1. Presented at the APS/URSI 2014 Conference at Memphis, TN.
Eigenproblem for an Ocean Acoustic Waveguide with Random Depth Dependent Sound Speed
C.P. Vendhan(1), A. Datta Chowdhury (1), Saba Mudaliar(2), and S.K. Bhattacharyya(1)
(1) Ocean Engineering Department, Indian Institute of Technology Madras, Chennai, Tamilnadu, India ,
(2) Sensors Directorate, AFRL, WPFAB, Dayton, Ohio, USA
Presented at the APS/URSI 2014 Conference at Memphis, TN.
(July 6-12, 2014)

2. Objective
The main objective of the study is to develop a finite element (FE) based numerical model for acoustic wave scattering and propagation in heterogeneous waveguides with random properties.
A code employing the deterministic FE model for depth and range dependent ocean acoustic waveguides (Vendhan et al, J. Acoustical Society of America, 127(6): , 2010) has been chosen to develop a random waveguide(WG) model.
The FE model requires a suitable absorbing boundary condition at the truncation boundary in the WG model. Bayliss type boundary dampers that employ the deterministic depth eigensolution have been adopted in the above code.

3. So, there is a need to study depth eigensolution with random variation of sound speed in the depth direction. This naturally leads to the study of random eigenproblems (see for example, Nair and Keane, J. Sound and Vibration 260: 45-65, 2003; Williams, Applied Mathematics and Communications 219: , 2013). In the ocean WG context there have been a few studies with specific reference to shallow water waveguides (see for example, Tielburger et al, J. Acoustical Society of America, 101(2): , 1997; Sazantov et al, IEEE J. Ocean Engineering, 27(3): , 2002)

4. General Methodology Adopted
There have been several studies on the Stochastic Finite Element Methods(SFEM)- ( see, Stefanou, Computer Methods in Applied Mechanics and Engineering, 198: , 2009).
The perturbation method has been chosen here to formulate the SFEM for a random heterogeneous WG. This has been widely used in many applications (see for example, Vonmarcke et al, Structural Safety 3: , 1986; Klieber and Hien, The stochastic finite element method, 1992, John Wiley)
In the present study, the SFEM for heterogeneous WG would be formulated using the perturbation approach. Consistent with this, it is required to set up the depth eigensolution using the perturbation approach. This forms the main focus of this presentation. As mentioned before, the above solution would be employed at the truncation boundary of the FE model for a ocean WG problem.

5. Depth Dependent Ocean Waveguide
Depth dependence of sound speed profile is responsible for many interesting features of ocean acoustic waveguide propagation (Jensen et al, Computational ocean acoustics, Springer 2011).
Exact solution of the Helmholtz equation governing waveguide acoustics is not feasible, in general, for such problems.
The classical Rayleigh-Ritz method offers a compact method for shallow water WG, to obtain the eigensolution of the depth equation. The eigensolution can be employed to obtain normal mode solution to waveguide propagation
Study of the depth eigenproblem using the perturbation approach is the main focus of the present paper.

6. Stochastic components in sound speed variation in a shallow
water ocean waveguide (Sazontov et al, 2002; Jensen et al, 2011):
Internal gravity waves;
Stratification in water column;
Surface roughness & subbottom inhomogeneties, etc.
In the present study only depth dependent sound speed variation
in the water column is considered.
(As mentioned in the beginning, this study is expected to be useful eventually in the solution of range and depth dependent waveguide problems with random inhomogeneities and embedded scatterers.)

7. Methodology and Scope
The variational formulation for the depth eigenproblem is chosen.
A Rayleigh-Ritz approximation is obtained using the depth eigenfunctions of isovelocity waveguide as trial functions.
The sound speed is assumed to be normally distributed.
The resulting algebraic random eigenproblem is solved using MATLAB.

8. Problem Domain
Consider a depth-dependent, cylindrically symmetric ocean waveguide.
ρ(z) , c(z) z r zs D
c(z)
ρ(z)
Water Depth = D
Source Depth zs
Source Frequency(Hz) f
Water Density ρ
Sound Speed c
Figure 1

9. Governing Equations
Inhomogeneous form of the pseudo Helmholtz equation for
acoustic pressure p(r,z) (Jensen et al, 2011):
(1)
denotes the Dirac delta function and ω = 2πf . Eq. (1) should be
augmented with appropriate boundary conditions at z = 0 and
z = D, in the form of Dirichlet, Neumann or mixed boundary
conditions.
Solution: (2)

10. Substituting (2) in (1), we have Radial equation : (3) Depth equation: (4) denotes the radial wave number. Eq. (4) should be solved first to obtain the radial wave number, using which the exact solution of Eq. (3) can be obtained. Solution of Eq. (4) with random sound speed is the main concern of the present study.

11. Variational Approximation
Approximate methods for solving Eq. (4):
* Finite difference and Finite element methods.
A variational formulation for Eq. (4) is adopted here to obtain
approximate solution employing classical form of the Rayleigh-Ritz
method .
We consider the following functional consistent with the boundary
value problem in Eq. (4):
(5)
denotes the prescribed Neumann boundary condition.

12. Rayleigh-Ritz approximation
(6)
denotes a known trial / coordinate function which should satisfy the kinematic boundary conditions, if any.
For convenience, the depth eigenfunctions of an isovelocity
waveguide are used as trial functions.
are the unknown constants.
Note that the eigenproblem in Eq. (4) is a Sturm-Liouville problem
having infinite number of real eigenvalues and orthogonal
eigenfunctions/modes.

13. Substituting Eq. (6) into Eq. (5) and assuming homogeneous
boundary conditions, we have the following discrete
approximation for the functional:
(7)
where,
(8a-c)
Applying the stationarity condition to Eq. (7), we
have the following algebraic (symmetric) eigenproblem:
(9)

14. Solution of Eq. (9) is given by the eigenpairs ( i.e. eigenvalues and
eigenvectors):
(10)
where the eigenvalues are real and the eigenvectors
orthogonal.
Using the above solution and Eq. (6), the depth eigenmodes/
eigenfunctions may be written as:
(11)
When an eigenvalue is positive, the corresponding radial mode
denotes a propagating/travelling wave, and for negative values
we have evanescent modes, whose contribution will be negligible
in the far field.

15. Random Algebraic Eigenproblem
Referring to Fig.1, consider a waveguide with pressure release
boundary at the top and rigid bottom. Then,
at z = 0 & at z = D (12)
Let the sound speed in Eq. (1) be written as
(13)
- mean sound speed;
- reference sound speed;
- zero mean normal random variable;
- first depth mode of the isovelocity waveguide,
with unit amplitude at z = 0.

16. In view of Eq. (13), Eq. (9) may be written as (14) Where. Eq
In view of Eq. (13), Eq. (9) may be written as (14) Where . Eq. (14) is evidently a random algebraic eigenproblem. The perturbation method which is known to be adequate for small random fluctuations, is chosen here to solve Eq. (14).

17. Perturbation Approach
An approximate solution to Eq. (14) may be written in the form of a
Taylor series about the mean as (Vonmarcke et al, Structural Safety 3:
, 1986)
(15a)
(15b)
Further,
(15c)
Where an over bar denotes deterministic quantity.
respectively denote the first and second order
derivatives of with respect to

18. Eq. (15) may be used in Eq. (14) to obtain the following algebraic equations: Zero-th order (16) First order: (17) Solution of Eq. (16) is straight forward. For the first order problem, setting the r.h.s. to zero, we have, (18)

19. Numerical Example
A shallow water waveguide of 100 m depth has been chosen, and
the depth dependent sound speed given in Sazontov et al., 2002
adopted (see figure). Source Frequency= 50 Hz.
Depth (m)
Sound Speed (m/s) 20 25 30 40 60 80
100

20. Random variable α is Normal with (0, 0.01) or (0, 0.1)
The matrices in Eq. (14) have been computed using the Rayleigh-Ritz approximation outlined above.
For Monte Carlo simulation (MCS), the matrices in Eq.(14) again have been computed using the Rayleigh-Ritz approximation. MCS has been done using the Latin Hypercube procedure in MATLAB.
The matrices in Eqs.(16) & (17) have also been obtained using the Rayleigh-Ritz approximation.
The eigenvalue problems in Eqs.(14) & (16) have been solved using MATLAB.
Eq.(18) has been used to obtain the eigenvalues of the first order problem.

21. Results: Radial Wavenumbers kri
Case 1 : Deterministic Case (α = 0)
Case 2 : Mean of kri. for α Normal with (0, 0.01) - simulation
Case 3 : Mean of kri. for α Normal with (0, 0.01) – perturbation
Case 4 : Mean of kri. for α Normal with (0, 0.1) - simulation
Case 5 : Mean of kri. for α Normal with (0, 0.1) – perturbation
kr1
kr2
kr3
kr4
kr5
kr6
kr7
Case I
Case 2
Case 3
Case 4
Case 5

22. Observations As expected, the perturbation technique is much faster.
For the case of α with (0, 0.01), the mean values of propagating wavenumbers obtained using the perturbation technique compare very well with simulation results.
For the case of α with (0, 0.1), the mean values of propagating wavenumbers obtained using the perturbation technique differ by about 1% from the simulation results.
The second order perturbation solution requires computation of the first order eigenvectors, which have not yet been computed.
The second order solution might improve the comparison of mean values.
The second order solution would also provide the variance of the wavenumbers.

23. Acknowledgement: Partial financial support for the
Further Work
Computation of the first & second order eigenvectors in the perturbation model
Use of the eigenvectors in the normal mode solution of a depth dependent ocean acoustic waveguide.
Explore the use of the depth eigensolution in the Rayleigh-Ritz/Finite element model for a range and depth dependent waveguide.
Acknowledgement: Partial financial support for the
research work has been provided by a AOARD project FA2386-