Membrane and Bulk Metamaterials Robert V. Kohn Courant Institute, NYU (1) Membrane metamaterials: work with Jens Jorgensen, Jianfeng Lu, Michael Weinstein (2) Bulk metamaterials: to put part (1) in context. PICOF , Paris, April 2012 Robert V. Kohn Courant Institute, NYU Membrane and Bulk Metamaterials
Outline (1) Membrane metamaterials - The phenomenon - A simplified model - An exactly-solvable special case - Further discussion (2) Bulk metamaterials - negative dynamic mass, as a substitute for damping - the mechanism: resonance in the microstructure Are these topics related? Yes: both involve classical wave theory; both have appln to sonic insulation. No: membrane metamaterials are not materials; mechanism is very different. Robert V. Kohn Courant Institute, NYU Membrane and Bulk Metamaterials
Membrane metamaterials – the phenomenon Basic phenomenon: a thin (suitably structured) elastic membrane can achieve near-total reflection of time-harmonic acoustic waves incident wave − → − → transmitted wave reflected wave ← − membrane The membrane: circular (or square, etc); thin elastic under tension; with a heavy weight in the middle. Different weights achieve total reflection at different frequencies. Initial work: Z. Yang et al, Membrane-type acoustic metamaterial with negative dynamic mass , PRL 101 (2008) Further work: Z. Yang et al, Appl Phys Lett 96 (2010); also Naify et al, J Appl Phys 108 (2010) and 109 (2011). Robert V. Kohn Courant Institute, NYU Membrane and Bulk Metamaterials
A simplified model Waveguide section Ω ; membrane is at x = 0 − → − → ← − V ( y , t ) = transverse displacement of membrane U ( x , y , t ) = pressure in waveguide ρ ( y ) V tt − div y ( σ ( y ) ∇ y V ) = ∂ x U + − ∂ x U − at x = 0 (the membrane) U tt − ∆ U = 0 for x < 0 and x > 0 (the waveguide) U ( 0 , y , t ) = V ( y , t ) continuity at membrane suitable bc at ∂ Ω and as x → ±∞ Membrane pde is just balance of forces; associated Hamiltonian is � � � t + |∇ U | 2 dx dy + t + σ |∇ V | 2 dy + t + |∇ U | 2 dx dy U 2 ρ V 2 U 2 x < 0 Ω ×{ 0 } x < 0 Robert V. Kohn Courant Institute, NYU Membrane and Bulk Metamaterials
Time-harmonic scattering − → − → ← − Take U ( x , y , t ) = u ( x , y ) e − i ω t , V ( y , t ) = v ( y ) e − i ω t Separation of vars in waveguide: U is superposn of special solns e ik n x − i ω t ϕ n ( y ) where − ∆ ϕ n = λ n ϕ n (with waveguide bdry conds) and − ω 2 + λ n + k 2 n = 0. Choose sign convention � √ if λ n < ω 2 (propagating) ω 2 − λ n √ k n ( ω ) = if λ n > ω 2 (evanescent) i λ n − ω 2 so that � outgoing as x → ∞ , if propagating e i ( k n x − ω t ) ϕ n ( y ) is exp decaying as x → ∞ , if evanescent Then: if incident wave is assoc n = 1, scattering expt has ∞ ∞ u x < 0 = e ik 1 x ϕ 1 ( y ) + � r n e − ik n x ϕ n ( y ); � t n e ik n x ϕ n ( y ) . u x > 0 = n = 1 n = 1 Robert V. Kohn Courant Institute, NYU Membrane and Bulk Metamaterials
A simple (but boring) exactly-solvable case Let membrane and waveguide share same bdry cond (say, Dirichlet) at ∂ Ω , and take ρ = 1 and σ = 1 (uniform). Then modes decouple. Time-harmonic membrane pde is − ω 2 v − ∆ v = [ u x ] x = 0 on Ω while in waveguide ∞ ∞ u x < 0 = e ik 1 x ϕ 1 ( y ) + � r n e − ik n x ϕ n ( y ); � t n e ik n x ϕ n ( y ) . u x > 0 = n = 1 n = 1 By continuity: ∞ � 1 + r 1 = t 1 ; r n = t n for n ≥ 2 ; v ( y ) = t n ϕ n ( y ) . n = 1 RHS of membrane PDE becomes ∞ � [ u x ] x = 0 = − 2 ik 1 φ 1 + 2 i k n t n ϕ n n = 1 Modes decouple, so ( − ω 2 + λ 1 − 2 ik 1 ) t 1 = − 2 ik 1 and t n = 0 for n ≥ 2 . In particular, t 1 ( ω ) � = 0, so total reflection is not achieved at any ω . Robert V. Kohn Courant Institute, NYU Membrane and Bulk Metamaterials
A more interesting exactly-solvable case As before, let membrane and waveguide share same bdry cond (say, Dirichlet) at ∂ Ω , and let σ = 1. But now take ρ = 1 + m δ y 0 , i.e. add mass m to membrane, at location y 0 . This makes sense only in 1D (we need v �→ v ( 0 ) to be continuous in H 1 norm), so take Ω = ( 0 , 1 ) . Waveguide is a strip. Weak form of membrane eqn − ρω 2 v − v yy = [ u x ] x = 0 is � � � − ω 2 vw dy − ω 2 mv ( y 0 ) w ( y 0 ) + v y w y dy = [ u x ] x = 0 w dy . So in spectral Galerkin representation, mass at y 0 introduces a rank-one perturbation to an otherwise-diagonal problem. Formally: v = � ∞ n = 1 t n ϕ n where t = ( t 1 , t 2 , · · · ) T solves ( D − ω 2 m ss T ) t = b with diag matrix with entries d n = − ω 2 + λ n − 2 ik n D = ss T = rank one matrix with n , m th entry ϕ n ( y 0 ) ϕ m ( y 0 ) ( − 2 ik 1 , 0 , 0 , 0 , · · · ) T . b = Exactly solvable using Sherman-Morrison formula. Robert V. Kohn Courant Institute, NYU Membrane and Bulk Metamaterials
The membrane with a point mass, cont’d Recall: if Ω = ( 0 , 1 ) and ρ = 1 + m δ y 0 then v = � ∞ n = 1 t n ϕ n where t solves ( D − ω 2 m ss T ) t = b with D = diag matrix with entries d n = − ω 2 + λ n − 2 ik n , s = ( ϕ 1 ( y 0 ) , ϕ 2 ( y 0 ) , · · · ) T , and b = ( − 2 ik 1 , 0 , 0 , 0 , · · · ) T . Now suppose there is just one propagating mode, i.e. λ 1 < ω 2 < λ 2 so that k 1 is real and k n = i | k n | for n ≥ 2. Then (after some algebra) 1 − m ω 2 � ∞ n = 2 d − 1 n | s n | 2 t 1 = d − 1 1 b 1 n = 1 d − 1 1 − m ω 2 � ∞ n | s n | 2 The fraction has real numerator (since d n = − ω 2 + λ n + 2 | k n | for n ≥ 2) and non-vanishing denominator (since d 1 is not real). Moreover ∞ | s n | 2 numerator = 1 − m ω 2 � √ λ n − ω 2 + 2 λ n − ω 2 n = 2 is a decreasing function of ω 2 , and a linear function of m . So There is at most one ω ∈ ( √ λ 1 , √ λ 2 ) at which total reflection occurs. For each ω , m can be chosen st t 1 ( ω ) = 0. Robert V. Kohn Courant Institute, NYU Membrane and Bulk Metamaterials
Some numerics 1 − m ω 2 � ∞ n = 2 d − 1 | s n | 2 | s n | 2 where d n = − ω 2 + λ n − 2 ik n , Recall: t 1 = d − 1 1 b 1 n n = 1 d − 1 1 − m ω 2 � ∞ n ω 2 − λ 1 . � s n = ϕ n ( y 0 ) , and b 1 = − 2 ik 1 = − 2 i Example: transmission | t 1 | graphed against ω 2 , when Ω = ( 0 , 1 ) , the bndry cond is Dirichlet, and the mass is midpoint y 0 = 1 / 2: Robert V. Kohn Courant Institute, NYU Membrane and Bulk Metamaterials
Is this an accident? Success is initially surprising: t 1 ( ω ) is complex; why can we make it vanish by varying a real parameter ( ω , or m )? Answer: actually, if there is just one propagating mode, then t 1 ( ω ) is restricted 1/2 to a circle in the complex plane. In fact: when there is one propagating mode, far-field reflected wave r 1 e − ik 1 x ϕ 1 ( y ) and far-field transmitted wave t 1 e ik 1 x ϕ 1 ( y ) are related by | r 1 | 2 + | t 1 | 2 = 1. But continuity at membrane gives r 1 + 1 = t 1 . So | t 1 − 1 | 2 + | t 1 | 2 = 1. (This holds for any ρ ( y ) and σ ( y ) ). A consequence: The phenomenon is robust – perturbing a system will perturb the freq where t 1 ( ω ) = 0, but it won’t eliminate the phenomenon. Robert V. Kohn Courant Institute, NYU Membrane and Bulk Metamaterials
Stepping back What is the essential mechanism? - Effect of the mass is to couple modes. Clearly visible in our example, where ( D − ω 2 mss T ) t = b with D diagonal. - If coupling is suitable, t 1 ( ω ) will pass 0 as it moves on the circle as a function of ω ∈ ( √ λ 1 , √ λ 2 ) . How general is this? - Hypothesis of one prop mode seems essential. For our solvable example with λ 2 < ω 2 < λ 3 (two prop modes), would need t 1 ( ω ) = t 2 ( ω ) = 0 for two indep incoming waves. Many (nonlinear, simultaneous) eqns! - Hypothesis that membrane and waveguide share same bdry condition is not physical. Closer to acoustics: Dirichlet bc for the membrane, Neumann for the waveguide. Effect persists numerically; can it be analyzed? Robert V. Kohn Courant Institute, NYU Membrane and Bulk Metamaterials
Stepping back, cont’d How useful is this? - Effect is intrinsically narrow-band: if t 1 ( ω ) passes through 0 at ω = ω 0 , then | t 1 ( ω ) | ∼ c | ω − ω 0 | . - To get broader-band effect, Yang et al (J Appl Phys 2010) explored using different membranes in series, and Naify et al (J Appl Phys 2011) explored using different membranes in parallel. − → − → − → ← − ← − Robert V. Kohn Courant Institute, NYU Membrane and Bulk Metamaterials
Transition (1) Membrane metamaterials - The phenomenon - A simplified model - An exactly-solvable special case - Further discussion (2) Bulk metamaterials - negative dynamic mass, as a substitute for damping - the mechanism: resonance in the microstructure Are these topics related? Yes: both involve classical wave theory; both have appln to sonic insulation. No: membrane metamaterials are not materials; mechanism is very different. Robert V. Kohn Courant Institute, NYU Membrane and Bulk Metamaterials
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