1. 16/03/2016 Quantum Physics-2016 1 Speaker Dr. Rab Nawaz Email: rabnawaz@comsats.edu.pk Speaker Dr. Rab Nawaz Email: rabnawaz@comsats.edu.pk On coupled wave scattering of structures involving flexible boundaries: Application in sound-structure interaction Department of Mathematics COMSATS Institute of Information technology Islamabad Pakistan

2. 16/03/2016 Quantum Physics-2016 2  Model Problem  An Overview Layout of Presentation  Mode-Matching Solution Non-Sturm Liouville system I.Rigid Strip II.Membrane Strip Matching Conditions  Results and Discussion  Concluding Remarks

3. 16/03/2016 Quantum Physics-2016 3 An overview  Unwanted sources of noise Water-air heat exchanger Water-water plate heat exchanger  Structural discontinuity with variable material properties  Structural acoustic, elasticity, electromagnetic theory and water waves

4. 16/03/2016 Quantum Physics-2016 4 An overview... Precisely the mode-matching (MM) technique together with an appropriate form of the orthogonality relation is viable tool to solve many interesting physical problems. It incorporates the structural discontinuity as well as the singularities at the corners. In such circumstances the mode-matching along with additional properties of its validity has been applied somewhat using the low frequency approximation (LFA). Rab Nawaz, M. Afzal, and M. Ayub, Acoustic propagation in two-dimensional waveguide for membrane bounded ducts, Communications in Non-Linear Science and Numerical Simulation 20, 421--433 (2015). Mahmood-Ul-Hassan, M., H. Meylan, and M. A. Peter, "Water-wave scattering by submerged elastic plates," Q. J. Mech. Appl. Math. 62, 321--344 (2009). Lawrie JB, Kirby R. Mode-matching without root finding: application to a dissipative silencer. J Acoust Soc Am ;119, 2050--61 (2006). Lawrie JB, Guled IMM, On tuning a reactive silencer by varying the position of an internal membrane. Journal of Acoustical Society of America 120(2): 780--790 (2006).

5. 16/03/2016 Quantum Physics-2016 5 N. Amitay and V. Galindo (1969) On energy conservation and the method of moments in scattering problems. IEEE Trans Antennas Propag 17(6): 747-751 G. A. Kriegsmann (1997) The Galerkin approximation of the Iris problem: Conservation of power. Appl Math Lett 10(1):41-44 G. A. Kriegsmann (1999) The flanged waveguide antenna: discrete reciprocity and conservation. Wave Motion 29:81--95. An overview... Problems involving the propagation of electromagnetic, water and sound waves generally satisfy a number of conservation laws and it is tempting to use these as a means of validating solution techniques.

6. 16/03/2016 Quantum Physics-2016 6 CRIGHTON, D. G., DOWLING, A. P., FFOWCS-WILLIAMS, J. E., HECKL, M. & LEPPINGTON, F. G., 1992. Modern Methods in Analytical Acoustics. Springer-Verlag. Rigid/Membrane a b x=0 Model Problem Assuming harmonic time dependence On non-dimensionalizing and

7. 16/03/2016 Quantum Physics-2016 7 The non-dimensional form of boundary value problem is Mathematical Formulation

8. 16/03/2016 Quantum Physics-2016 8 and Mode-Matching Solution

9. 16/03/2016 Quantum Physics-2016 9 Non-Sturm Liouville system D.P. Waren and J.B. Lawrie, Acoustic scattering in wave guides with discontinuities in height and material property, Wave Motion, 36, 119-142 (2002). J. B. Lawrie and D. Abraham, An orthogonality condition for a class of problem with high order boundary conditions; applications in sound/structure interaction, Q. J. Mech appl. Math. 52, 161-181 (1999).

10. 16/03/2016 Quantum Physics-2016 10 Green's function can be constructed J. B. Lawrie, On eigenfunction expansions associated with wave propagation along ducts with wave-bearing boundaries. IMA.J. Appl. Math, 72,376-394 (2007)..

11. 16/03/2016 Quantum Physics-2016 11 D.P. Waren and J.B. Lawrie, Acoustic scattering in waveguides with discontinuities in height and material property, Wave Motion, 36, 119-142 (2002). Expressions for powers

12. 16/03/2016 Quantum Physics-2016 12 Matching Conditions

13. 16/03/2016 Quantum Physics-2016 13 Rigid Strip

14. 16/03/2016 Quantum Physics-2016 14 To obtain the conservation of energy

15. 16/03/2016 Quantum Physics-2016 15 D.P. Warren and J.B. Lawrie, Acoustic scattering in wave guides with discontinuities in height and material property, Wave Motion, 36, 119-142 (2002). On using the edge condition and collecting the real part yields

16. 16/03/2016 Quantum Physics-2016 16 Membrane Strip

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21. 16/03/2016 Quantum Physics-2016 21 J. B. Lawrie, On eigenfunction expansions associated with wave propagation along ducts with wave-bearing boundaries. IMA.J. Appl. Math, 72,376-394 (2007). The parameters involved are chosen to be as: The system is truncated up to N The system is suitably convergent Results and Discussion

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23. 16/03/2016 Quantum Physics-2016 23 Rab Nawaz and J. B. Lawrie, Scattering of a fluid-structure coupled wave at a flanged junction between two flexible waveguides, Journal of Acoustical Society of America 134, 1939--1949 (2013).

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29. 16/03/2016 Quantum Physics-2016 29  With the inclusion of rigid vertical strip at matching interface, the solution preserve the power identity.  With the inclusion of flexible vertical strip at matching interface, the solution does not preserve the power identity but it can be achieved by taking.  The results hold for generalized edge conditions.  The effect of flexible junction is not confined to the power identity.  The reflected and transmitted rates vary significantly for both the situations  The use of Lanczos filter aids the convergence and smoothens out Gibb’s phenomenon thus providing a more accurate approximation to the function. Concluding remarks

30. 16/03/2016 Quantum Physics-2016 30 Thanks for your attention