the equivalent mass is found to be m
eq
=
2896 kg/m2 and it seems a little too high. For a submarine with a length of 113 m, considering about 400 tons added at the ends to simulate the ballast tanks a more realistic value should be around m
eq
= ~2150 kg/m2. If this distributed mass is used, √
γ
=
2.51 and f
1
(
0
) becomes 9.6 Hz. However by considering the realistic model of the submarine with the conical end caps and the fluid loading with m
eq
=  ~2150 kg/m2, the match of the frequencies is still achieved, as can be seen in Table 1. Therefore we will keep using the simplified model of a submarine in vacuum with m
eq
=
2896 kg/m2 for a quick guide. It has been noted that the adjusting dimensional parameters of length L, radius a, hull thickness h, distributed mass m
eq, and so forth within 10% can still give harmonics of 8.5 Hz for the natural frequencies vibrations of the models. This result means the range of the uncertainty of the dimension of this model structure is less than 10%. The solution of the second equation in (7) for the second branch of the eigenvalues gives under the condition of small (
k
a
) and small value of β
2
~
10
-
5
(9)
Ω
2
≈
(
1
+
μ
+
2
χ
)
.
Here μ
≈
χ
=
0.387 and we get f
=
81.6 Hz. The accurate calculation of the corresponding frequencies gives 81.6 Hz (m
=
1), 81.7 Hz (m
=
2), 81.8 (m
=
3), and 85.0 Hz. Since these frequencies are greater than the cut-off frequency f
c
=
40 Hz of the seismic signals reported for this event, we will not consider this branch of frequencies any more.Table 1
Comparison of the natural frequency of the simplified model structure (m
eq
=
2
896 kg/m2) with the resonant frequency of the realistic model structure (m
eq
=
2150 kg/m2) for the n
=
0 mode and the n
=
1 mode vibrations. L
=
113 m. (U
,
W) for n
=
0 and (U
,
V
,
W) for n
=
1 are the normalized eigenvectors.
(n, m)
Simplified model
Realistic model
Natural frequency (Hz) (U
,
W)
Resonant frequency (Hz)
(0, 1)
8.5 (0.999, 0.022)
8.6
(0, 2)
17.1 (0.999, 0.046)
17.4
(0, 3)
25.5 (0.997, 0.046)
25.7
(0, 4)
34.0 (0.994, 0.105)
35.5
(U
,
V
,
W)
(1, 1)
0.9 (0.104, 0.705, 0.702)
1.4
(1, 2)
3.3 (0.175, 0.701, 0.691)
3.5
(1, 3)
6.6 (0.210, 0.698, 0.684)
6.3
(1, 4)
10.4 (0.219, 0.696, 0.684)
9.3
(1, 5)
14.3 (0.212, 0.693, 0.689)
12.4
(1, 6)
18.2 (0.197, 0.686, 0.700)
14.7


### 2.3. Asymmetric Modes of Vibration

For then
=
1 bending modes the solution of Ω
2 in (4) gives three branches of eigenvalues and the corresponding three eigenvectors. The frequency for each m can be calculated from the eigenvalues of Ω
2. The lowest frequency branch represents mainly radial motion and is of interest here; the first 6 frequencies and the corresponding normalized eigenvectors are listed in Table 1. The corresponding resonant frequencies calculated with the realistic model are also listed in Table 1. The frequencies of the second and the third branches of Ω
2 are greater than the frequency range investigated and they will not be considered in this study. The vibrations with n
>
1 are out of consideration in this work because those give little contribution to the response as it will be shown later. It should be noted that in Table 1 the difference between the frequencies calculated using the simplified model with m
eq
=
2
896 kg/m
2 and the realistic model with m
eq
=
2
150 kg/m2 is small supporting the usefulness of the simplified model.


## 2.1. Natural Vibrations of a Submarine Hull

A model of a submarine is presented according to the approach of [11, 12], where the submarine was modeled as having a main cylindrical hull with internal bulkheads and ring stiffeners. The cylindrical shell is closed by truncated conical shells, which are closed by circular plates at each end as shown in Figure 2(a). Hereafter we call this the realistic model of a submarine; however we also consider a simplified version, as shown in Figure 2(b), where the hull is modeled as a cylindrical shell with ring stiffeners closed by circular plates at ends of the shell. This is because the effect of the end cones on the structural response is minimal [11] and the purpose of using the simplified model is to get a closed form solution for the main parameters of the hull and get a first matching with the recorded seismic frequencies. The detailed model as in [12] will be used later to calculate a more realistic structural and acoustic response of the submarine.Figure 2
(a) Schematic diagram of the more realistic model of a submarine for the free-free cylindrical shell with bulkheads, stiffeners, and closed by plates following conical end caps. (b) Cylindrical coordinate system (z, θ, and r) and the corresponding displacements (u, ν, and w) for a thin walled shell. The figure indicates the simplified model of a submarine for the cylindrical shell, stiffeners, and closed by plates at ends of the shell.At first, let us consider a cylindrical shell of lengthL with shear diaphragm boundary conditions to model the main part of the hull (Figure 2(b)). The equations of motion and the solutions and the boundary condition equation are given in Appendix B. The solution of the free vibration of the cylindrical shell satisfying the Flügge differential equations of motion can be written as given by Caresta et al. [12, 13]:
(1)
u
n
z
,
θ
,
t
=
U
n
cos
⁡
n
θ
cos
⁡
k
z
e
-
j
w
t
,
v
n
z
,
θ
,
t
=
V
n
sin
n
θ
sin
k
z
e
-
j
w
t
,
w
n
z
,
θ
,
t
=
W
n
cos
⁡
n
θ
sin
k
z
e
-
j
w
t
.
Here u
(
n
), v
(
n
), and w
(
n
) are the orthogonal components of shell displacements in the z-axial, the θ-circumferential, and the r-radial directions, respectively. n is the circumferential mode number (n
=
0, 1, 2…), j is the imaginary unit, ω is the angular frequency, and k is the axial wave number. The shear diaphragm boundary conditions are given by
(2)
v
n
z
,
θ
,
t
=
0
,
w
n
z
,
θ
,
t
=
0
,
N
z
=
M
z
=
0
,
z
=
0
,
L
,
where N
z is the membrane force and the M
z is the bending moment. With these boundary conditions, the axial wave number k is given by
(3)
k
m
=
m
π
L
,
m
=
0,1
,
2
…
.
Substituting (1) and (3) into the Flügge equations gives the eigenvalue equation in the matrix form
(4)
L
11
-
Ω
2
L
12
L
13
L
21
L
22
-
Ω
2
L
23
L
31
L
32
L
33
-
Ω
2
U
(
n
)
V
(
n
)
W
(
n
)
=
0
.
In (4)   Ω is the dimensionless frequency parameter given by Ω
=
ω
γ
a
/
c
L. The elements of the matrix are given by
(5a)
L
11
=
k
a
2
+
(
1
-
υ
)
2
n
2
(
1
+
β
2
)
(5b)
L
12
=
-
(
1
+
υ
)
2
n
k
a
(5c)
L
13
=
-
υ
k
a
-
β
2
k
a
3
-
1
-
υ
2
n
2
k
a
(5d)
L
21
=
L
12
(5e)
L
22
=
(
1
-
υ
)
2
k
a
2
(
1
+
3
β
2
)
+
n
2
(
1
+
μ
)
(5f)
L
23
=
n
(
1
+
μ
+
χ
)
-
χ
n
3
+
β
2
(
3
-
υ
)
2
n
k
a
2
(5g)
L
31
=
L
13
(5h)
L
32
=
L
23
(5i)
L
33
=
(
1
+
μ
+
2
χ
)
-
2
χ
n
2
+
β
2
k
a
2
+
n
2
2
+
1
-
2
n
2
,
where
(5j)
Ω
2
=
γ
a
2
c
L
2
ω
2
,
γ
=
1
+
A
h
b
+
m
eq
ρ
h
,
c
L
2
=
E
ρ
(
1
-
υ
2
)
,
β
2
=
h
2
12
a
2
.
μ
=
(
1
-
υ
2
)
A
h
b
,
χ
=
(
1
-
υ
2
)
A
z
e
h
b
a
.
In the previous equations, a is the mean radius of the cylindrical shell, h is the shell thickness, A is the cross-section area of the ring stiffener, b is the stiffener spacing, and z
e is the distance between the shell mid surface and the center of a ring. E, ρ, and υ are Yong’s modulus, density, and Poisson’s ratio of the shell, respectively. c
L is the longitudinal wave speed in the shell and m
eq is the equivalent distributed mass on the shell to take into account the onboard equipment and ballast tanks. For the each set of (n, m) the matrix equation of (4) gives three real eigenvalues of Ω
2 and the corresponding eigenvectors (U
(
n
), V
(
n
), and W
(
n
)) under the usual normalization condition U
(
n
)
2
+
V
(
n
)
2
+
W
(
n
)
2
=
1.


## 2.2. Axisymmetric Modes of Vibrations

Ifn
=
0, the wave modes are independent of θ and if we do not consider the torsional modes, we can set v
(
n
)
(
z
,
θ
,
t
)
=
0. The matrix equation (4) simplifies in a 2
×
2 matrix equation:
(6a)
L
11
-
Ω
2
L
12
L
21
L
22
-
Ω
2
U
(
n
)
W
(
n
)
=
0
,
where
(6b)
L
11
=
k
a
2
,
L
12
=
-
υ
k
a
-
β
2
k
a
3
(6c)
L
21
=
L
12
,
L
22
=
(
1
+
μ
+
2
χ
)
+
β
2
{
k
a
4
+
1
}
.
Under the condition, (
k
a
)
<
1, the off-diagonal elements become small values and then the solutions of (6a) approximately give for the nonzero of (
U
n
,
V
(
n
)
)
(7)
L
11
-
Ω
2
≈
0
,
L
22
-
Ω
2
≈
0
.
The first equation (7) with (3) and the first relation in (5j) and L
11
=
(
k
a
)
2in (6b) give
(8)
f
m
(
0
)
≈
m
2
L
c
L
γ
,
where
m
=
1,2
…
.
It is very important to note that (8) is a harmonic series with a fundamental frequency of f
1
(
0
), which can be used to match the characteristic frequencies of the seismic signals reported in [3, 9]. The validity of (8) in the approximation can be seen up to the wave number m
=
8 from the solution of (6a). The first 4 frequencies and the corresponding normalized eigenvectors of (6a) are listed in Table 1. Using (8) and (5j), we can obtain analytically the length L of a submarine giving f
1
(
0
)
=
8
.
5 Hz as to match the recorded seismic signature. The submarine hull is assumed to be made of steel with physical properties of ρ
≈
7800 kg/m
3, E
≈
2.1
×
10
11 N/m
2, and υ
≈
0.3; c
L
=
5
.
450 km/s. It is widely known that the typical dimension of submarines has the ratio of 2
a
/
L
= ~0.1 and h
/
a
=  ~0.01. In γ
=
(
1
+
A
/
h
b
+
m
eq
/
ρ
h
), A
=
0.08 m
×
0.15
m and b
=
0.5
m can be used as in [12]. The stiffeners parameters A and b give minor contribution in γ so it is not necessary to know accurate values. m
eq is supposed to be less than the value estimated by the relation m
eq
=
ρ
f
V
s
/
S
s where V
s and S
s are the volume and the surface area of submarine, respectively. ρ
f is the density of seawater, about 1025 kg/m3. If V
s
=
π
a
2
L and S
s
=
2
π
a
L, m
eq
=
ρ
f
a
/
2. Inserting these values in (8) gives L
=
113
m. This value is the typical length of a large submarine. In summary we have L
=
113
m, a
=
5.65
m (radius of the cylindrical model), h
=
0.0565
m (thickness of the hull), and m
eq
=
2896 kg/m2 giving √
γ
=
2.84. This model of submarine according to (8) has the natural frequencies as a harmonic series of 8.5 Hz. In this result the equivalent mass is found to be m
eq
=
2896 kg/m2 and it seems a little too high. For a submarine with a length of 113 m, considering about 400 tons added at the ends to simulate the ballast tanks a more realistic value should be around m
eq
= ~2150 kg/m2. If this distributed mass is used, √
γ
=
2.51 and f
1
(
0
) becomes 9.6 Hz. However by considering the realistic model of the submarine with the conical end caps and the fluid loading with m
eq
=  ~2150 kg/m2, the match of the frequencies is still achieved, as can be seen in Table 1. Therefore we will keep using the simplified model of a submarine in vacuum with m
eq
=
2896 kg/m2 for a quick guide. It has been noted that the adjusting dimensional parameters of length L, radius a, hull thickness h, distributed mass m
eq, and so forth within 10% can still give harmonics of 8.5 Hz for the natural frequencies vibrations of the models. This result means the range of the uncertainty of the dimension of this model structure is less than 10%. The solution of the second equation in (7) for the second branch of the eigenvalues gives under the condition of small (
k
a
) and small value of β
2
~
10
-
5
(9)
Ω
2
≈
(
1
+
μ
+
2
χ
)
.
Here μ
≈
χ
=
0.387 and we get f
=
81.6 Hz. The accurate calculation of the corresponding frequencies gives 81.6 Hz (m
=
1), 81.7 Hz (m
=
2), 81.8 (m
=
3), and 85.0 Hz. Since these frequencies are greater than the cut-off frequency f
c
=
40 Hz of the seismic signals reported for this event, we will not consider this branch of frequencies any more.Table 1
Comparison of the natural frequency of the simplified model structure (m
eq
=
2
896 kg/m2) with the resonant frequency of the realistic model structure (m
eq
=
2150 kg/m2) for the n
=
0 mode and the n
=
1 mode vibrations. L
=
113 m. (U
,
W) for n
=
0 and (U
,
V
,
W) for n
=
1 are the normalized eigenvectors.
(n, m)
Simplified model
Realistic model
Natural frequency (Hz) (U
,
W)
Resonant frequency (Hz)
(0, 1)
8.5 (0.999, 0.022)
8.6
(0, 2)
17.1 (0.999, 0.046)
17.4
(0, 3)
25.5 (0.997, 0.046)
25.7
(0, 4)
34.0 (0.994, 0.105)
35.5
(U
,
V
,
W)
(1, 1)
0.9 (0.104, 0.705, 0.702)
1.4
(1, 2)
3.3 (0.175, 0.701, 0.691)
3.5
(1, 3)
6.6 (0.210, 0.698, 0.684)
6.3
(1, 4)
10.4 (0.219, 0.696, 0.684)
9.3
(1, 5)
14.3 (0.212, 0.693, 0.689)
12.4
(1, 6)
18.2 (0.197, 0.686, 0.700)
14.7


## 2.3. Asymmetric Modes of Vibration

For then
=
1 bending modes the solution of Ω
2 in (4) gives three branches of eigenvalues and the corresponding three eigenvectors. The frequency for each m can be calculated from the eigenvalues of Ω
2. The lowest frequency branch represents mainly radial motion and is of interest here; the first 6 frequencies and the corresponding normalized eigenvectors are listed in Table 1. The corresponding resonant frequencies calculated with the realistic model are also listed in Table 1. The frequencies of the second and the third branches of Ω
2 are greater than the frequency range investigated and they will not be considered in this study. The vibrations with n
>
1 are out of consideration in this work because those give little contribution to the response as it will be shown later. It should be noted that in Table 1 the difference between the frequencies calculated using the simplified model with m
eq
=
2
896 kg/m
2 and the realistic model with m
eq
=
2
150 kg/m2 is small supporting the usefulness of the simplified model.


## 3. Structural and Acoustic Responses

### 3.1. Structural Response

Figures3 and 4 show the structural frequency response function (FRF) of the complete submarine model for the axial and radial displacement at the driving point, as applying a point unit force at one end of the cylindrical hull in axial direction. For the reciprocity principle the FRFs shown can be also interpreted as the response for the radial and axial displacement in response to a radial force applied at the same point. The calculations in these plots were carried out using the modeling approach presented in [12]. The contribution of the n
=
0 and n
=
1 modes is also shown in the same figure; modes with n
>
1 were excluded by the simulation since it did not give a significant contribution to the response. It has to be noted that those results are for a point force, in case the excitation force is well distributed around the hull, such as in the case of a front collision, the contribution on the response would be given only by the n
=
0 modes. The natural frequencies corresponding to the sharp peaks in the FRF are summarized in Table 1.Figure 3
Frequency response function for an axial force applied to the submarine end.Figure 4
Frequency response function for a radial force applied to the submarine end.The striking feature in Figure3 is the resonant frequencies of the n
=
0 mode, being around 8.6 Hz, 17.0 Hz, 25.0 Hz, and 36.0 Hz. The contribution of the n
=
1 modes is also important and an increase in response is observed at around 35.0 Hz where the waves of Class II (the second branch of W2) cut on, as it was also observed in [12]. These results show a qualitative similarity between the seismic signature and the natural frequencies of a large submarine, validating the possibility of a collision discussed before. The result also suggest a collision with the front part of the submarines with a main excitation of the axisymmetric (n
=
0) modes of vibration. It can be seen in Figure 4 that the contribution of n
=
0 to the radial displacement is much lower than the axial displacements as shown in Figure 3. This aspect is consisting with the fact of low ratios of W
/
U in the eigenvectors (
U, W
) as in Table 1. Also it can be noted that the n
=
1 modes have a mainly radial displacement. This aspect can be expected by noting the ratios of W
/
U in the eigenvectors (
U, V, and W
) in Table 1.It has to be mentioned that the purpose of this calculation is to show the FRF of the submarine model and its natural frequencies, and a possible qualitative correlation with the seismic data recorded. In reality the exact impact point on the hull, the actual force transmitted to the submarine, and its frequency content would surely affect the quantitative results but an exact reconstruction of the impact it is surely a huge and uncertain task and is out of the scopes of this work.


### 3.2. Radiated Sound at Far Field

The sound pressure radiated at far field was also calculated using the model in [12] and the corresponding wave displacement is shown in Figure 5 normalized as a displacement per unit force and per km. The far field location is chosen to be in front of the submarine as a standard point for discussion. The trend of the figure is not much different from the seismic spectra reported in [3, 9] as discussed in Section 4. However this matching does not mean that the front of the submarine was toward to the BAR station, where Korea Meteorological Administration Station (37.9771N, 124.7142E) in Baekryong Island is [9] from where the seismic data analyzed in [3, 9] was obtained. The reason is that the waves from the vibration source of the conical shells of the submarine will propagate spherically in the far field through mediums. The path of this spherically propagated waves arrived at the BAR station was suggested in Figure 6 of [3] (see Figure 6 in this paper).Figure 5
Far field wave displacement for axial force acting on the submarine end.Figure 6
s: The source of the vibration of signals; RS: the recording system at BAR station. Table 4 of [3] lists that the sound velocity in sea water is 1.5 km/s; the velocities of p-wave in crust and mantle are 6.3 km/s and 7.95 km/s, respectively, while those of s-wave are 3.50 km/s and 4.41 km/s. The densities of crust and mantle are 2.58 kg/m3 and 3.30 kg/m3, respectively.


## 3.1. Structural Response

Figures3 and 4 show the structural frequency response function (FRF) of the complete submarine model for the axial and radial displacement at the driving point, as applying a point unit force at one end of the cylindrical hull in axial direction. For the reciprocity principle the FRFs shown can be also interpreted as the response for the radial and axial displacement in response to a radial force applied at the same point. The calculations in these plots were carried out using the modeling approach presented in [12]. The contribution of the n
=
0 and n
=
1 modes is also shown in the same figure; modes with n
>
1 were excluded by the simulation since it did not give a significant contribution to the response. It has to be noted that those results are for a point force, in case the excitation force is well distributed around the hull, such as in the case of a front collision, the contribution on the response would be given only by the n
=
0 modes. The natural frequencies corresponding to the sharp peaks in the FRF are summarized in Table 1.Figure 3
Frequency response function for an axial force applied to the submarine end.Figure 4
Frequency response function for a radial force applied to the submarine end.The striking feature in Figure3 is the resonant frequencies of the n
=
0 mode, being around 8.6 Hz, 17.0 Hz, 25.0 Hz, and 36.0 Hz. The contribution of the n
=
1 modes is also important and an increase in response is observed at around 35.0 Hz where the waves of Class II (the second branch of W2) cut on, as it was also observed in [12]. These results show a qualitative similarity between the seismic signature and the natural frequencies of a large submarine, validating the possibility of a collision discussed before. The result also suggest a collision with the front part of the submarines with a main excitation of the axisymmetric (n
=
0) modes of vibration. It can be seen in Figure 4 that the contribution of n
=
0 to the radial displacement is much lower than the axial displacements as shown in Figure 3. This aspect is consisting with the fact of low ratios of W
/
U in the eigenvectors (
U, W
) as in Table 1. Also it can be noted that the n
=
1 modes have a mainly radial displacement. This aspect can be expected by noting the ratios of W
/
U in the eigenvectors (
U, V, and W
) in Table 1.It has to be mentioned that the purpose of this calculation is to show the FRF of the submarine model and its natural frequencies, and a possible qualitative correlation with the seismic data recorded. In reality the exact impact point on the hull, the actual force transmitted to the submarine, and its frequency content would surely affect the quantitative results but an exact reconstruction of the impact it is surely a huge and uncertain task and is out of the scopes of this work.


## 3.2. Radiated Sound at Far Field

The sound pressure radiated at far field was also calculated using the model in [12] and the corresponding wave displacement is shown in Figure 5 normalized as a displacement per unit force and per km. The far field location is chosen to be in front of the submarine as a standard point for discussion. The trend of the figure is not much different from the seismic spectra reported in [3, 9] as discussed in Section 4. However this matching does not mean that the front of the submarine was toward to the BAR station, where Korea Meteorological Administration Station (37.9771N, 124.7142E) in Baekryong Island is [9] from where the seismic data analyzed in [3, 9] was obtained. The reason is that the waves from the vibration source of the conical shells of the submarine will propagate spherically in the far field through mediums. The path of this spherically propagated waves arrived at the BAR station was suggested in Figure 6 of [3] (see Figure 6 in this paper).Figure 5
Far field wave displacement for axial force acting on the submarine end.Figure 6
s: The source of the vibration of signals; RS: the recording system at BAR station. Table 4 of [3] lists that the sound velocity in sea water is 1.5 km/s; the velocities of p-wave in crust and mantle are 6.3 km/s and 7.95 km/s, respectively, while those of s-wave are 3.50 km/s and 4.41 km/s. The densities of crust and mantle are 2.58 kg/m3 and 3.30 kg/m3, respectively.


## 4. Discussion

Figure 7 of [3] showed the frequency spectrum of the p-wave signals in Z-direction which was the Fourier transform of the first arrived p-waves at the BAR recording system in one second time window. Also Figure 3(b) of [9] showed the normalized spectra of seismic signals in a time window of 15 seconds for each Z, EW (east-west direction) and NS (north-south direction), separately using the same BAR recordings of the seismic signals. In this case the analyzed signals contain not only the first arrived p-wave, but the later arrived s-wave, Rayleigh, Love waves, and so forth. The Fourier transforms of all these waves gives the spectrum which is the sum of the spectrum of each signal travelled via several independent paths. As expected the spectrum in each direction of Z, EW, and NS in Figure 3 of [9] showed similar patterns to each other. For the comparison of the spectrum in Figure 5 with these reported spectra, the data of the spectra peaks in Figure 7 of [3] and in Figure 3(b) of [9] are summarized in Tables 2 and 3, respectively. The frequencies of the peaks of Figure 5 are compared with the peaks of the reported spectra of Figure 7 of [3] in Table 2 and with the peaks of Figure 3(b) of [9] in Table 3.Table 2
The amplitude data of the peaks in the frequency spectrum reported in Figure7 of [3]. A-column is for meter unit scale, B-column is for nm unit, C-column is for the relative amplitudes divided by 8.5 Hz amplitude, and D-column is for the relative amplitudes of pressure converted from C-column displacement amplitudes.
f (Hz)
A (m)
B (nm)
C
D
Cal.*
8.5
10−8.22
6.0
1.0
1.0+
1.0
17.7
10−8.88
1.3
0.22
0.45
0.23
26.0
10−9.40
0.4
0.07
0.20
0.04
34.6
10−10.17
0.07
0.01
0.05
0.016
+5.2 Pa. *Cal.: the estimated peak ratios from Figure 5, which are comparable with those in C-column.Table 3
The amplitude data of the peaks in the normalized frequency spectra forZ, EW, and NS seismic signals reported in Figure 3(b) of [9].
f (Hz)
Z
EW
NS
Av.*
8.5
1.0
1.0
1.0
1.0
17.0
0.30
0.38
0.22
0.30
25.5
0.13
0.17
0.10
0.13
34.0
0.03
0.03
0.02
0.03
*Av. is for the average of peak amplitudes in Z, EW, and NS.In [9] the original unit scale was not specified in the spectra, but supposed to be in Pascal. Thus the difference between those in D column in Table 2 and those in the average in Table 3 may indicate uncertainty of amplitudes in the recorded seismic spectra. On the other hand, the peak ratios in Figure 5 are estimated to be 1.0 for 8.6 Hz, 0.23 for 17.0 Hz, 0.04 for 25.0 Hz, and 0.016 for 36.0 Hz as listed in Cal.-column in Table 2. These ratios are close to those in Table 2: C-column, implying good fitting for both frequencies and amplitudes ratios too with the reported spectra. We also noted that the frequencies of natural vibrations of the hull frame of ROKS Cheonan were calculated and listed in 〈Table III-6-2〉 (p160) in [1], as 2.32 Hz, 4.74 Hz, 7.71 Hz, 10.41 Hz, and 13.40 Hz in the bending motion only. (No axial vibration is expected for the hull frame of the warship.) Clearly these natural frequencies could not be correlated with the characteristic harmonic frequencies (8.5 Hz and the multiples) of the seismic signals recorded.In summary the main peaks at 8.5 Hz and its multiples in the reported spectra of the seismic signals are reasonably consistent with the natural frequencies of the submarine with a length of around 113 m. This result suggests a possibility of the collision between the Cheonan warship and the submarine. Then one might wonder about the damage that the submarine would have following a collision. We believe that the submarine would be negligibly damaged by the collision since the thickness of the hull of a large submarine is supposed to be more than 6 cm with and made of high strength steels, while the hull thickness of the ROKS Cheonan is known to be about 1.2 cm with steels and aluminum alloys in the upper parts. Besides we have found that it is no difficult with this collision theory discussed in this paper to illustrate damage aspects including deformations on the recovered bow and the stern of the split Cheonan warship from the sinking.


## 5. Conclusion

This study shows that the characteristic frequencies at spectral peaks of the seismic signals recorded at the time of the Cheonan incident are consistent with the natural frequencies of vibrations of a large submarine with a length of around 113 m. The matching is particularly good with the axisymmetric (n
=
0) modes of vibrations suggesting a front collision with the submarine that might have caused the ROKS Cheonan sinking. The authors hope this work may be the starting point for new investigations to shine some light on the mysterious causes of the Cheonan sinking that caused the death of 46 people and for which a clear and definitive explanation has not been given yet.



---

*Source: 514346-2014-11-20.xml*