Factorized dispersion relations for two coupled systems

F actorized disp ersion relations for t w o coupled systems Alexander Figotin ∗ Dep artment of Mathematics, University of California, Irvine, CA 92697, USA W e establish that the disp ersion relations of an y ph ysical system comp osed of t wo coupled sub- systems, go verned b y a space-time homogeneous Lagrangian, admit a factorized form G 1 G 2 = γ G c , where G 1 and G 2 are the subsystem disp ersion functions, G c is the coupling function, and γ is the coupling parameter. The result follows from a determinant expansion theorem applied to the blo c k structure of the coupled system matrix, and is illustrated through three examples: the trav- eling w av e tub e, vibrations of an airplane wing, and the Mindlin-Reissner plate theory . F or the Mindlin-Reissner example we carry out a complete asymptotic analysis of the coupled dispersion branc hes, establishing that the factorized form provides a precise quantitativ e measure of mode h ybridization: all four branc hes carry the imprint of b oth subsystem factors for any nonzero cou- pling, while asymptotically reco vering the identit y of pure uncoupled mo des at large frequencies and w av en umbers. W e further analyze the univ ersal local geometry of the coupled dispersion branc hes near their intersection — the cross-p oint mo del — showing it is generically hyperb olic, and present a mec hanical analog in whic h the w av en umber is replaced b y a scalar parameter, exhibiting the same factorized structure and av oided crossing. I. INTR ODUCTION Disp ersion relations — equations relating frequency ω to wa v enum ber k — are among the most fundamental ob jects in w a ve physics. They encode the propagation prop erties of a ph ysical system and go vern phenomena as v aried as wa ve pack et spreading, group velocity , band gaps, and instabilities. The structure of disp ersion relations for systems comp osed of t w o in teracting subsystems is therefore of broad ph ysical interest. Classical treatments of w a ve propagation in elastic media Ac hen [ 1 ], GerRix [ 19 ], disp ersive w av es Whith1 [ 47 ], Whith2 [ 48 ], and coupled-mode theory HausHua [ 23 ] pro vide the conceptual bac kground for the presen t w ork. In our recent work FigFDT1 [ 14 ] on the ﬁeld theory of tra v eling wa v e tub es (TWT) we discov ered a physically app ealing factorized form of the disp ersion relations, based on the fact that the TWT can b e viewed as tw o interacting coupled subsystems: the electron beam and the metal w av e-guiding structure containing it. The factorized form G 1 G 2 = γ G c expresses the disp ersion relation of the coupled system as a product of the t wo subsystem disp ersion functions p erturbed by the coupling, where G 1 and G 2 are the disp ersion functions of the ﬁrst and second subsystems resp ectiv ely , G c is the coupling function, and γ is the coupling parameter. Since decomp osition into t w o in teracting subsystems arises in a wide v ariet y of physical con texts, it is natural to ask whether this factorized structure is a general prop ert y of tw o-subsystem Lagrangian ﬁeld theories. The answer is aﬃrmativ e, and the present pap er establishes this in full generality . A striking consequence of this factorized structure, developed in detail for the Mindlin-Reissner plate example, is that it provides a precise and quantitativ e account of mo de hybridization induced by coupling: every branc h of the coupled disp ersion relation carries the imprint of both subsystem factors for any nonzero coupling, and the degree of mixing is directly controlled b y the coupling parameter. The Lagrangian framework pro vides the natural setting for this in v estigation. Physical systems furnished with Lagrangians dep ending on ﬁelds and their partial deriv ativ es ov er space-time, under the assumption of space-time homogeneit y , p ossess dispersion relations through the F ourier-domain eigenv alue condition det { ˆ L ( k ) } = 0 . When the system Lagrangian admits a decomp osition in to t wo subsystem Lagrangians coupled b y a single coupling parameter b , the determinan t condition factors in a precise algebraic sense go verned by Theorem 1 b elow, which is based on Markus’s determinan t formula Markus [ 33 ]. W e note that related asymptotic approaches to factorizing plate disp ersion relations ha ve b een dev elop ed b y Kapluno v and collaborators KapNolRog [ 26 ], KapNob [ 27 ], ChebKapRog [ 9 ], AlzKapPri [ 3 ], where p olynomial approximations of the Rayleigh-Lam b and Mindlin plate equations isolate individual wa v e branches; the present approac h diﬀers in that the factorization is exact and algebraic, deriv ed d irectly from the Lagrangian coupling structure. The av oided-crossing and cross-p oin t phenomena that arise in the coupled disp ersion branches are closely related to the non-crossing rule of quantum mec hanics Noh [ 35 ] and to coupled-mo de theory HausHua [ 23 ]. F or ph ysical systems that do not possess disp ersion relations, a natural substitute is the dep endence of the system eigenfrequencies on a physical parameter p , which w e refer to as fr e quency-p ar ameter r elations . W e sho w that such relations admit the same factorized structure for tw o coupled systems, as illustrated explicitly in Section V B . ∗ aﬁgotin@uci.edu 2 The pap er is organized as follo ws. Section I I pro vides a concise review of the Lagrangian v ariational framew ork for ﬁelds with higher-order deriv atives and the asso ciated dispersion relations, establishing the notation used through- out. Section I I I introduces the coupled t wo-subsystem framew ork, deﬁnes the coupling parameter, and develops the factorized form of the disp ersion relation, with the k ey result given b y Theorem 1 on the determinant of the coupled system matrix. Section IV illustrates the theory through three physically app ealing examples: the trav eling wa v e tub e (Section IV A ), vibrations of an airplane wing (Section IV B ), and the Mindlin-Reissner plate theory (Section IV C ), including a detailed asymptotic analysis of mo de h ybridization and a comparison with the classical Kirc hhoﬀ plate theory . Section V in troduces the cross-point model as the univ ersal local description of tw o coupled dispersion branc hes near their intersection, derives the asso ciated h yp erb olic geometry , constructs the Lagrangian underlying the cross-p oint disp ersion relation, presents a ﬁnite-dimensional mec hanical analog in whic h the wa v enum ber is re- placed by a scalar parameter, and demonstrates that hybridization is spatially concentrated near the cross-p oint with the coupled branches recov ering their individual mo de c haracter asymptotically at large frequencies and wa v enum bers. The App endix (Section VI ) collects auxiliary material including the F ourier transform conv en tions and the Markus determinan t formula used in the pro ofs. I I. REVIEW OF LA GRANGIANS WITH HIGHER DERIV A TIVES AND THE DISPERSION RELA TIONS W e provide here a concise review of the Lagrangian v ariational framework that inv olves higher deriv ativ es follo wing mostly GelF om [ 18 , Sec. 11] and GiaqHild [ 20 , Chap. 1, Sec. 5,6] (see also Carath [ 8 , Sec. 18], Hass [ 22 , Sec. 33]). Our motiv ation for considering Lagrangian dep enden t on higher order partial deriv ativ es is that some problems of mec hanics of contin ua related to bending and t wisting in volv e the second order deriv ativ es, see for instance Langh [ 28 , Sec. 2.1, 8.7, 8.8]. A. The Lagrangian and the Euler equations Supp ose that conceiv able conﬁgurations of the ph ysical system are describ ed by a set of real-v alued ﬁelds u i ( x ) , 1 ≤ i ≤ N , o ver space-time R n +1 , that is u ( x ) = ( u 1 ( x ) , . . . , u N ( x )) , x ∈ R n +1 , (I I.1) where N is the total n um b er of ﬁeld v ariables. In the cases of in terest the space-time vector x ∈ R n +1 represen ts the space R n so that x = ( x 0 , x 1 , · · · , x n ) , x 0 = ct, (I I.2) where x 1 , · · · , x n are Cartesian spatial coordinates, t is time and constan t c is a “natural” to the system velocity . In this setting x 0 represen ts the time v ariable. In the most of ph ysical applications of in terest n = 1 , 2 , 3 . W e assume further that the physical system is furnished with its Lagrangian L . Commonly the system Lagrangian L depends on the relev an t ﬁelds u i ( x ) and their ﬁrst order partial deriv atives, which are ∂ j u i ( x ) , ∂ j = ∂ ∂ x j , 0 ≤ j ≤ n, 1 ≤ i ≤ N . (I I.3) In mec hanics of con tinua though the system Lagrangian L may dep end may also dep end on the partial deriv ativ es of the second order. The physical origins of the presence of the second order deriv ativ es are often bending and twisting, Langh [ 28 , Sec. 2.1, 8.7, 8.8]. So to cov er all cases of interest we allow the system Lagrangians to b e dep enden t on the higher order partial deriv atives. T o deal with system Lagrangians that ma y b e dep enden t on the higher order partial deriv atives w e in troduce notations for them using multi-indic es µ : µ = ( µ 0 , µ 1 , . . . , µ n ) , µ j = 0 , 1 , 2 , . . . , 0 ≤ j ≤ n ; | µ | = µ 0 + µ 1 + · · · + µ n . (I I.4) The corresponding partial deriv ativ es ∂ µ then are deﬁned as follo ws: ∂ µ def = ∂ | µ | ∂ x µ 1 1 · · · ∂ x µ n n = n Y j =0 ∂ µ j j , (I I.5) 3 where | µ | is the or der of the p artial derivative . Note that for µ = 0 = (0 , . . . , 0) w e hav e ∂ 0 u = u . Let us denote by I L the set of all m ulti-indices µ suc h that the Lagrangian L in equation ( I I.5 ) dep ends on ∂ µ u i ( x ) for at least one i . F ollo wing the general v ariational setup pro cedure w e consider a set of real-v alued v ariables { v iµ } def = { v iµ : i = 1 , . . . , N , µ ∈ I L } , (I I.6) with index structure matching exactly the same for the partial deriv ativ es ∂ µ u i ( x ) . W e in tro duce then: (i) the system real-v alued Lagrangian function L ( x, { v iµ } ) assuming that it is inﬁnitely diﬀerentiable with resp ect to v ariables { v iµ } (often it is just a p olynomial function of relev ant v ariables); (ii) the corresp onding action inte gr al L using substitution v iµ = ∂ µ u i ( x ) in the L agr angian function L ( x, { v iµ } ) : L ( u ) = ˆ Ω L ( x, { ∂ µ u i ( x ) } ) d x, (I I.7) where Ω ⊆ R n +1 is an open domain in R n . Then the extrema of the action in tegral L ( u ) satisfy the following Euler e quations : X µ ∈ I L ( − 1) | µ | ∂ µ  L v iµ ( x, { ∂ µ u i ( x ) } )  = 0 , i = 1 , . . . , N , (I I.8) where L v iµ def = ∂ v iµ L, i = 1 , . . . , N , µ ∈ I L . (I I.9) A particularly imp ortant case when the Euler-Lagrange equations are linear, that is when the Lagrangian L is a quadratic function of { v iµ } , namely L ( x, { v iµ } ) = 1 2 N X i,j =1 X µ,η ∈ I L a iµ ; j η ( x ) v iµ v j η , (I I.10) where w e ma y assume without loss of generalit y that a iµ ; j η ( x ) = a j η ; iµ ( x ) , i, j = 1 , . . . N , µ, η ∈ I L . (I I.11) In this case the partial deriv ativ es L v iµ deﬁned b y equations ( I I.9 ) tak e the form L v iµ = N X j =1 X η ∈ I L a iµ ; j η ( x ) v j η , (I I.12) and then the corresponding Euler equations ( I I.8 ) turn in to X η ∈ I L ( − 1) | η | ∂ η   N X j =1 X γ ∈ I L a iη ; j γ ( x ) ∂ γ u j ( x )   = 0 , i = 1 , . . . , N . (I I.13) In the case when the system is time and space homogeneous with co eﬃcients a iη ; j γ ( x ) = a iη ; j γ b eing indep endent of x constan ts the Lagrangian ( II.12 ) and the Euler equations ( I I.13 ) turn in to the follo wing respective equations: L = L ( { v iµ } ) = 1 2 N X i,j =1 X µ,η ∈ I L a iµ ; j η v iµ v j η , (I I.14) N X j =1 L ij ( ∂ ) u j ( x ) = 0 , L ij ( ∂ ) def = X η ,γ ∈ I L ( − 1) | η | a iη ; j γ ∂ γ + η , i = 1 , . . . , N . (I I.15) Note that the Euler equations ( II.15 ) are linear as a consequence of the quadratic dep endence of the Lagrangian L on { v iµ } according to equations ( II.14 ). The Euler equations ( II.15 ) can b e also recast in to the follo wing matrix form: L ( ∂ ) u ( x ) = 0 , L ( ∂ ) = { L ij ( ∂ ) } i,j =1 ,...,N , u ( x ) = [ u 1 ( x ) , · · · , u N ( x )] T , (I I.16) where L ( ∂ ) is N × N matrix with each entry b eing diﬀerential op erator L ij ( ∂ ) deﬁned b y equations ( II.15 ) and u ( x ) is a column v ector. 4 B. The disp ersion relations T o use the well-kno wn approac h for analyzing time and space homogeneous systems we consider vector functions u ( x ) of the form u ( x ) = exp    i   k 0 x 0 − n X j =1 k j x j      ˆ u ( k ) , k = ( k 0 , k 1 , · · · , k n ) ∈ R n +1 , k 0 = ω c . (I I.17) Substituting the abov e form v ector function u ( x ) into equation ( I I.16 ) w e obtain the follo wing equation: ˆ L ( k ) ˆ u ( k ) = 0 , k = ( k 0 , k 1 , · · · , k n ) ∈ R n +1 , k 0 = ω c , (I I.18) where ˆ L ( k ) = n ˆ L ij ( k ) o is N × N matrix deﬁned b y ˆ L ( k ) = h ˆ L ij ( k ) i i,j =1 ,...,N , k = ( k 0 , k 1 , · · · , k n ) ∈ R n +1 (I I.19) ˆ L ij ( k ) def = X η ,γ ∈ I L ( − 1) | η | a iη ; j γ ( − 1) k 0 ( − i) | k | k γ + η , i = 1 , . . . , N , (I I.20) k µ def = n Y j =0 k µ j j , | k | = k 0 + k 1 + · · · + k n . (I I.21) Note that equation ( I I.18 ) can b e viewed as a generalized eigenv alue problem with k 0 = ω c b eing an eigenv alue and non trivial ˆ u ( k ) b eing a generalized eigenv ector. According to a w ell kno wn statement from the linear algebra v ector equation ( II.18 ) has a nontrivial (nonzero) solution ˆ u ( k )  = 0 if and only if det n ˆ L ( k ) o = 0 , k = ( k 0 , k 1 , · · · , k n ) ∈ R n +1 , k 0 = ω c , (I I.22) and this equation can b e viewed as the disp ersion r elation b etw een k 0 = ω c and ¯ k = ( k 1 , · · · , k n ) ∈ R n . Note also that equation ( II.22 ) relates the angular frequency ω to angular wa v evector ¯ k and it is a justiﬁcation for calling it the disp ersion relation. I II. COUPLED SYSTEMS AND THE F ACTORIZED FORM OF THE DISPERSION RELA TIONS Quite often a decomp osition of giv en system S in to say tw o interacting (coupled) subsystems S 1 and S 2 is rather clear based on physical grounds. Nevertheless there could b e alternative mathematical formulations of a basis for such a decomp osition. One ph ysically sound approach to the system decomp osition is based on the system Lagrangian assuming that it is av ailable. Being given such a Lagrangian we ﬁrst split the relev an t ﬁelds describing the system S conﬁguration in to t wo groups say U 1 ( x ) = { u i ( x ) : 1 ≤ i ≤ m < N } , U 2 ( x ) = { u i ( x ) : 1 + m ≤ i ≤ N } . (I II.1) W e asso ciate then ﬁelds U 1 ( x ) and U 2 ( x ) with resp ectively subsystems S 1 and S 2 and consider the system Lagrangian decomp osition in to the sum L = L 1 { ∂ µ U 1 ( x ) } + L 2 { ∂ µ U 2 ( x ) } + L 12 , (I II.2) where Lagrangians L 1 and L 2 represen t resp ectiv ely subsystems S 1 and S 2 and L 12 = L − L 1 − L 2 represen ts the in teraction b etw een subsystems S 1 and S 2 . W e exp ect then that there exists a parameter b of the system S that is in volv ed in the in teraction Lagrangian L 12 so that if b = 0 then L 12 = 0 . If that is the case w e refer to such a parameter b as a c oupling p ar ameter. 5 If we can not identify the desired coupling parameter b but still insist on ha ving subsystems S 1 and S 2 as a basis of the system S decomp osition w e set up a family of Lagrangians L ( b ) = L 1 { ∂ µ U 1 ( x ) } + L 2 { ∂ µ U 2 ( x ) } + bL 12 , (I II.3) where Lagrangians L 1 , L 2 and L 12 are the same as deﬁned abov e. In other w ords we in tro duced an additional parameter b into the system Lagrangian. Then eviden tly the original Lagrangian is reco v ered for b = 1 , and b can b e view ed as a coupling parameter since for b = 0 we ha v e L (0) = L 1 { ∂ µ U 1 ( x ) } + L 2 { ∂ µ U 2 ( x ) } , (I II.4) indicating that the subsystems S 1 and S 2 are decoupled. Hence without loss of generalit y we may assume that we alw ays hav e a coupling parameter associated with the decomp osition of a giv en system S in to sa y tw o coupled subsystems S 1 and S 2 . Ha ving a natural to the system S coupling parameter that can b e con trolled b y us is physically preferable of course. In this case we can control the lev el of coupling exp erimen tally . A. Setting up the coupled system Rather often the ph ysical systems at hand can b e naturally decomp osed in to t wo interacting, coupled subsystems. This situation can be sp eciﬁed and quan tiﬁed as follo ws. Based on our studies of TWT systems and the factorized form of the relev an t disp ersion relations w e in tro duce here a general model for factorized disp ersion relations of a system comp osed of some t wo coupled systems. The disp ersion relations emerge when we recast the original homogeneous problem in the frequency-wa v evector domain with ω b eing the frequency and k ∈ R n b eing the w a v evector. Supp ose we hav e tw o initially non-interacting systems. Supp ose also that the systems are go verned by linear ev olution equations and such that the corresponding eigen v alue problem for each of them can b e written in the follo wing form: Λ j Q j = 0 , Λ j = Λ j ( k , ω ) j = 1 , 2 , (II I.5) where Λ j is a n j × n j square matrix and Q j is a n j dimensional column vector for j = 1 , 2 . Then the systems disp ersion relations are det { Λ j ( k , ω ) } = 0 , j = 1 , 2 . (I II.6) Assume no w that the t wo systems in teract and the system composed of these in teracting subsystems is describ ed b y the follo wing linear problem: AQ = 0 , A = Λ + B , Λ = Λ ( k , ω ) =  Λ 1 ( k , ω ) 0 0 Λ 2 ( k , ω )  , (I II.7) B = B ( k , ω ) =  B 11 ( k , ω ) B 12 ( k , ω ) B 21 ( k , ω ) B 22 ( k , ω )  , Q = Q ( k , ω ) =  Q 1 ( k , ω ) Q 2 ( k , ω )  where B is referred to as c oupling matrix where submatrices B ij , j = 1 , 2 may dep end on k and ω . It is conv enien t to mo dify the deﬁnition of coupling matrix B by scaling it with a scalar real v alued factor b . Consequen tly , the eigen v alue problem ( II I.7 ) turns in to AQ = 0 , A = A ( b ) = Λ + bB , Λ =  Λ 1 0 0 Λ 2  , B =  B 11 B 12 B 21 B 22  , Q =  Q 1 Q 2  . (I II.8) Note then that matrix A ( b ) deﬁned by equations ( I II.8 ) dep ends linearly on b and A (0) = Λ and A (1) = Λ + B . In other words, b = 0 corresp onds to the case when subsystems are completely decoupled and describ ed b y the matrix Λ is in equations ( II I.7 ) whereas for b = 1 w e get the original coupling matrix B is in equations ( II I.7 ). Then the dispersion relations of the coupled system are consequen tly det { A ( b ) } = det { Λ ( k , ω ) + bB ( k , ω ) } = 0 . (I II.9) 6 B. F actorized form of the dispersion relation The factorized disp ersion relation assumes that the original system is comp osed of tw o coupled (in teracting) sub- systems. Mathematical representation of the coupling comes through a particular form of the system matrix M k ( b ) where b is a scalar-v alued c oupling c o eﬃcient . Namely , we assume the system matrix M k ( b ) to be of the form M k ( b ) = Λ + bB ( b ) , Λ =  Λ 1 0 0 Λ 2  B ( b ) =  B 11 ( b ) B 12 ( b ) B 21 ( b ) B 22 ( b )  . (I II.10) where matrix B ( b ) is assumed to depend on b polynomially . Note that when the coupling coeﬃcient b = 0 then according to equations ( I I I.10 ) M k (0) = Λ where Λ is a blo c k-diagonal matrix. The fact that Λ is block-diagonal manifests the decomp osition of the original system into t wo n on-in teracting subsystems with resp ective system matrices Λ 1 and Λ 2 . The particular choice bB ( b ) in equations ( I I I.10 ) to represent the subsystems in teraction is justiﬁed by t wo requiremen ts: (i) B ( b ) dep ends on b polynomially and (ii) the coupling/in teraction has to v anish as b = 0 . W e start oﬀ with the following implication of Markus’s formula ( II I.12 ) for det { A + bB ( b ) } . Theorem 1 (determinant of the coupled systems matrix) . L et A and B ( b ) b e two n × n matric es with n ≥ 2 . Assume also that b is a c omplex numb er and matrix B ( b ) dep ends on b p olynomial ly, that is B ( b ) = m X s =0 B ( s ) b s , (I II.11) wher e m ≥ 0 is an inte ger and B ( s ) ar e n × n matric es. Then det { A + bB ( b ) } is a p olynomial function of b satisfying the fol lowing r epr esentation det { A + bB ( b ) } = det { A } + n − 1 X r =1 c r ( b ) b r + b n det { B ( b ) } , (I II.12) wher e c r ( b ) = X α,β ∈ Q n − r,n ( − 1) | α | + | β | det { A [ α | β ] } det { B ( b ) [ α c , β c ] } , 1 ≤ r ≤ n − 1 , n ≥ 2 , (I II.13) Co eﬃcient c 1 ( b ) satisﬁes the fol lowing r epr esentation c 1 ( b ) = tr  A A B ( b )  , (I II.14) wher e A A is the adjugate to A matrix deﬁne d by e quations ( VI.12 ). In the c ase when A is a diagonal matrix e quations ( III.14 ) the fol lowing r epr esentation holds for tr  A A B ( b )  tr  A A B ( b )  =   n X i =1   Y j  = i A j,j   B i,i ( b )   . (I II.15) Equations ( III.12 )-( III.14 ) r e adily imply det { A + bB ( b ) } = det { A } + tr  A A B (0)  b + O  b 2  , b → 0 . (I II.16) Pr o of. F orm ula ( II I.12 ) for det { A + bB ( b ) } follo ws straightforw ardly from Markus’s form ula ( VI.23 ). As to equation ( I II.14 ) for c 1 ( b ) it is v eriﬁed b y using (i) equation ( II I.13 ) for r = 1 and (ii) the deﬁnition ( VI.12 ) of adjugate matrix A A . Finally , asymptotic formula ( I II.16 ) follows readily from equations ( I I I.12 ) and ( I I I.14 ). Note that for any tw o n × n matrices C and D we hav e tr { C D } = n X i =1 C i,j D j,i . (I II.17) Equation ( I I I.15 ) follows from (i) the deﬁnition ( VI.12 ) of adjugate A A applied to the special case of a diagonal matrix A and (ii) equation ( II I.17 ) applied for C = A A and D = B ( b ) . 7 R emark 2 (factorized disp ersion relation) . Applying Theorem 1 to the system matrix M k ( b ) = Λ + bB ( b ) deﬁned in equation ( I II.10 ), with A = Λ , yields the dispersion relation det { M k ( b ) } = 0 . The key step connecting this to the factorized form G 1 G 2 = γ G c is the observ ation that, since Λ = diag ( Λ 1 , Λ 2 ) is block-diagonal, det { Λ } = det { Λ 1 } det { Λ 2 } = G 1 G 2 , (I II.18) where G j = det { Λ j } is the disp ersion function of subsystem S j , j = 1 , 2 . The remaining terms in the expansion ( I I I.12 ) then play the role of γ G c , so that the disp ersion relation det { M k ( b ) } = 0 takes precisely the factorized form ( V.2 ). Th us Theorem 1 is the algebraic engine b ehind the factorization, and the block-diagonal structure of Λ is its physical driv er. In particular, the coupling parameter b of the system matrix M k ( b ) plays the role of the coupling coeﬃcient γ in the factorized dispersion relation ( V.2 ): setting b = 0 decouples the tw o subsystems and recov ers G 1 G 2 = 0 , while increasing b from zero in tro duces the in teraction term γ G c that p erturbs the pro duct of the bare disp ersion functions. IV. PHYSICALL Y APPEALING EXAMPLES W e illustrate here the eﬃciency of our theory by implementing it in a num b er of ph ysically appealing examples. A. T rav eling wa v e tube The TWT-system Lagrangian L TB is deﬁned similarly to its expression in [ 13 , Chap. 4, 24] with the only diﬀerence that there is an additional term related to serial c ap acitanc e C c , namely L TB = L B + L Tb , L B = 1 2 β ( ∂ t q + ˚ v ∂ z q ) 2 − 2 π σ B q 2 , (IV.1) L Tb = L 2 ( ∂ t Q ) 2 − 1 2 C ( ∂ z Q + b∂ z q ) 2 − 1 2 C c Q 2 , where b is the so-called coupling constan t which is a dimensionless phenomenological parameter and other parameters are discussed in [ 13 , Chap. 4, 24]. Constan t b is assumed often to satisfy 0 < b ≤ 1 eﬀectiv ely reducing the inductiv e input of the e-beam current into the shun t current, see [ 13 , Chap. 3] for more details. Note that coupling b et w een the GTL and e-b eam is introduced through term − 1 2 C ( ∂ z Q + b∂ z q ) 2 indicating that the GTL distributed shun t capacitance C is shared with e-b eam. F ollowing the dev elopmen ts in [ 13 , Chap. 4, 24] we in tro duce the TWT princip al p ar ameter γ deﬁned by γ = b 2 C β = b 2 C σ B 4 π ω 2 rp , ω 2 rp = R 2 sc 4 π ˚ ne 2 m . (IV.2) The Euler-Lagrange (EL) equations corresp onding to the Lagrangian L TB deﬁned by equations ( IV.1 ) are the follo wing system of the second-order diﬀeren tial equations L∂ 2 t Q − 1 C ∂ 2 z ( Q + bq ) + 1 C c Q 2 = 0 , (IV.3) 1 β ( ∂ t + ˚ v ∂ z ) 2 q + 4 π σ B q − b C ∂ 2 z ( Q + bq ) = 0 , β = σ B 4 π ω 2 rp . (IV.4) The F ourier transformation (see App endix VI A ) in time t and space v ariable z of equations ( IV.3 ) and ( IV.4 ) yields  k 2 C − ω 2 L + 1 C c  ˆ Q + k 2 b C ˆ q = 0 , (IV.5) bk 2 C ˆ Q + ( b 2 k 2 C + 4 π σ B " 1 − ( ω − ˚ v k ) 2 ω 2 rp #) ˆ q = 0 , (IV.6) where functions ˆ Q = ˆ Q ( k , ω ) and ˆ q = ˆ q ( k , ω ) are the F ourier transforms of the system v ariables Q ( t, z ) and q ( t, z ) . W e will refer to equations ( IV.5 ), ( IV.6 ) as tr ansforme d EL e quations . The TWT-system eigenmo des are naturally assumed to be of the form Q ( z , t ) = ˆ Q ( k , ω ) e − i( ω t − kz ) , q ( z , t ) = ˆ q ( k , ω ) e − i( ω t − kz ) , (IV.7) 8 where ω and k = k ( ω ) are the frequency and the wa v enum ber, resp ectively . Multiplying the EL equations ( IV.5 ), ( IV.6 ) by C we can recast them in to the follo wing matrix form: M kω x = 0 , M kω = " k 2 − ω 2 w 2 + C C c bk 2 bk 2 b 2 k 2 + 4 π C σ B h 1 − ( ω − ˚ vk ) 2 ω 2 rp i # , x =  ˆ Q ˆ q  . (IV.8) Note that equations ( IV.8 ) can be view ed as an eigen v alue type problem for k and x assuming that ω and other parameters are ﬁxed. T aking in to accoun t expressions w = 1 √ C L , ω c = w k c = 1 √ C c L for w and ω c as w ell as expression ( IV.2 ) for the TWT principle parameter γ w e can rewrite equations ( IV.8 ) as M kω x = 0 , M kω = M kω ( b ) =   k 2 + ω 2 c − ω 2 w 2 bk 2 bk 2 b 2 h k 2 + ω 2 rp − ( ω − ˚ vk ) 2 γ i   , x =  ˆ Q ˆ q  . (IV.9) Y et another equiv alen t form of equations ( IV.9 ) can b e obtained b y using phase velocity u = ω k instead of wa ven um b er k in equation ( IV.9 ), namely M uω x = 0 , M uω = M uω ( b ) = " ω 2 u 2 + ω 2 c − ω 2 w 2 bω 2 u 2 bω 2 u 2 h ω 2 u 2 + 1 γ  ω 2 rp − ω 2 ( u − ˚ v ) 2 u 2 i b 2 # , x =  ˆ Q ˆ q  , (IV.10) where w e use once again the principal TWT parameter γ = b 2 C β = b 2 C σ B 4 π ω 2 rp deﬁned b y equations ( IV.2 ). Note that matrices M kω ( b ) and M uω ( b ) satisfy the follo wing factorized represen tation: M kω ( b ) = D b M kω (1) D b , M uω ( b ) = D b M uω (1) D b D b =  1 0 0 b  , (IV.11) where matrices M kω (1) and M uω (1) eviden tly do not depend on b . B. Vibration of an airplane wing A simpliﬁed one-dimensional mo del that accounts for vibrations of an airplane is a beam with v ariable section prop erties and v ariable mass distribution, Langh [ 28 , Sec. 2.1, 8.6]. When the wing vibrates, the segmen t included b et w een tw o neighboring cross-sectional planes is displaced in its plane as a rigid lamina (thin layer, plate) . The displacemen t of the lamina can b e describ ed by rotation angle θ ab out a c hosen p oint P and its translation motion. The rotation θ is indep enden t of the lo cation of point P . It is con venien t to c hoose p oint P to b e the cen ter of mass of the lamina. The wing model parameters and inv olv ed v ariables are as follows: • x is the axial (horizontal) coordinate along the wing; • w = w ( x ) is the vertical displacement, z -axis of the cen ter of mass P = P ( x ) of the lamina at x ; • θ = θ ( x ) is the rotation (twisting, torsion) of the lamina at x ; • m = m ( x ) is the linear mass densit y of the lamina, that is m d x is the mass of a lamina of thickness d x ; • I m = I m ( x ) is the linear density of mass moment, that is I m = I m d x is inertia of the lamina of thic kness d x ab out point P ; • P ′ = P ′ ( x ) is the cen troid of the lamina deﬁned in Remark 3 b elow. Note that p oin ts P and P ′ ordinarily do not coincide; • I = I ( x ) is the linear densit y of the moment of inertia of the cross-section of the structural parts of the wing ab out the principal axis of inertia through p oint P ′ , that is I = I d x is the corresp onding momen t of inertia of the lamina of thic kness d x ; 9 • a = a ( x ) is the algebraic distance b etw een the cen ter of mass P and the centroid P ′ in the direction of y -axis (orthogonal to z and x axes); • E is the Y oung mo dule, J is a the so called p olar moment of inertia , G is the she ar mo dulus , and the pro duct GJ is the called “torsional stiﬀness”, Langh [ 28 , Sec. 2.1, 8.6]. In general quantities J , G and GJ functions of x . F or a solid shaft of circular cross section or for a hollow shaft of ann ular cross section, J is the polar momen t of inertia of the cross section ab out its center. F or any other shap e of cross section, J is less than the p olar momen t of inertia; • The v ertical deﬂection of the cen troid P ′ is appro ximately w + aθ . R emark 3 (centroid) . In mathematics and ph ysics, the centroid, also known as “geometric center” or “center” of ﬁgure, of a plane ﬁgure or solid ﬁgure is the arithmetic mean p osition of all the p oin ts in the ﬁgure. The same deﬁnition extends to an y ob ject in n -dimensional Euclidean space. The cen troid C of a subset S of R n is deﬁned as C = ´ S x d x ´ S d x . (IV.12) The centroid coincides with the center of mass or the center of gra vity only if the material of the b o dy is homogeneous. A geometric cen troidal axis is an axis that passes through the cen troid of a cross section. The concept of centroid arises naturally in man y areas of ph ysics, in particularly in ﬂuid mec hanics, namely the centroid of a bo dy is its buo yancy cen ter, White [ 46 , Sec. 2.8]. The kinetic energy densit y T of the wing can b e represen ted as follo ws: T = 1 2 h m ( ∂ t w ) 2 + I m ( ∂ t θ ) 2 i . (IV.13) By the elemen tary beam theory the strain energy densit y U 1 of bending of the wing and the strain energy U 2 due to t wisting are U 1 = 1 2 E I  ∂ 2 x ( w + aθ )  2 , U 2 = 1 2 GJ ( ∂ x θ ) 2 . (IV.14) Consequen tly , the total strain energy densit y U of the wing is U = U 1 + U 2 = 1 2 E I  ∂ 2 x ( w + aθ )  2 + 1 2 GJ ( ∂ x θ ) 2 . (IV.15) In view of equations ( IV.13 ) and ( IV.15 ) we get the following expression for wing Lagrangian L = T − U = 1 2  m ( ∂ t w ) 2 + I m ( ∂ t θ ) 2 − 1 2 E I  ∂ 2 x ( w + aθ )  2 − 1 2 GJ ( ∂ x θ ) 2  , (IV.16) where vertical displacemen t w = w ( x ) and θ = θ ( x ) is the rotation of the lamina the ﬁelds that determine the wing conﬁguration. Since we interested in the disp ersion relations we hav e to assume from now on that the wing parameters ρ , I m , E , I , a , G and J are constants independent of x . W e can clearly see from the expression ( IV.16 ) for the wing Lagrangian that the wing system is naturally comp osed of tw o subsystems. The ﬁrst subsystem depends on vertical displacemen t w = w ( x ) and the second one depends on rotation θ = θ ( x ) . Indeed, let us introduce a dimensionless coupling parameter b and the Lagrangian L ( b ) = 1 2  m ( ∂ t w ) 2 + I m ( ∂ t θ ) 2 − 1 2 E I  ∂ 2 x ( w + baθ )  2 − 1 2 GJ ( ∂ x θ ) 2  . (IV.17) One can readily v erify that Lagrangian L ( b ) can be decomp osed as follo ws: L ( b ) = L w + L θ + 1 2 ba∂ 2 x θ  ba∂ 2 x θ + 2 ∂ 2 x w  L w = 1 2  m ( ∂ t w ) 2 − 1 2 E I  ∂ 2 x w  2  , L θ = 1 2  I m ( ∂ t θ ) 2 − 1 2 GJ ( ∂ x θ ) 2  10 Note that L (1) = L, L (0) = 1 2  m ( ∂ t w ) 2 − 1 2 E I  ∂ 2 x w  2  + 1 2  I m ( ∂ t θ ) 2 − 1 2 GJ ( ∂ x θ ) 2  . (IV.18) F or b = 1 the Lagrangian L ( b ) turns into the original wing Lagrangian L and b = 0 the Lagrangian L (0) eviden tly represen ts tw o decoupled subsystems with Lagrangians that dep endent resp ectiv ely on v ertical displacement w = w ( x ) and rotation θ = θ ( x ) . This is an example of coupling b et ween v ertical displacement w = w ( x ) and rotation θ = θ ( x ) . The EL equations associated with Lagrangian ( IV.17 ) are as follo ws, Langh [ 28 , Sec. 2.1, 8.6]: m∂ 2 t w + ∂ 2 x ( E I ϕ ) = 0 , ϕ = ∂ 2 x w + b∂ 2 x ( aθ ) , (IV.19) I m ∂ 2 t θ + E I ∂ 2 x ϕ − ∂ x ( GJ ∂ x θ ) − 2 ∂ x ( E I ϕb∂ x a ) + ∂ 2 x ( E I baϕ ) = 0 . (IV.20) In particular in the case when all inv olv ed system parameters E , I , G , J , a and I m are constant the ab ov e EL equations turn in to m∂ 2 t w + E I  ∂ 4 x w + ab∂ 4 x θ  = 0 , (IV.21) I m ∂ 2 t θ − GJ ∂ 2 x θ + E I ba  ∂ 4 x w + ba∂ 4 x θ  = 0 . (IV.22) T o obtain the disp ersion relations asso ciated with the Euler-Lagrange equations ( IV.21 ), ( IV.22 ) we consider the system eigenmodes represented as follo ws: θ ( x, t ) = ˆ θ ( k , ω ) e − i( ω t − kx ) , w ( x, t ) = ˆ w ( k , ω ) e − i( ω t − kx ) , (IV.23) where ω and k = k ( ω ) are the frequency and the wa v en umber, resp ectively . The F ourier transformation (see App endix VI A ) in time t and space v ariable x of the Euler-Lagrange equations ( IV.21 ), ( IV.22 ) can be written in the follo wing matrix form: B b X = 0 , B b =  I m ω 2 −  b 2 a 2 E I k 2 + GJ  k 2 baE I k 4 baE I k 4 mω 2 − E I k 4  , X =  ˆ θ ( k , ω ) ˆ w ( k , ω )  . (IV.24) The abov e form ula readily implies the follo wing equation for the T aylor series of matrix B b at b = 0 : B b =  I m ω 2 − GJ k 2 0 0 mω 2 − E I k 4  + baE I k 4  0 1 1 0  − b 2 a 2 E I k 4  1 0 0 0  , (IV.25) The dispersion relations asso ciated with equations ( IV.24 ) are det { B b } =  I m ω 2 − GJ k 2   mω 2 − E I k 4  − b 2 a 2 E I k 2 mω 2 = 0 , (IV.26) or equiv alently  I m ω 2 − GJ k 2   mω 2 − E I k 4  = b 2 a 2 E I k 2 mω 2 . (IV.27) C. Mindlin-Reissner theory for plates The Mindlin-Reissner is a plate theory for rectangular and circular plates of constant thic kness, see original papers Reiss [ 39 ], Mindlin [ 34 ] and a review pap er Liew [ 30 ]. According to E. Magrab the Mindlin-Reissner theory is an impro ved plate theory which is “the direct equiv alen t of using th e Timoshenk o b eam theory as an impro ved theory with resp ect to the Euler–Bernoulli b eam theory .”, Magrab [ 32 , Sec. 7.1]. W e provide b elo w a concise review of the Mindlin-Reissner following mostly Magrab [ 32 , Sec. 7], Leissa [ 29 , Sec. 12.3], RaoV CS [ 37 , Sec. 14.9], ReddyPS [ 38 , Chap. 10.1]. J. Reddy refers to the Mindlin-Reissner plate theory as the ﬁrst-order shear deformation plate theory (FSDT), ReddyPS [ 38 , Chap. 10.1]. He developed a more accurate third-order shear deformation plate theory (TSDT), ReddyPS [ 38 , Chap. 10.3]. Consider a rectangular plate of constan t thic kness h whose top and bottom surfaces are parallel to the ( x, y ) -plane with the co ordinate system lo cated midwa y b etw een these surfaces. The plate has a length a in the x -direction, a length b in the y -direction. The thickness h , whic h is in the z -direction, is such that h ≪ a and h ≪ b . The plate has a density ρ , a Y oung’s mo dulus E and a Poisson ’s r atio ν . Let u and v b e respectively the in-plane displacements in 11 the x and y directions and w b e the transv erse displacement in the z -direction. The displacement w is assumed to be indep enden t of z and the surfaces of the plate are stress-free; that is σ z z = 0 . As with the Timoshenk o b eam, let us assume that the in-plane displacements are prop ortional to the z coordinate as follows, Graﬀ [ 21 , Sec. 8.3.1], Magrab [ 32 , Sec. 7.1.1], u = z ψ x ( x, y , t ) , v = z ψ y ( x, y , t ) , w = w ( x, y , t ) , (IV.28) where ψ x is the r otation of the cr oss se ction ab out a line p ar al lel to the y -axis and ψ y is the rotation of the cross section about a line parallel to the x -axis. The Lagrangian L for the Mindlin-Reissner theory is deﬁned as follo ws, Magrab [ 32 , Sec. 7.1.2], RaoVCS [ 37 , Sec. 14.9.2], Szil [ 42 , Sec. 4.6]: L = T − U, T = ρh 2  h 2 12 ( ∂ t ψ x ) 2 + h 2 12 ( ∂ t ψ y ) 2 + ( ∂ t w ) 2  , (IV.29) U = D 2  ( ∂ x ψ x ) 2 + ( ∂ y ψ y ) 2 + 2 ν ( ∂ x ψ x ) ( ∂ y ψ y ) + 1 − ν 2 ( ∂ y ψ x + ∂ x ψ y ) 2  (IV.30) + κhG 2 h ( ψ x + ∂ x w ) 2 + ( ψ y + ∂ y w ) 2 i , where T is the kinetic energy density p er unit of area and U is the strain energy densit y p er unit of area. Constant κ th at app ears in expression ( IV.30 ) for the strain energy U is a she ar c orr e ction c o eﬃcient introduced for the Timoshenk o b eam Magrab [ 32 , Sec. 5.2.1]. A typical v alue of shear correction co eﬃcient κ is κ = 5 6 , Magrab [ 32 , Sec. 5.2.1]. Constan t G is the she ar mo dulus and constant D is the ﬂexur al rigidity of the plate deﬁned as follows, Magrab [ 32 , Sec. 6.2.1, 7.1.2], Langh [ 28 , Sec. 5.1], GerRix [ 19 , Sec. 4.4.5]: G = E 2 (1 + ν ) , D = E h 3 12 (1 − ν 2 ) . (IV.31) One recov ers the Kirc hhoﬀ (classical) plate theory Lagrangian L deﬁned by equation ( IV.85 ) from the Mindlin- Reissner plate theory Lagrangian L deﬁned by equations ( IV.29 ) and ( IV.30 ) by (i) setting ψ x = − ∂ x w , ψ y = − ∂ y w in the strain energy densit y U expression, that is no shear strain con tribution; (ii) removing terms inv olving ∂ t ψ x and ∂ t ψ y from the kinetic energy T expression, that is no rotary motion con tribution, Graﬀ [ 21 , Sec. 8.1.1]. The Euler-Lagrange equations corresp onding Lagrangian L deﬁned b y equations ( IV.29 ) and ( IV.30 ) are, Magrab [ 32 , Sec. 7.1.3], RaoV CS [ 37 , Sec. 14.9.2] : ρh 3 12 ∂ 2 t ψ y + κhG ( ψ y + ∂ y w ) − D 2 [(1 − ν ) ∆ ψ y + (1 + ν ) ∂ y Φ] = 0 , (IV.32) ρh 3 12 ∂ 2 t ψ x + κhG ( ψ x + ∂ x w ) − D 2 [(1 − ν ) ∆ ψ x + (1 + ν ) ∂ x Φ] = 0 , (IV.33) ρh∂ 2 t w − κhG (∆ w + Φ) = 0 , Φ = ∂ x ψ x + ∂ y ψ y . (IV.34) W e will refer to the EL ( IV.32 )-( IV.34 ) as Mindlin-Reissner plate equation or MR equations for short. T o construct a factorized form of the dispersion relations related to the MR equations ( IV.32 )-( IV.34 ) w e w ould lik e to em b ed Lagrangian L in to a family of Lagrangians L b where b is real-v alued parameter as follo ws: L b = T − U b , T = ρh 2  h 2 12 ( ∂ t ψ x ) 2 + h 2 12 ( ∂ t ψ y ) 2 + ( ∂ t w ) 2  , (IV.35) U b = D 2  ( ∂ x ψ x ) 2 + ( ∂ y ψ y ) 2 + 2 ν ( ∂ x ψ x ) ( ∂ y ψ y ) + 1 − ν 2 ( ∂ y ψ x + ∂ x ψ y ) 2  (IV.36) + κhG 2 h ( bψ x + ∂ x w ) 2 + ( bψ y + ∂ y w ) 2 i , 12 Note that Lagrangian L 0 represen ts a system for which ﬁeld w and ﬁelds ψ x , ψ y don’t in teract and Lagrangian L 1 is exactly Lagrangian L for the Mindlin-Reissner theory deﬁned b y equations ( IV.29 ) and ( IV.30 ). These facts justiﬁes naming b a c oupling p ar ameter . The presence of coupling parameter b in expressions for quantities of in terest is helpful in assessing the eﬀect of in teraction betw een ﬁeld w and ﬁelds ψ x , ψ y on those quan tities. The EL equations for Lagrangian L b are as follo ws: ρh 3 12 ∂ 2 t ψ y + κhG ( bψ y + ∂ y w ) − D 2 [(1 − ν ) ∆ ψ y + (1 + ν ) ∂ y Φ] = 0 , (IV.37) ρh 3 12 ∂ 2 t ψ x + κhG ( bψ x + ∂ x w ) − D 2 [(1 − ν ) ∆ ψ x + (1 + ν ) ∂ x Φ] = 0 , (IV.38) ρh∂ 2 t w − κhG (∆ w + b Φ) = 0 , Φ = ∂ x ψ x + ∂ y ψ y . (IV.39) Note that in the case of b = 1 the EL equations ( IV.37 )-( IV.39 ) are iden tical to the MR equations ( IV.32 )-( IV.34 ) and from no w on w e refer to them as the Mindlin-Reissner equations. T o obtain the disp ersion relations asso ciated with the MR equations ( IV.37 )-( IV.39 ) we consider the system eigen- mo des represen ted as follo ws: w ( x, y , t ) = ˆ w ( k , ω ) e − i( ω t − k x x − k y y ) , k = ( k x , k y ) , (IV.40) ψ x ( x, y , t ) = ˆ ψ x ( k , ω ) e − i( ω t − k x x − k y y ) , ψ y ( x, t ) = ˆ ψ y ( k , ω ) e − i( ω t − k x x − k y y ) , where ω and k = k ( ω ) are the frequency and the wa v en umber, resp ectively . The F ourier transformation (see App endix VI A ) in time t and space v ariables x, y of the Euler-Lagrange equations ( IV.37 )-( IV.39 ) can b e written in the following matrix form: B b X = 0 , B b def =  A b − i bκk y hG k i bκhG k T h  ρω 2 − κGk 2   , X def =  ˆ Ψ ( k , ω ) ˆ w ( k , ω )  , ˆ Ψ def =  ˆ ψ y ˆ ψ x  , (IV.41) where B b is 3 × 3 is a Hermitian matrix, A is 2 × 2 matrix and k is 2 × 1 matrix (v ector) deﬁned as follo ws: A b def = " ρh 3 12 ω 2 − κhGb 2 − D  1 − ν 2 k 2 x + k 2 y  − D (1+ ν ) 2 k x k y − D (1+ ν ) 2 k x k y ρh 3 12 ω 2 − κhGb 2 − D  1 − ν 2 k 2 y + k 2 x  # , k def =  k y k x  . (IV.42) It turns out that matrix B b has a blo c k diagonal form which is as follows. Let us introduce the follo wing orthonormal basis in R 3 : τ 1 =   0 0 1   , τ 2 =   k x k k y k 0   , τ 3 =   − k x k k y k 0   , ( τ j , τ m ) = δ j m , j, m = 1 . . . 3 , (IV.43) where ( · , · ) is the scalar product in C 3 . Then using v ectors ( IV.43 ) w e deﬁne the follo wing 3 × 3 matrix T k def = [ τ 1 , τ 2 , τ 3 ] =   0 k y k − k x k 0 k x k k y k 1 0 0   , k = q k 2 x + k 2 y . (IV.44) Note that vector τ 2 corresp onds to longitudinal (irrotational, dilational) mo de of oscillations, whereas vectors τ 1 and τ 3 represen t transverse (equivoluminal, distortional) modes of oscillations. It is straigh tforw ard to verify that for any b Hermitian matrix B b satisﬁes the follo wing represen tation: B b = T k C b T − 1 k , T − 1 k = T T k =   0 0 1 k y k k x k 0 − k x k k y k 0   , k = q k 2 x + k 2 y , (IV.45) where C b is the block-diagonal Hermitian matrix 3 × 3 deﬁned b y C b def =   h  ρω 2 − κGk 2  i bκhk G 0 − i bκhk G g ( k , ω ) 0 0 0 f ( k , ω )   , (IV.46) g ( k , ω ) def = ρh 3 12 ω 2 − D k 2 − b 2 hκG, f ( k , ω ) def = ρh 3 12 ω 2 − D (1 − ν ) 2 k 2 − κb 2 hG. (IV.47) 13 Note that equations ( IV.43 ) and ( IV.45 )-( IV.47 ) show that matrix B b can b e blo ck-diagonalized and that v ector τ 3 is an eigen vector of matrix B b , namely B b τ 3 = f ( k , ω ) τ 3 =  ρh 3 12 ω 2 − D (1 − ν ) 2 k 2 − b 2 κhG  τ 3 , τ 3 =   − k x k k y k 0   . (IV.48) Note also that ( B b τ j , τ 3 ) = ( τ j , B b τ 3 ) = f ( k , ω ) ( τ j , τ 3 ) = 0 , j = 1 , 2 , (IV.49) implying that B b T ⊆ T , T = span { τ 1 , τ 2 } , (IV.50) that is space T is an inv ariant under action of matrix B b subspace of R 3 . The latter is consistent with equations ( IV.45 )-( IV.47 ). Equation of in terest B b X = 0 in ( IV.40 ) has a nonzero solution X if and only if det { B b } = 0 , and the latter equation determines the disp ersion relations asso ciated with the Mindlin-Reissner equations ( IV.37 )-( IV.39 ). A tedious but elemen tary analysis of equation det { B b } = det { C b } = 0 and equations ( IV.45 )-( IV.47 ) yield the following factorize d form of the disp ersion r elations for the Mind lin-R eissner plate the ory : f ( k , ω ) A ( k , ω ) = 0 , f ( k , ω ) = ρh 3 12 ω 2 − D (1 − ν ) 2 k 2 − b 2 κhG, (IV.51) A ( k , ω ) =  ρh 3 12 ω 2 − D k 2   ρω 2 − κGk 2  − b 2 κGρhω 2 . (IV.52) Consequen tly , each pairs ( k , ω ) which is a solution to the disp ersion equations ( IV.51 ), ( IV.52 ) must b e also a solution to at least one of the following equations: f ( k , ω ) = 0 or equiv alen tly ρh 3 12 ω 2 − D (1 − ν ) 2 k 2 − b 2 κhG = 0 , (IV.53) A ( k , ω ) = 0 or equiv alen tly  ρh 3 12 ω 2 − D k 2   ρω 2 − κGk 2  = b 2 κGρhω 2 . (IV.54) It is instructiv e to consider the following alternative approac h for obtaining dispersion relations ( IV.51 )-( IV.52 ). Let us solve the ﬁrst tw o equations of the system B b X = 0 for ﬁelds ˆ ψ y and ˆ ψ x and obtain their represen tation in terms of ˆ w . If we then plug in the obtained expressions into the third equation of the system B b X = 0 we ﬁnd that the resulting equation to b e of the form A ( k , ω ) ˆ w = 0 , where A ( k , ω ) is deﬁned b y equation ( IV.52 ). The latter equation eviden tly has a nontrivial solution ˆ w if and only if A w ( k , ω ) = 0 . Rep eating similar developmen ts for v ariables ˆ ψ x and ˆ w and for v ariables ˆ ψ y and ˆ w w e obtain equation f ( k , ω ) A ( k , ω ) = 0 for the both cases. The described alternative approac h for obtaining the disp ersion relations ( IV.51 )-( IV.52 ) is consisten t with somewhat diﬀerent approach used b y S. Rao, RaoV CS [ 37 , Sec. 14.9]. Namely the author solv es the MR equations ( IV.37 )-( IV.39 ) for ﬁelds ψ x , ψ y and w for case of free vibrations assuming the ﬁelds to b e time-harmonic. Approac h pursued by Rao in RaoV CS [ 37 , Sec. 14.9.3] is based on a representations of ﬁelds ψ x , ψ y in terms of the Lame p oten tials that correspond to the dilatation and shear components of motion of the plate, ErinSuh [ 11 , Sec. 7.3, 7.9]. The dispersion equation ( IV.54 ), that is A ( k , ω ) = 0 , can b e readily recast into the velocity disp ersion relation, namely 1 12 h 2 k 2  1 − c 2 κc 2 T   c 2 P c 2 − 1  = b 2 , c = ω k , (IV.55) where c L and c T are resp ectively the longitudinal and the transverse wa v e sp eeds for 3D homogeneous isotropic elastic medium free of bo dy forces deﬁned as follo ws c L = s λ + 2 G ρ = s E (1 − ν ) ρ (1 + ν ) (1 − 2 ν ) , c T = s G ρ = s E 2 ρ (1 + ν ) . (IV.56) 14 As to velocity c P it is the v elo cit y of the so-called extensional wa v es in thin plates deﬁned by the following form ula, Bish [ 7 ], A c hen [ 1 , Sec. 2.7, 6.12.3], ErinSuh [ 11 , Sec. 7.3], Graﬀ [ 21 , Sec. 4.3.2, 8.3.1]: c P = s E ρ (1 − ν 2 ) . (IV.57) Indeed, an analysis of the plane-stress problem for thin plates shows that the passage from the plane-strain problem to the corresp onding plane-stress problem can b e made by simply replacing the Lame constant λ with its mo diﬁed v alue λ ′ , namely , Bish [ 7 ], Ac hen [ 1 , Sec. 2.7, 6.12.3], ErinSuh [ 11 , Sec. 7.3, 7.9]: λ ′ = 2 Gλ λ + 2 G = ν E 1 − ν 2 . (IV.58) Using this modiﬁed v alue λ ′ of the Lame constan t in place of λ and the expression for longitudinal w a v e speed c L = q λ +2 G ρ one obtains expression ( IV.57 ) of the phase velocity c P , namely c P = s λ ′ + 2 G ρ = s E ρ (1 − ν 2 ) . (IV.59) Equations ( IV.56 ) and ( IV.57 ) readily imply the following relations betw een v elo cities c T and c P , c 2 T c 2 P = 1 − ν 2 , (IV.60) Note that disp ersion relation ( IV.55 ) matches exactly the v elo cit y disp ersion relation in Graﬀ [ 21 , Sec. 8.3.1] for the special case when b = 1 corresponding to the Mindlin-Reissner plate theory . The dispersion relations ( IV.53 ) can b e recast as the velo city disp ersion r elation as follo ws c = c T hk p h 2 k 2 + 12 κb 2 = c T r 1 + 12 κb 2 h 2 k 2 , c = ω k . (IV.61) Note that equation ( IV.61 ) implies the following asymptotic form ulas: c = c T " 2 b √ 3 κ h k − 1 + h 4 b √ 3 κ k + O  k 3  # , c = ω k , k → 0 , (IV.62) c = c T  1 + 6 κb 2 h 2 k 2 + O  k − 4   , c = ω k , k → ∞ . (IV.63) It is instructiv e to consider a sp ecial case b = 0 when ﬁelds ψ x , ψ y and w are decoupled. In this case equations ( IV.41 ), ( IV.42 ), ( IV.46 ) turn in to B 0 X = 0 , B 0 def =  A 0 0 0 h  ρω 2 − κGk 2   , X def =  ˆ Ψ ( k , ω ) ˆ w ( k , ω )  , ˆ Ψ def =  ˆ ψ y ˆ ψ x  , (IV.64) where A 0 is 2 × 2 matrix and k is 2 × 1 matrix (v ector) deﬁned as follo ws: A 0 def = " ρh 3 12 ω 2 − D  1 − ν 2 k 2 x + k 2 y  − D (1+ ν ) 2 k x k y − D (1+ ν ) 2 k x k y ρh 3 12 ω 2 − D  1 − ν 2 k 2 y + k 2 x  # , k def =  k y k x  , (IV.65) C 0 =    h  ρω 2 − κGk 2  0 0 0 ρh 3 12 ω 2 − D k 2 0 0 0 ρh 3 12 ω 2 − D (1 − ν ) 2 k 2    , (IV.66) 15 The velocity dispersion equations ( IV.55 ) and ( IV.61 ) yield resp ectiv ely the following non disp ersive v alues of the c haracteristic velocities when b = 0 : T = span { τ 1 , τ 2 } = span      0 0 1   ,   k x k k y k 0      : √ κc T = s E κ 2 ρ (1 + ν ) , c P = s E ρ (1 − ν 2 ) ; (IV.67) τ 3 =   − k x k k y k 0   : c T = s E 2 ρ (1 + ν ) . (IV.68) W e remind that vector τ 2 corresp onds to longitudinal (irrotational, dilational) mo de of oscillations associated with v elo cit y c P , whereas vectors τ 1 and τ 3 represen t transverse (equiv oluminal, distortional) mod es of oscillations, ha ving resp ectiv ely v elo cities √ κc T and c T . In the case of arbitrary b v ector τ 3 represen ts oscillations propagating at the sp eed satisfying the velocity disp ersion equation ( IV.61 ), whereas vectors in the inv ariant subspace T = span { τ 1 , τ 2 } hav e characteristic velocities that satisfy the v elo cit y disp ersion equation ( IV.55 ). 1. Hybridization of mo des W e analyze here the eﬀect of coupling co eﬃcient b on dispersion relations ( IV.53 ) and ( IV.54 ). W e start oﬀ with illustrating graphically the analytical dev elopments of Section IV C by plotting the disp ersion relations ( IV.53 ) and ( IV.54 ) for three sets of data diﬀering only in the v alue of the coupling co eﬃcient b : ρ = 1 , h = 1 , D = 1 , ν = 1 2 , κ = 1 , G = 1 , b = 0 , b = 0 . 1 , b = 0 . 2 . (IV.69) The ﬁrst set in ( IV.69 ) has b = 0 (no coupling), while the other tw o sets hav e b = 0 . 1 and b = 0 . 2 respectively . The dashed blue curve corresp onds to b = 0 , the solid crimson curv e to b = 0 . 1 , and the solid dark green curv e to b = 0 . 2 . Disp ersion r elation f ( k , ω ) = 0 . Figure IV.1 sho ws the dispersion relation f ( k , ω ) = 0 , namely ρh 3 12 ω 2 − D (1 − ν ) 2 k 2 − b 2 κhG = 0 . (IV.70) F or b = 0 this reduces to the straight-line pair ω = ± p 6 D (1 − ν ) / ( ρh 3 ) k passing through the origin. F or b > 0 the coupling term b 2 κhG shifts the branches aw a y from the origin: the intercept at k = 0 becomes ω 0 = ± b p 12 κG/ ( ρh 2 ) , so the f = 0 branc hes are lifted oﬀ the origin b y the coupling. FIG. IV.1. Dispersion relation f ( k , ω ) = 0 for data sets ( IV.69 ). Dashed blue: b = 0 ; solid crimson: b = 0 . 1 ; solid dark green: b = 0 . 2 . The coupling lifts the solid branches oﬀ the origin. 16 Disp ersion r elation A ( k , ω ) = 0 . Figure IV.2 shows the disp ersion relation A ( k , ω ) = 0 , namely  ρh 3 12 ω 2 − D k 2   ρω 2 − κGk 2  = b 2 κGρh ω 2 , (IV.71) together with a zo omed view near the origin. F or b = 0 the equation factors into t w o pairs of straight lines through the origin: ω = ± p 12 D / ( ρh 3 ) k (from the ﬁrst factor) and ω = ± p κG/ρ k (from the second factor), giving four branc hes all pinned at ( k , ω ) = (0 , 0) . F or b > 0 the picture changes markedly: the t wo upp er branches (larger | ω | ) are lifted oﬀ the origin, while the tw o lower branches (smaller | ω | ) appear to remain pinned. The zo omed plot (right panel of Figure IV.2 ) conﬁrms that the low er branches do indeed pass through the origin, approaching it parab olically rather than linearly . (a) (b) FIG. IV.2. Dispersion relation A ( k , ω ) = 0 for data set ( IV.69 ). Dashed blue: b = 0 ; solid crimson: b = 0 . 1 ; solid dark green: b = 0 . 2 . (a) F ull view, k ∈ ( − 0 . 3 , 0 . 3) ; (b) zoomed view of low er branches near origin, k ∈ ( − 0 . 05 , 0 . 05) , with dotted curves sho wing the parab olic approximation ω = k 2 /b . The tw o upp er branches are lifted oﬀ the origin b y coupling, while the tw o lo wer branches are pinned at the origin with parabolic tangency . Asymptotic analysis: hybridization of mo des. W e now carry out an asymptotic analysis of A ( k , ω ) = 0 near the origin to determine precisely which factors — and hence which mo des — gov ern each branch. Dividing equation ( IV.71 ) b y ω 4 and in tro ducing S = k 2 /ω 2 w e obtain the quadratic κGD · S 2 −  ρh 3 κG 12 + ρD  S + ρ 2 h 3 12 = b 2 κGρh ω 2 . (IV.72) In tro ducing the shorthand P = ρh 3 κG 12 + ρD , Q = ρh 3 κG 12 − ρD , R = 2 bκG p D ρh, (IV.73) the t wo solutions are S ± = P ± p Q 2 + R 2 /ω 2 2 κGD . (IV.74) Note that P and Q inv olv e parameters from b oth factors: ρD from the ﬁrst and ρh 3 κG/ 12 from the second, while R in v olves the coupling b together with parameters from b oth factors. Pulling out 1 / | ω | from the square ro ot and expanding √ 1 + ε = 1 + 1 2 ε − 1 8 ε 2 + · · · with ε = Q 2 ω 2 /R 2 , w e obtain the Lauren t series for small | ω | : S + = R 2 κGD · 1 | ω | + P 2 κGD + Q 2 4 κGD R | ω | − Q 4 16 κGD R 3 | ω | 3 + · · · , (IV.75) S − = − R 2 κGD · 1 | ω | + P 2 κGD − Q 2 4 κGD R | ω | + Q 4 16 κGD R 3 | ω | 3 + · · · . (IV.76) 17 Both series ha ve b een veriﬁed b y direct substitution in to the quadratic ( IV.72 ): ev ery coeﬃcient from | ω | − 2 through | ω | 3 v anishes identically . The tw o series hav e opp osite signs in the singular leading term, with decisive physical consequences. Since k 2 = ω 2 S , w e analyze eac h branch in turn: S + br anch. The leading term + R/ (2 κGD | ω | ) > 0 dominates as ω → 0 , so S + > 0 for all small ω and k 2 = ω 2 S + ∼ R | ω | / (2 κGD ) → 0 : this is the lower pinne d br anch , approac hing the origin parab olically . Inv erting k 2 = ω 2 S + as a Puiseux series ω = c 1 k 2 + c 2 k 4 + c 3 k 6 + · · · and solving order by order (v eriﬁed b y direct substitution into A ( k , ω ) = 0 ) giv es: c 1 = √ D b √ ρh , c 2 = − √ D (12 D + κGh 3 ) 24 κG b 3 h 3 / 2 √ ρ , c 3 = √ D (4 D + κGh 3 )(36 D + κGh 3 ) 384 κ 2 G 2 b 5 h 5 / 2 √ ρ , (IV.77) so that ω = ±  c 1 k 2 + c 2 k 4 + c 3 k 6 + · · ·  . (IV.78) S − br anch. The leading term − R/ (2 κGD | ω | ) < 0 dominates as ω → 0 , so S − < 0 for small | ω | , giving k 2 = ω 2 S − < 0 — no real k exists. The branch is absent near the origin and only becomes physical (real k ) once | ω | exceeds the threshold ω 0 found b y setting k = 0 in ( IV.71 ) and dividing b y ω 2  = 0 : ω 2 0 = 12 b 2 κG ρh 2 , ω 0 = 2 b √ 3 κG √ ρ h . (IV.79) This is the upp er lifte d br anch . Expanding around ω 0 as ω = ω 0 + d 1 k 2 + d 2 k 4 + · · · and solving order by order giv es: d 1 = √ 3  D + κGh 3 12  √ κG b h 2 √ ρ , d 2 = − √ 3 (144 D 2 + 72 D κGh 3 + κ 2 G 2 h 6 ) 576 ( κG ) 3 / 2 b 3 h 3 √ ρ , (IV.80) so that ω = ±  ω 0 + d 1 k 2 + d 2 k 4 + · · ·  . (IV.81) Sev eral conclusions follo w from the series ( IV.78 ) and ( IV.81 ). Pinning c onﬁrme d. The low er t wo branc hes ( IV.78 ) are pinned at the origin with parab olic tangency ω ∼ c 1 k 2 . The upp er tw o branches ( IV.81 ) are lifted oﬀ the origin to ± ω 0 , where ω 0 → 0 as b → 0 : they exist only b ecause of the coupling. Both mo des c ontribute at every or der. In the low er branch ( IV.78 ), the leading co eﬃcient c 1 = √ D / ( b √ ρh ) in volv es only D , ρ , h from the ﬁrst factor, so the parab olic curv ature is set by the ﬁrst mode alone. How ev er, c 2 already in volv es κG from the se c ond factor, and all higher co eﬃcients mix b oth. In the upp er branch ( IV.81 ), the threshold ω 0 in volv es κG (second factor) and ρh (ﬁrst factor), and every co eﬃcien t d 1 , d 2 , . . . mixes parameters from b oth factors. Hybridization. The coupling b  = 0 results in hybridization of the tw o uncoupled mo des in all four branc hes of A ( k , ω ) = 0 : the low er branches are go v erned to leading order b y the ﬁrst factor but receive second-factor corrections at every higher order, while the upp er branches are a genuinely hybrid phenomenon whose very existence requires the in teraction of b oth modes. L ar ge- | ω | and lar ge- | k | asymptotics: r e c overy of pur e mo des. The quadratic ( IV.72 ) reveals an elegant complemen- tary picture in the opp osite limit. As | ω | → ∞ the righ t-hand side b 2 κGρh/ω 2 → 0 , which is algebraically identical to setting b = 0 . The asymptotic equation for S is therefore simply κGD · S 2 − P · S + ρ 2 h 3 12 = 0 , (IV.82) with t wo ro ots S ∞ ± = P ± | Q | 2 κGD =      ρ κG , (+) ρh 3 12 D , ( − ) (IV.83) yielding the asymptotic slopes ω ∼ ± s κG ρ k and ω ∼ ± s 12 D ρh 3 k , | ω | , | k | → ∞ . (IV.84) 18 These are precisely the slop es of the tw o uncoupled ( b = 0 ) straight-line branches. Hence all four coupled branc hes are asymptotically straigh t lines at large | ω | and | k | , with slop es en tirely determined by the individual factors — the coupling term b 2 κGρh/ω 2 deca ys as 1 /ω 2 and b ecomes negligible. The hybridization is therefore a low-fr e quency, smal l-wavenumb er phenomenon : it is most pronounced near the origin and fades a wa y as | ω | , | k | → ∞ , where each branc h asymptotically reco vers the identit y of a single pure mode. Comp arison with the cr oss-p oint mo del and gr owth of hybridization with c oupling. The structural parallel b et ween A ( k , ω ) = 0 and the cross-p oint mo del ( V.10 ) is illuminating. In both cases the factorized left-hand side is a pro duct of the tw o individual mode disp ersion functions, and the righ t-hand side is the coupling term. Setting the righ t-hand side to zero recov ers the uncoupled straigh t-line branc hes; any nonzero right-hand side forces every branch to carry the imprint of both factors. The cross-p oint model is in fact the local (linearized near the crossing) approximation to the general story: it applies near (0 , 0) in the Mindlin-Reissner case just as it do es near any cross-p oint ( ω 0 , k 0 ) in the general factorized system. The degree of hybridization grows monotonically with the coupling and is directly readable from the plots. In Figure V.1 the coupled branches progressively depart from the dashed uncoupled reference lines as γ increases: the a voided-crossing gap widens and the branches curve more strongly , mixing the t w o modes ever more thoroughly . F or large | κ | and | δ | the coupled branc hes visibly return to the reference lines, conﬁrming the asymptotic reco v ery of pure mo des. The same tendency app ears in Figure IV.2 : as b increases from 0 to 0 . 1 to 0 . 2 , the upp er branc hes are lifted higher ( ω 0 ∝ b ) and the parab olic lo wer branc hes op en more slo wly ( ω ∼ k 2 /b , so the curv ature decreases with b ) — b oth are signatures of stronger hybridization near the origin. Contr asting dir e ctions of hybridization gr owth. In the cross-point mo del larger γ alwa ys widens the av oided-crossing gap, while in th e Mindlin-Reissner model larger b pushes the lo w er parabolic branc hes closer to the k -axis (smaller ω for ﬁxed k ) — a subtler but equally unam biguous signature of increased mode mixing. In b oth cases the asymptotic straigh t-line b eha vior at large | ω | and | k | is indep enden t of the coupling strength, conﬁrming that h ybridization is conﬁned to the neigh b orhoo d of the cross-point. Summary. The factorized form G 1 G 2 = γ G c mak es mo de hybridization not merely a qualitative statement but a quan titatively precise one: the coupling parameter b controls the degree of mixing at ev ery order of the asymptotic expansions, and the deviation of each coupled branc h from the uncoupled reference curves pro vides a direct measure of hybridization that grows with b near the origin and v anishes asymptotically at large frequencies and w a ven um b ers. 2. Classic al Kir chhoﬀ ’s plate the ory W e concisely review here the classical Kirchhoﬀ ’s plate theory whic h is analogous of the Bernoulli-Euler beam theory . In the case of small deﬂections its Lagrangian is, Langh [ 28 , Sec. 5.1, 8.8], Magrab [ 32 , Sec. 6.2.2], GerRix [ 19 , Sec. 4.4] L = 1 2 ρh ( ∂ t w ) 2 + D (1 − ν )  ∂ 2 x w ∂ 2 y w −  ∂ 2 xy w  2  − 1 2 D  ∂ 2 x w + ∂ 2 y w  2 , (IV.85) where w = w ( x, y , t ) is the plate deﬂection, h is the plate thickness, ν is Poisson ’s r atio , ρ is the mass density and D is the ﬂexur al rigidity deﬁned by , Magrab [ 32 , Sec. 6.2.2], Langh [ 28 , Sec. 5.1], GerRix [ 19 , Sec. 4.4.5] D = E h 3 12 (1 − ν 2 ) . (IV.86) The Euler-Lagrange equation corresp onding to Lagrangian L deﬁned by equation ( IV.85 ) is, Langh [ 28 , Sec. 5.1, 8.8], Magrab [ 32 , Sec. 6.2.3], GerRix [ 19 , Sec. 4.4.9] ρh∂ 2 t w + D ∆ 2 w = 0 , ∆ 2 =  ∂ 2 x + ∂ 2 y  2 = ∂ 4 x + 2 ∂ 2 x ∂ 2 y + ∂ 4 y . (IV.87) The fundamental diﬀeren tial equation ( IV.87 ) in the classical theory of vibration of plates w as derived b y Lagrange, Langh [ 28 , Sec. 8.8]. Note that the second term of the Lagrangian L in equation ( IV.85 ) mak es no contribution to the Euler-Lagrange equation ( IV.87 ). W e brieﬂy review here the theory of b ending of plates follo wing Tim W oi [ 43 , Chap. 1.1]: “ ...the simple problem of the b ending of a long rectangular plate that is sub jected to a transverse load that do es not v ary along the length of the plate. The deﬂected surface of a p ortion of suc h a plate at a considerable distance from the ends can b e assumed cylindrical, with the axis of the cylinder parallel to the 19 length of the plate. W e can therefore restrict ourselves to the in vestigation of the bending of an elemen tal strip cut from the plate by tw o planes perp endicular to the length of the plate and a unit distance (say 1 in.) apart. The deﬂection of this strip is giv en b y a diﬀerential equation whic h is similar to the deﬂection equation of a bent beam. ” Supp ose that the plate has uniform thickness h and let xy b e the middle plane of the plate b efore loading. Let the y -axis coincide with one of the longitudinal edges of the plate and let the p ositive direction of the z axis be down w ard and the plate is b end down w ards under the load. Supp ose that w is the deﬂection of the plate in the z direction and w is assumed to b e small. Then the disp ersion relations that corresp ond to equations ( IV.87 ) are, Graﬀ [ 21 , Sec. 4.2.3] ρhω 2 = D k 4 , k 2 = k 2 x + k 2 y . (IV.88) The Kirchhoﬀ theory inv olves a single ﬁeld w and its dispersion relation ( IV.88 ) has no tw o-subsystem factorized structure; it serves here as the classical reference theory against which the richer Mindlin-Reissner framework, with its coupling b etw een the transverse deﬂection w and the rotational ﬁelds ψ x , ψ y , is to b e compared. W e remark that asymptotic approaches to factorizing plate disp ersion relations, in a spirit related to the present work, ha v e been dev elop ed by Kaplunov and collaborators, see e.g. KapNolRog [ 26 ], KapNob [ 27 ], ChebKapRog [ 9 ], AlzKapPri [ 3 ], where p olynomial approximations of the Rayleigh-Lam b and Mindlin plate disp ersion relations are deriv ed that eﬀec- tiv ely isolate individual wa v e branc hes. The present approac h diﬀers in that the factorization is achiev ed algebraically through the Lagrangian coupling-parameter framew ork rather than b y asymptotic expansion in a small parameter. V. CR OSS-POINT MODEL FOR F A CTORIZED DISPERSION RELA TIONS The cross-p oin t mo del introduced by us in FigFDT1 [ 14 ] is arguably the simplest mo del illustrating the eﬀect of coupling on the disp ersion relations of a system composed of tw o in teracting subsystems. W e provide here a brief review of this model. Let us assume that there are tw o initially non-in teracting systems with the disp ersion relations deﬁned by equations G 1 ( k , ω ) = 0 , G 2 ( k , ω ) = 0 . (V.1) Supp ose then that the tw o systems are coupled and the disp ersion relations for this in teracting system is of the follo wing factorized form G 1 ( k , ω ) G 2 ( k , ω ) = γ G c ( k , ω ) , (V.2) where γ is the c oupling c o eﬃcient and w e refer to G c ( k , ω ) as the c oupling function . W e assume v ariables k and ω b e real-v alued or complex-v alued. Supp ose no w that ( ω 0 , k 0 ) is a “cr oss-p oint” of the graphs of functions G 1 and G 2 , that a p oin t satisfying the tw o disp ersion relations ( V.1 ), namely G 1 ( ω 0 , k 0 ) = 0 , G 2 ( ω 0 , k 0 ) = 0 . (V.3) Supp ose also coupling parameter γ to b e small, and consider solutions to equation ( V.2 ) in a small vicinit y of p oin t ( ω 0 , k 0 ) , that is ( k , ω ) = ( k 0 + κ, ω 0 + δ ) , | δ | , | κ | ≪ 1 , (V.4) Assuming that equations ( V.3 ) and ( V.4 ) hold and that | δ | , | κ | are small, that is | κ | ≪ 1 , | δ | ≪ 1 , (V.5) w e arrive at the following princip al appr oximation to the disp ersion equation ( V.2 ) ( g 1 ω δ + g 1 k κ ) ( g 2 ω δ + g 2 k κ ) = γ g c , (V.6) where the constan ts g j ω , g j k and g γ are deﬁned b y g j ω = ( ∂ ω G j ) ( ω 0 , k 0 ) , g j k = ( ∂ k G j ) ( ω 0 , k 0 ) , j = 1 , 2; g c = G c ( ω 0 , k 0 ) . (V.7) F or generic v alues of co eﬃcients g j ω , g j k and g γ for whic h g 1 ω δ + g 1 k κ and g 2 ω δ + g 2 k κ are linearly indep endent, w e can transform equations ( V.6 ) in to a simple special form by the following c hange of coordinates g 1 ω δ + g 1 k κ = δ ′ + κ ′ , g 2 ω δ + g 2 k κ = δ ′ − κ ′ . (V.8) 20 Indeed equation ( V.6 ) can be recast in terms of these v ariables as δ ′ 2 − κ ′ 2 = γ g c . (V.9) Note now that the graph of equation ( V.9 ) is a hyperb ola implying that the graph of original equation ( V.6 ) is a linear transformation of the h yperb ola asso ciated with special form ( V.9 ). In summary, we may c onclude that generic al ly the gr aph of the disp ersion r elations of two inter acting systems in a vicinity of the r elevant interse ction p oint is a line ar tr ansformation of the hyp erb ola if the c oupling p ar ameter γ is smal l. In case when g 1 ω  = 0 and g 2 ω  = 0 we can divide b oth sides of equation ( V.6 ) b y g 1 ω g 2 ω obtaining the follo wing equiv alent equation ( δ + g 1 κ ) ( δ + g 2 κ ) = γ g γ , g 1 def = g 1 k g 1 ω , g 2 def = g 2 k g 2 ω , g γ def = g c g 1 ω g 2 ω . (V.10) W e will refer to disp ersion equation ( V.10 ) as the cr oss-p oint principle mo del disp ersion r elations , and Figure V.1 sho ws the plots of these relations for g 1 = 1 , g 2 = 10 , γ = 0 . 4 , 2 , 4 , and b oth signs of g γ . In the ﬁgure: solid curv es sho w the disp ersion curves for the indicated v alues of γ (ro y al blue: γ = 0 . 4 , crimson: γ = 2 , dark green: γ = 4 ); dashed blue straigh t lines sho w the uncoupled ( γ = 0 ) reference curv es. FIG. V.1. Cross-p oin t principle model disp ersion relations ( V.10 ) for g 1 = 1 , g 2 = 10 , γ = 0 . 4 , 2 , 4 : (a) g γ = +1 > 0 ; (b) g γ = − 1 < 0 . Dashed blue: uncoupled reference lines ( γ = 0 ); solid ro yal blue, crimson, dark green: coupled curv es for γ = 0 . 4 , 2 , 4 respectively . Curv es farther from the dashed reference lines correspond to larger v alues of γ . Note that coupled mo de theory is y et another example that yields frequency dep endence on a parameter (detuning frequency) with graphical represen tation HausHua [ 23 , Fig. 1] similar to Figure V.1 . The asymptotic b eha vior of the cross-p oint model is also worth noting. F or large | κ | and | δ | the right-hand side γ g γ of ( V.10 ) b ecomes negligible compared to the left-hand side, and the coupled branc hes asymptotically approach the uncoupled reference lines δ = − g 1 κ and δ = − g 2 κ . More precisely , the deviation of each branc h from the nearer reference line deca ys as O (1 /κ ) for large | κ | , since from ( V.10 ) the deviation ∆ δ satisﬁes ∆ δ ∼ γ g γ / (( g 2 − g 1 ) κ ) . This is the cross-p oint analog of the large- | ω | recov ery of pure mo des established for A ( k, ω ) = 0 in Section IV C 1 (equations ( IV.82 )–( IV.84 )), where the coupling term deca ys as b 2 /ω 2 : in b oth models the coupling term on the righ t-hand side of the factorized equation b ecomes relatively small far from the cross-p oin t, so the h ybridization is spatially concen trated near the crossing and the branches reco v er their individual mo de c haracter at large | κ | , | δ | (or equiv alently large | k | , | ω | ). This b ehavior is clearly visible in Figure V.1 : all coupled branc hes, regardless of the v alue of γ , asymptotically conv erge to the tw o dashed reference lines. 21 A. Lagrangian framework for the cross-p oin t mo del Let us consider the follo wing general form of the disp ersion function and the corresponding disp ersion relations G ( k , ω ) def = Aω 2 − 2 ω k B − C k 2 − D , G ( k , ω ) = 0 . (V.11) The choice of signs before co eﬃcien ts in equations ( V.11 ) is motiv ated by its applications to the GTL, the e-b eam and other ph ysical systems. It is natural and important to ask if the dispersion relations ( V.11 ) can b e asso ciated with a “real physical system”, that is with the Euler-Lagrange equations of a Lagrangian. The answer to this question is p ositive, and an expression for suc h a Lagrangian L G is as follo ws: L G ( ∂ t Q, ∂ z Q, Q ) def = Q 2 2 G  ∂ z Q Q , ∂ t Q Q  ≡ 1 2 h A ( ∂ t Q ) 2 + 2 B ∂ t Q∂ z Q − C ( ∂ z Q ) 2 − D Q 2 i , (V.12) where Q = Q ( z , t ) . Indeed, the EL equations for Lagrangian L G deﬁned b y equations ( V.12 ) are  A∂ 2 t + 2 B ∂ t ∂ z − C ∂ 2 z + D  Q = 0 . (V.13) T o ﬁnd the disp ersion relations associated with the EL equation ( V.13 ) we pro ceed in the standard fashion and consider the system eigenmodes of the form Q ( z , t ) = ˆ Q ( k , ω ) e − i( ω t − kz ) . (V.14) Plugging in expression ( V.13 ) for Q ( z , t ) in the EL equation ( V.13 ) after elemen tary ev aluations we obtain e − i( ω t − kz ) ˆ Q ( k , ω )  − Aω 2 + 2 ω k B + C k 2 + D  = 0 . (V.15) Assuming naturally that ˆ Q ( k , ω ) b eing an amplitude of an eigenmo de is not zero we recov er from equation ( V.15 ) the follo wing disp ersion relation associated with the EL equation ( V.13 ) − Aω 2 + 2 ω k B + C k 2 + D = 0 , whic h is evidently equiv alent to the original disp ersion relation ( V.11 ). Hence indeed the Lagrangian L G deﬁned by equation ( V.12 ) yields indeed the EL equation having the desired disp ersion relation ( V.11 ). Motiv ated by the cross-point disp ersion relations ( V.10 ) w e introduce cross-p oin t disp ersion relations G crp ( k , ω ) def = ( ω + g 1 k ) ( ω + g 2 k ) − γ g γ , G crp ( k , ω ) = 0 . (V.16) Then according to form ula ( V.12 ) the corresponding to disp ersion relations ( V.16 ) Lagrangian L crp is of the form L crp = 1 2  ( ∂ t Q + g 1 ∂ z Q ) ( ∂ t Q + g 2 ∂ z Q ) − γ g γ Q 2  . (V.17) The expression ( V.17 ) can b e also readily obtained from the last expression of relations ( V.12 ) b y setting up there the follo wing v alues of co eﬃcients: A = 1 , B = g 1 + g 2 2 , C = − g 1 g 2 , D = γ g γ . (V.18) B. Mec hanical analog of the cross-p oin t mo del The cross-p oin t disp ersion relation ( V.10 ) describ es the b ehavior of t wo coupled con tin uum subsystems near a crossing p oin t in the ( ω , k ) plane. W e present here a ﬁnite-dimensional mechanical analog in whic h the wa v enum ber k is replaced b y a scalar parameter p , yielding a system whose eigenfrequencies exhibit the same factorized structure and crossing b ehavior as ( V.10 ). The construction is based on the coupled-oscillator framework of Likh [ 31 , Sec. 6.1], mo diﬁed to in troduce a p -dep enden t Lagrangian. Consider t wo harmonic oscillators with masses m j and b -dependent spring constan ts κ j ( b ) = κ j − bκ, j = 1 , 2 , (V.19) 22 where κ j are the bare spring constants, κ > 0 is a coupling spring constant, and b ≥ 0 is a dimensionless coupling amplitude. The uncoupled Lagrangian is L = L 1 + L 2 with L j = m j 2 ˙ x 2 j − κ j ( b ) + pα j κ j 2 x 2 j , j = 1 , 2 , (V.20) where p is a real parameter with | p | ≤ 1 / 5 and α j are ﬁxed dimensionless coeﬃcients. The full Lagrangian includes the ph ysically meaningful relativ e-displacement coupling L = L 1 + L 2 + L int , L int = − bκ 2 ( x 1 − x 2 ) 2 . (V.21) Expanding L int and com bining with ( V.20 ), the eﬀectiv e diagonal potential for oscillator j is κ j ( b ) + pα j κ j + bκ 2 x 2 j = (1 + pα j ) κ j 2 x 2 j , (V.22) where the b -dependent terms cancel exactly . The full Lagrangian therefore reduces to L = 2 X j =1  m j 2 ˙ x 2 j − (1 + pα j ) κ j 2 x 2 j  + bκx 1 x 2 , (V.23) whic h is equiv alen t to a system with bare spring constan ts (1 + pα j ) κ j and a purely oﬀ-diagonal coupling bκx 1 x 2 . The Euler–Lagrange equations of ( V.23 ) are m 1 ¨ x 1 + (1 + pα 1 ) κ 1 x 1 = bκx 2 , (V.24) m 2 ¨ x 2 + (1 + pα 2 ) κ 2 x 2 = bκx 1 . (V.25) Deﬁning the p -dependent partial frequencies e Ω 2 j ( p ) = (1 + pα j ) κ j m j , j = 1 , 2 , (V.26) and seeking solutions x j = c j e iω t , the c haracteristic determinan t of ( V.24 )–( V.25 ) yields the factorize d char acteristic e quation  ω 2 − e Ω 2 1 ( p )   ω 2 − e Ω 2 2 ( p )  = b 2 κ 2 m 1 m 2 , (V.27) the mechanical analog of the cross-p oint disp ersion relation ( V.10 ), with the wa v enum ber k replaced by the parameter p and the coupling coeﬃcient γ g γ replaced b y b 2 κ 2 / ( m 1 m 2 ) . The t w o real eigenfrequency branc hes are ω 2 ± ( p, b ) = e Ω 2 1 + e Ω 2 2 2 ± v u u t  e Ω 2 1 − e Ω 2 2  2 4 + b 2 κ 2 m 1 m 2 . (V.28) Note that for p = 0 equation ( V.28 ) in view of ( V.26 ) turns into ω 2 ± (0 , b ) = Ω 2 1 + Ω 2 2 2 ± s (Ω 2 1 − Ω 2 2 ) 2 4 + b 2 κ 2 m 1 m 2 , Ω 2 j = e Ω 2 1 (0) = κ j m j , j = 1 , 2 . (V.29) Equation ( V.29 ) in turn readily implies ω 2 ± (0 , 0) =  Ω 2 1 + Ω 2 2  ±   Ω 2 1 − Ω 2 2   2 , Ω 2 j = κ j m j , j = 1 , 2 . (V.30) Since in view of ( V.26 ) e Ω 2 j is indep enden t of b , the partial-frequency crossing condition e Ω 2 1 ( p ∗ ) = e Ω 2 2 ( p ∗ ) is lik ewise b -indep enden t, and yields p ∗ = κ 2 /m 2 − κ 1 /m 1 α 1 κ 1 /m 1 − α 2 κ 2 /m 2 . (V.31) 23 A t p = p ∗ b oth branc hes split symmetrically about the common v alue Ω ∗ = e Ω j ( p ∗ ) : ω 2 ± ( p ∗ , b ) = Ω ∗ 2 ± bκ √ m 1 m 2 , (V.32) so that neither branch is pinned to Ω ∗ for b > 0 . This symmetric av oided crossing is the direct mechanical coun terpart of the h yp erbolic geometry of the cross-p oin t disp ersion relation ( V.9 ). T o illustrate these results numerically w e set m 1 = m 2 = 1 , κ 1 = 1 , κ 2 = 6 5 , κ = 1 , α 1 = 1 , α 2 = − 1 , (V.33) giving p ∗ = 1 / 11 and Ω ∗ = p 12 / 11 ≈ 1 . 044 . Figure V.2 sho ws ω ± ( p ) o v er the range p ∈ ( − 1 / 20 , 23 / 100) , chosen so that p ∗ lies near the center, for four v alues of the coupling parameter b = 0 , 0 . 2 , 0 . 4 , 0 . 6 . F or b = 0 the t w o branc hes cross at p ∗ ; for b > 0 the crossing is replaced by an a v oided crossing whose gap ω + − ω − at p ∗ gro ws with b , in precise analogy with the cross-point dispersion relation ( V.10 ). FIG. V.2. Eigenfrequencies ω ± ( p ) for the parameter set ( V.33 ) and b = 0 (blac k), 0 . 2 (blue), 0 . 4 (green), 0 . 6 (orange). Solid lines: upp er branch ω + ; dashed lines: low er branc h ω − . The open circle marks the bare crossing p oin t at p ∗ = 1 / 11 , ω = Ω ∗ ≈ 1 . 044 . The av oided crossing for b > 0 is the mechanical analog of the cross-point disp ersion relation ( V.10 ). A CKNO WLEDGMENT: This researc h was supp orted b y AFOSR MURI Grant F A9550-20-1-0409 administered through the Univ ersit y of New Mexico. VI. APPENDIX A. F ourier transform There are sev eral common con ven tions for the F ourier transform, diﬀering in signs and constants. Our preferred form of the F ourier tr ansform b f = f ∧ of f and the inverse F ourier tr ansform f ∧ of f follows to AdamHed [ 2 , Sec. 1.1.7], ArfW eb [ 4 , Sec. 20.2], DauLio1 [ 10 , Notations], F oll [ 15 , Sec. 7.2, 7.5], T reB [ 44 , Sec. 25]: b f ( k ) def = ˆ ∞ −∞ f ( z ) e − i kz dz , f ( z ) = h b f ( k ) i ∨ = 1 2 π ˆ ∞ −∞ b f ( k ) e i kz d k (VI.1) b f ( ω ) def = ˆ ∞ −∞ f ( t ) e i ω t d t, f ( t ) = 1 2 π ˆ ∞ −∞ b f ( ω ) e − i ω t d ω , (VI.2) 24 b f ( k , ω ) def = ˆ ∞ −∞ f ( z , t ) e i( ω t − kz ) dz d t, (VI.3) f ( z , t ) = h b f ( k , ω ) i ∨ = 1 (2 π ) 2 ˆ ∞ −∞ b f ( k , ω ) e − i( ω t − kz ) d k d ω . Note the diﬀer enc e of the choic e of the sign for time t and sp atial variable z in the ab ove formula. It is motivate d by the desir e to have “wave” form for exp onential e − i( ω t − kz ) when b oth variables t and z ar e pr esent . F or m ulti-dimensional space v ariable x ∈ R n the F ourier transform b f of f and the in verse F ourier transform f ∧ of f are deﬁned b y , AdamHed [ 2 , Sec. 1.1.7], DauLio1 [ 10 , Notations], F oll [ 15 , Sec. 7.5]: b f ( k ) def = ˆ R n b f ( x ) e − i k · x d x, f ( x ) = h b f ( k ) i ∨ = 1 (2 π ) n ˆ R n b f ( k ) e i k · x d k , k, x ∈ R n , (VI.4) whic h is consistent with equations ( VI.1 ). Then the Planc herel-Parsev al formula reads, Ev ans [ 12 , Sec. 4.3.1], F oll [ 15 , Sec. 7.5], F olPDE [ 16 , Sec. 0.26]: ( f , g ) = (2 π ) − n  b f , b g  , ∥ f ∥ = (2 π ) − n/ 2    b f    , (VI.5) ( f , g ) def = ˆ R n f ( x ) g ( x ) d x, ∥ f ∥ def = p ( f , f ) . This preference was motiv ated b y the fact that the so-deﬁned F ourier transform of the con v olution of tw o functions has its simplest form. Namely , the con volution f ∗ g of tw o functions f and g is deﬁned b y Ev ans 12 , Sec. 4.3.1, F oll [ 15 , Sec. 7.2, 7.5], [ f ∗ g ] ( t ) = [ g ∗ f ] ( t ) = ˆ ∞ −∞ f ( t − t ′ ) g ( t ′ ) d t ′ , (VI.6) [ f ∗ g ] ( z , t ) = [ g ∗ f ] ( z , t ) = ˆ ∞ −∞ f ( z − z ′ , t − t ′ ) g ( z ′ , t ′ ) d z ′ d t ′ . (VI.7) Then its F ourier transform as deﬁned by equations ( VI.1 )-( VI.3 ) satisﬁes the following prop erties: [ f ∗ g ] ∧ ( ω ) = b f ( ω ) b g ( ω ) , (VI.8) [ f ∗ g ] ∧ ( k , ω ) = b f ( k , ω ) b g ( k , ω ) . (VI.9) B. A few facts ab out determinan ts W e present here a few important statements for determinants following mostly ArnoODE [ 5 ], BernS [ 6 ], HorJohn [ 25 ], PizOde [ 36 ]. The theory of determinants is an imp ortant part of the linear algebra and its geometric applications. Concepts of Grassmann exterior and Cliﬀord algebras give a deep insigh t in to the prop erties of determinan ts, V eiDal [ 45 , Sec. 1.2, 3.3], HesSob [ 24 , Sec. 1.4], SnyggN [ 41 , Chap. 4]. In particular, according to HesSob [ 24 , Sec. 1.4]: “ ... a determinan t is nothing more nor less than the scalar pro duct of t wo blades.”. W e in tro duce ﬁrst basic notations. Let M m,n ( F ) is a set of m × n matrices with entries in ﬁeld F . W e also use an abbreviation M n ( F ) = M n,n ( F ) . T o describ e submatrices of a giv en matrix w e in tro duce ﬁrst index sequences Q r,m = { ( i 1 , i 2 , . . . , i r ) | 1 ≤ i 1 < i 2 < · · · < i r ≤ m } , 1 ≤ r ≤ m. (VI.10) Then if A = { A ij } ∈ M m,n ( F ) , α ∈ Q r,m and β ∈ Q s,n then A [ α, β ] ∈ M r,s ( F ) stands for a submatrix of A with ro w indexes coming from α and column indexes coming from β . It useful to introduce also a complimen tary to A [ α, β ] ∈ M r,s ( F ) submatrix A [ α c , β c ] ∈ M r,s ( F ) where α c is the complimen tary to α sequence, namely α c = { 1 , . . . , m } \ α ∈ Q m − r,m , (VI.11) or, in other words, α c is obtained b y remov al sequence α from sequence { 1 , . . . , m } , and β c ∈ Q n − s,n is deﬁned similarly . Supp ose no w A = { A ij } ∈ M m,n ( F ) where A ij , 1 ≤ i, j ≤ n are the entries of matrix A . F or any pair 1 ≤ i, j ≤ n w e in tro duce a ( n − 1) × ( n − 1) submatrix A [ i c , j c ] obtained by deleting i -th ro w and j -th column from A and refer 25 to it c ofactor of A ij . W e introduce also the so-called adjugate to A matrix A A (sometimes called adjoint) deﬁned using cofactors A [ i c , j c ] as follo ws, BernS [ 6 , Sec. 3.8], PizOde [ 36 , App. C.3.3]:  A A  i,j def = ( − 1) i + j det { A [ j c , i c ] } , 1 ≤ i, j ≤ n. (VI.12) The adjugate matrix satisfy the follo wing iden tities, BernS [ 6 , Sec. 3.8, 3.19], PizOde [ 36 , App. C.3.3]: n X m =1 A ij  A A  m,i =  AA A  i,i =  A A A  i,i = det { A } , (VI.13) n X m =1 A ij  A A  m,j =  AA A  i,j =  A A A  i,j = 0 , i  = j, (VI.14) A A A = AA A = det { A } I , (VI.15) where I is the identit y matrix. Note in case when A is not degenerate the identit y ( VI.15 ) readily implies the following represen tation: A A = 1 det { A } A − 1 , det { A }  = 0 . (VI.16) 1. L aplac e exp ansion The Laplace expansion represen ts the determinant of a square matrix in terms of the pro duct of determinants of certain submatrices. Here is its main statemen t, V eiDal [ 45 , Sec. 3.3], HesSob [ 24 , Sec. 1.4], HorJohn [ 25 , Sec. 0.8.9], PizOde [ 36 , App. C.3.3]. Theorem 4 (Laplace expansion theorem) . L et A ∈ M n ( F ) and α ∈ Q r,n for 1 ≤ r ≤ n b e ﬁxe d. Then the fol lowing L aplac e exp ansion of det { A } by r ows holds det { A } = X β ∈ Q r,n ( − 1) | α | + | β | det { A [ α | β ] } det { A [ α c , β c ] } , (VI.17) wher e for α = ( i 1 , i 2 , . . . , i r ) quantity | α | is deﬁne d by | α | = | ( i 1 , i 2 , . . . , i r ) | = i 1 + i 2 + · · · + i r . (VI.18) Similarly, if β ∈ Q r,n for 1 ≤ r ≤ n is ﬁxe d, then the fol lowing L aplac e exp ansion of det { A } by c olumns holds det { A } = X α ∈ Q r,n ( − 1) | α | + | β | det { A [ α | β ] } det { A [ α c , β c ] } . (VI.19) 2. The Liouvil le-Jac obi formula Supp ose that A ( t ) and M ( t ) is n × n matrices satisfying the following Cauch y problem dM ( t ) dt = A ( t ) M ( t ) , M (0) = I , (VI.20) where A ( t ) is n × n matrix. Then the following Liouvil le-Jac obi formula holds, ArnoODE [ 5 , Sec. 27.6], GantMa2 [ 17 , Sec. XIV.1], Y akSta [ 49 , Sec. I I.1.2]: det { M ( t ) } = exp   t ˆ 0 T r { A ( τ ) } d τ   . (VI.21) In the case when A ( t ) = A is a constan t matrix the Liouville-Jacobi form ula readily yields the following identit y , ArnoODE [ 5 , Sec. 16.3, 16.4, 27.6], BernS [ 6 , Section 15.2] det { exp [ A ] } = exp [ A ] . (VI.22) The follo wing statemen t holds, ReeSim4 [ 40 , Sec. XII I.16 Lemma 6] 26 Lemma 5. F or any matrix A = { a ij } and τ smal l det { I + τ A } = exp ( − ∞ X m =1 τ k T r ( − A ) k k ) , wher e I is the identity matrix and T r ( A ) = P n i =1 a ii is the tr ac e of matrix A , that is the sum of its diagonal entries. In p articular, det { I + τ A } = 1 + τ T r ( A ) + O  τ 2  , τ → 0 , . 3. 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