The amplitude is then found to be w = w0β. With solutions to these last two equations, the field is developed:

All quantities in (7.167) are evaluated on the ray path, that is, are functions of s only. It was stated previously that these equations can be derived for any ray parameter; in particular one can take advantage of the nature of the ray paths to scale them by any function of coordinates. In particular, using the parameterization introduced in the first section for describing rays as geodesics leads to a form of (7.165) and (7.166) that is identical to the equation for geodesic deviation used in the paraxial ray trace. Clearly the meaning of the parameters is different when used for a Gaussian beam calculation, but the form of the equations already developed can be reused to evaluate these parameters.

7.10Exercises

1.Given the following sound speed profile

cz = Aexpαz + Bexpβz

where A, B, α, and β are constants,

(a)Determine the ray paths and travel times using the ray path integrals.

(b)Evaluate for A = B and α = − β.

(c)Either prove or demonstrate by example the ideal focusing feature of this profile.

2.Generalize the ray path integrals for the case when the background fluid velocity has all three components: υ0x, υ0y, and υ0z.

(a)Analyze the special case when the flow is perpendicular to the initial ray path, for example, the ray starts in the x–z plane and υ0y is the only nonzero component.

(b)Derive the ray path integrals for 2-dim rays in the x–z plane with c, υ0y, and υ0z, all functions of z.

(c)Starting from the conservation laws along the ray path, derive a result for the case where the environmental parameters c, υ0 are independent of coordinates but functions of time.

3.Assuming a purely 2-dim spatial environment with sound speed profile c(x, z),

(a)Derive the Christoffel symbols using the definition from Chapter 4.

(b)Derive the components of the Riemann tensor (7.34).

4.Using MATLAB or an open-source equivalent, write a routine that will apply the RK algorithm to the ray equations and test out on the ideal waveguide example presented in this chapter. Compare results with the exact solution.

References

[1]Bergman, D. R., Internal symmetry in acoustical ray theory, Wave Motion, Vol. 43, pp. 508–516, 2006.

[2]Cerveny, V., Seismic Ray Theory, Cambridge University Press, Cambridge, 2001.

[3]Guenther, R., Modern Optics, John Wiley & Sons, Inc., New York, 1990, (specifically Chapter 5, see section entitled “Propagation in a Graded Index Optical Fiber” for a wave guide paraxial ray approximation).

[4]Abromowicz, A. and Kluzniak, W., Epicyclic orbital oscillations in Newton’s and Einstein’s dynamics, Gen. Relativ. Gravit., Vol. 35, No. 1, pp. 69–77, 2003.

[5]Bazanski, S. L., Kinematics of relative motion of test particles in general relativity, Ann. Inst. Henri Poincare A, Vol. 27, No. 2, pp. 115–144, 1977.

[6]Hawking, S. W. and Ellis, G. F. R., The Large Scale Structure of Space: Time, Cambridge University Press, Cambridge, 1973.

[7]Schneider, P., Ehlers, J., and Falco, E. E., Gravitational Lenses, Astronomy and Astrophysics Library, Springer-Verlag, Berlin, 1992.

[8]Blokhintzev, D., The propagation of sound in an inhomogeneous and moving medium, I, J. Acoust. Soc. Am., Vol. 18, No. 2, pp. 322–328, 1945.

[9]Boone, M. M. and Vermaas, E. A., A new ray-tracing algorithm for arbitrary inhomogeneous and moving media, including caustics, J. Acoust. Soc. Am., Vol. 90, No. 4 (Part 1), pp. 2109–2117, 1991.

[10]Ludwig, D., Uniform asymptotic expansion at a caustic, Commun. Pure Appl. Math., Vol. 19, pp. 215– 250, 1966.

[11]Foreman, T., A Frequency Dependant Ray Theory, Technical Report, ARL-TR-88-17, Applied Research Laboratories, The University of Texas at Austin, Austin, 1988.

[12]Brekhovskikh, L. M., Waves in Layered Media, Academic Press, New York, 1960.

[13]Kravtsov, Yu. A. and Orlov, Yu. I., Caustics, Catastrophes, and Wave Fields, Second Edition, Springer Series on Wave Phenomena, Lotsch, H. K. V., Managing Editor, Springer, Heidelberg, 1999.

[14]Franchi, E. and Jacobson, M., Ray propagation in a channel with depth: Variable sound speed and current, J. Acoust. Soc. Am., Vol. 52, No. 1 (Part 2), pp. 316–331, 1972.

[15]Heller, G. S., Propagation of acoustic discontinuities in an inhomogeneous moving liquid medium, J. Acoust. Soc. Am., Vol. 25, No. 5, p. 950, 1953.

[16]Rudenko, O. V., Sukhorukova, A. K., and Sukhorukov, A. P., Full solutions to the equations of geometrical acoustics in stratified moving media, Acoust. Phys., Vol. 43, No. 3, pp. 339–343, 1997.

[17]Thompson, R. J., Ray theory for an inhomogeneous moving medium, J. Acoust. Soc. Am., Vol. 51, No. 5 (Part 2), p. 1675, 1972.

[18]Ugincius, P., Acoustic-ray equations for a moving, inhomogeneous medium, J. Acoust. Soc. Am., Vol. 37, No. 3, pp. 476–479, 1965.

[19]Ugincius, P., Ray acoustics and Fermat’s principle in a moving inhomogeneous medium, J. Acoust. Soc. Am., Vol. 51, No. 5 (Part 2), pp. 1759–1763, 1972.

[20]Kornhauser, E. T., Ray theory for moving fluids, J. Acoust. Soc. Am., Vol. 25, No. 5, pp. 945–949, 1953.

[21]Bergman, D. R., Generalized space-time paraxial acoustic ray tracing, Waves Random Complex Media, Vol. 15, No. 4, pp. 417–435, 2005.

[22]Iserles, A., A First Course in the Numerical Analysis of Differential Equations, Cambridge University Press, Cambridge, 2002.

[23]Butcher, J. C., The Numerical Analysis of Ordinary Differential Equations, Runge-Kutta and General Linear Methods, John Wiley & Sons, Ltd, Chichester, 1987.

[24]Gottlieb, S., Ketcheson, D., and Shu, C., Strong Stability Preserving Runge–Kutta and Multistep Time Discretizations, World Scientific, River Edge, 2011.

[25]Press, W. H., Teukolsky, S. A., Vetterling, W. T., and Flannery, B. P., Numerical Recipes in C++: The Art of Scientific Computing, Second Edition, Cambridge University Press, Cambridge, 2005.

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8

Finite Difference and Finite

Difference Time Domain

8.1 Introduction

This chapter presents the finite difference (FD) technique in the frequency domain and the finite difference time domain (FDTD) method. The former is applied in the frequency domain primarily to the second-order scalar wave equation with spatial inhomogeneities. The FD method involves discretizing the differential equation by creating finite difference representations of the derivatives in the spatial variable on a discrete lattice of points in space. The continuum is replaced by points, and the continuous operators replace with couplings between the discrete points designed to provide estimates for those operators to some desired accuracy. The resulting equation is a matrix equation Ax = b, where the components of x are the values of the unknown scalar field at the lattice points, b is the source term evaluated on the lattice, and the matrix A is a sparse matrix containing terms related to the discretized differential operators and evaluations of the sound speed, density, and other environmental fields evaluated on the lattice.

The application of finite difference techniques to the time derivative introduces a new set of issues to deal with. In this case a first-order system is preferable. The basic equations required for propagating an initial field configuration forward in time are presented for the first-order linear equations of acoustics, with pressure and particle velocity as the field variables. The primary FD scheme is still valid, but splitting of the variables is required. A method borrowed from computational electrodynamics, known as Yee’s method, is presented as a procedure for an FDTD approach to acoustics. For the FDTD approach an initial field is propagated forward in time in tiny steps. In this case processing time is spent on the numerous applications, possibly tens of thousands, of the matrix to the updated source terms in a feedback loop. This last statement depends on the method being explicit in time. For implicit methods matrix inversion is required at each step. This chapter presents the basic theory behind developing FD and FDTD operators and the building up of FD/FDTD simulations.