Wave Scattering of Structures with Flexible Boundaries

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

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

8_radiator An overview Structural acoustic, elasticity, electromagnetic theory and water waves Structural discontinuity with variable material properties Unwanted sources of noise Water-water plate heat exchanger http://www.airforce-technology.com/contractor_images/systecon/systecon1.jpg Water-air heat exchanger 3 16/03/2016 Quantum Physics-2016

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). 4 16/03/2016 Quantum Physics-2016

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. 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. 5 16/03/2016 Quantum Physics-2016

Model Problem b 2 tot x2 2 tot y2 2 tot t2 1 Rigid/Membrane c2 a 2H x2 1 2H t2 1 T P 2 cm x=0 H t P x,y,t y tot t tot and e i t Assuming harmonic time dependence x kx y ky and t t On non-dimensionalizing CRIGHTON, D. G., DOWLING, A. P., FFOWCS-WILLIAMS, J. E., HECKL, M. & LEPPINGTON, F. G., 1992. Modern Methods in Analytical Acoustics. Springer-Verlag. 6 16/03/2016 Quantum Physics-2016

Mathematical Formulation The non-dimensional form of boundary value problem is 2 1 0, 1 x,y , x 0, 0 y a 2 x,y , x 0, 0 y b x,y j y 0, y 0, x , j 1,2. 2 x2 2 1y 1 0, y a, x 0 2 x2 2 2y 2 0, y b, x 0 cmand 2 c Tk3. # 7 16/03/2016 Quantum Physics-2016

Mode-Matching Solution The eigenfunction expansions formof velocity potentials: 1 x,y F cosh y ei x Ancosh ny e i nx, n 0 2 x,y Bncosh ny eisnx. n 0 F / C n 0,1,2,...... 2 1 sn n and 2 1 n n The quantities nand n, n 0,1,2,......are the roots of the dispersion relation K ,p 0 K ,p 2 1 2 sinh p cosh p , 8 16/03/2016 Quantum Physics-2016

Non-Sturm Liouville system For the left hand duct the OR is acosh my cosh ny dy mnCm m nsinh ma sinh na 0 2 1 2 2 m 3 m Cm a 2. msinh ma 2 2 On replacing a with b , m with m and Cm with Dm, we get the OR for right hand duct 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). 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). 9 16/03/2016 Quantum Physics-2016

The eigenfunctions Yn y ,n 0,1,2... are linearly dependent a Yn y Cn Yn n 0 0, 0 y a # a Cn 2 Yn n 0 1. # . Green's function can be constructed Yn Yn y Cn n 0 y y y 2a ,0 ,y a, # J. B. Lawrie, On eigenfunction expansions associated with wave propagation along ducts with wave- bearing boundaries. IMA.J. Appl. Math, 72,376-394 (2007). 10 16/03/2016 Quantum Physics-2016

Expressions for powers In addition, the non-dimensional scattered powers are expressed as: J1 m 0 |Am|2Cm m, E1 1 J2 m 0 |Bm|2Dmsm, E2 1 where J1/J2is the number of cut-on modes in the left/right hand duct. The choice of F , 0,1 is such that E1 E2 1 This is a conservative law of energy. D.P. Waren and J.B. Lawrie, Acoustic scattering in waveguides with discontinuities in height and material property, Wave Motion, 36, 119-142 (2002). 11 16/03/2016 Quantum Physics-2016

Matching Conditions At x 0, 0 y a the fluid pressure is continuous 0 y a. 1 2, The zero displacement edge condition is 1y 0,a 0 Cm Am F m msinh ma E1 BnRmn, Cm n 0 E1 1y 0,a acosh my cosh ny dy. Rmn 0 E1 0 12 16/03/2016 Quantum Physics-2016

Rigid Strip 1 0, x, 0 y a a y b 2 x . The zero displacement edge condition is 2y 0,b 0 Dmsm msinh mb E2 Dmsm F R m Dmsm An nRnm, Bm n 0 E2 i 2xy 0,b . S1 m 0 F R m msinh mb Dmsm E2 An nRnm , n 0 2 S1 msinh mb Dmsm . m 0 13 16/03/2016 Quantum Physics-2016

To obtain the conservation of energy N m 0 |Bm|2Dmsm |Am|2Cm m N 0,b |F |2 C N 0,b 2xy C F A i 2y N 0,a 1xy N 0,a F C A . i 1y 14 16/03/2016 Quantum Physics-2016

Note that will be real for an incident mode therefore and C and F will also be real. On using the edge condition and collecting the real part yields E1 E2 1 Thus for N 0, the obtained solution preserves the energy conservative law. But we get physical power balance only when N 1 J1,J2and is sufficiently large to ensure that Am, m 0,1,2,...J1and Bm, m 0,1,2,...J2 have converged adequately. 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). 15 16/03/2016 Quantum Physics-2016

Membrane Strip The rigid vertical strip at x 0, a y b is replaced by membrane 1 M y x, 0 y a 2 x , a y b 1 M y 2 2 2xyy 2E3 y a 2E4 y b . The zero displacement edge conditions at each membrane edge 2x 0,a 2x 0,b 2y 0,b 0. 16 16/03/2016 Quantum Physics-2016

1 Dm m cosh ma E3 cosh mb E4 msinh mb E2 Bm Dm m F R m 2 Qmn i 2 , Bn isn n An nRnm n 0 n 0 E2 Bn n nsinh nb , n 0 2 2 m ism m acosh my cosh ny dy. Qmn 0 17 16/03/2016 Quantum Physics-2016

2x 0,y smcosh my Dm m cosh ma E3 cosh mb E4 m 0 msinh mb E2 n 0 smcosh my Dm m 2 Qmn Bn isn n m 0 An nRnm . i 2F R m i 2 n 0 Similarly, we can get an expression for 2y 0,y to impose the edge conditions. 18 16/03/2016 Quantum Physics-2016

N m 0 |Bm|2Dmsm |Am|2Cm m i 2xy N 0,b 2y N 0,b 1xy E3 i E4 i 1y i N 0,a N 0,b N 0,a N 0,a 2 2 2 2 b 2 2xyy 2 i N 0,y N N 0,y dy. 0,y 2 a E1 E2 I N 1 N Re i E3 i N 0,a N 0,b E4 I 2 2 2 2 2 b 2 i 2xyy N 0,y N 0,y dy a 19 16/03/2016 Quantum Physics-2016

Now let N , the we recollect the vertical membrane condition 2xyy 0,y 2 2x 0,y 2 0,y 2E3 y a 2E4 y b , a y b. b 2 limN IN Re i 0,y 2x 0,y dy a b limN IN Re i 0,y a 2x 0,y 2xy b 2xy 0,y 2xy i 0,y dy . a N 0 as N ,and thus E1 E2 1. Clearily, I 20 16/03/2016 Quantum Physics-2016

Results and Discussion The system is truncated up to N The system is suitably convergent The parameters involved are chosen to be as: c 343ms 1, 1.2043kgm 3, m 0.1715kgm 2and T 350Nm, while the duct heights are fixed at a 0.06m and a 0.085m. J. B. Lawrie, On eigenfunction expansions associated with wave propagation along ducts with wave- bearing boundaries. IMA.J. Appl. Math, 72,376-394 (2007). L. Huang, Parametric study of a drumlike silencer, J. Sound Vib. 269, 467 488 (2004) 21 16/03/2016 Quantum Physics-2016

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T = n = ( ) cosh( ) f y A y n n n 0 sinh( / ) a / T = n n a T n 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). 23 16/03/2016 Quantum Physics-2016

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Concluding remarks 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. N 29 16/03/2016 Quantum Physics-2016

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