text: of higher order contributions in the two perturbation parameters, namely the perturbation strength and the coupling strength. We then turned to obtaining explicit formulas for our model. Using the Magnus expansion, we derived an explicit expression for the operator sine function that gov- erns the dynamics of the acoustic wave equation and calculated the lowest order eﬀects of the perturbation operator in the integral formulation of the acoustic wave equation. For the boundary vibrations equations, we used the observation that the relevant operator algebra is commutative for all points in time and derived a closed form expression for the operator sine function that governs the dynamics of the solution to the boundary vibrations equation. We have checked that this expression reduces to the expression we have found in our previous papers concentrating exclusively on the treatment of the internally coupled ears (ICE) model. These equations can be used in the iteration scheme presented in the preceding paragraph to the explicit calculations. Examples of how the formalism interweaves in a more concrete setting can be found in our previous papers on acoustic boundary conditions dynamics (ABCD). The ﬁnal point we have addressed for the derivation of the perturbation theory is the question of convergence of semigroups. Since partial diﬀerential operators are in general unbounded operators and the Laplace-Beltrami operators in particular are unbounded linear operators, we needed to restrict their domain such that they turn into bounded and by compactness of the geometrical domains of the operators even compact operators. Otherwise, we could not have used the already existent results on convergence of the Magnus expansion. By linearity of the Laplace-Beltrami operators it followed from boundedness that the Laplace-Beltrami operators are also continuous and thus closed linear operators. Because the Laplace-Beltrami operators are then no longer self-adjoint but just symmetric operators, we needed for the check on the relevant items of well-posedness of the problem, i.e., existence, uniqueness, and continuous dependence on the source term of the solution, at least essentially self-adjoint operators, i.e., symmetric diﬀerential operators that admit a unique (maximal) self-adjoint extension, the Friedrichs extension. We have used the notion of analytic vectors combined with the Sobolev spaces embedding theorems to perform the conceptually instructive exercise that the Laplace-Beltrami operators admit self-adjoint extensions. Investigating the regularity properties of the overall source term to the partial diﬀerential equation system, we could check convergence of the iteration scheme with a positive result. This agrees with a literature result on convergence properties of the Magnus expansion. After having ﬁnished the analysis of out geometric perturbation theory, we turned to the investigation of an applied question. Namely, one might want to simplify the solutions of the equation by neglecting the local dependencies of the boundary vibrations in the solution to the acoustic wave equation. This corresponds geometrically with replacing the boundary vibrations with pistons and is formalized in the notation of piston bundles associated to a perturbation bundle. The pistons are displaced from their equilibrium position by the geometrical mean of the boundary vibrations. While it is clear that such an approximation can be performed, we asked what the approxima- tion means geometrically. We have found that the approximation corresponds to the mean curvature of the boundaries of the perturbed ﬁbers diﬀering from a constant mean curvature only by a small additional perturbation operator. From the experimental viewpoint, the local mean curvature can be assessed easily. One simply looks whether the mean curvature is approximately constant, i.e., the boundary vibrations (e.g. plates or mem- branes in the physical setting) only vibrate very mildly. More precisely, the local mean curvature should only diﬀer from the local mean curvature constant by a quantity in the order of the perturbation strength. Under this circumstances, it is consistent with the perturbation theory developed before to replace the boundary vibrations by piston boundary vibrations or, in geometrical language, one can associate to the original perturbation bundle a piston perturbation bundle of the same perturbation strength such that one loses spatial dependencies of the perturbations of the ﬁbers around the unperturbed ﬁber without 129

text: of

text: higher

text: order

text: contributions

text: in

text: the

text: two

text: perturbation

text: parameters,

text: namely

text: the

text: perturbation

text: strength

text: and

text: the

text: coupling

text: strength.

text: We

text: then

text: turned

text: to

text: obtaining

text: explicit

text: formulas

text: for

text: our

text: model.

text: Using

text: the

text: Magnus

text: expansion,

text: we

text: derived

text: an

text: explicit

text: expression

text: for

text: the

text: operator

text: sine

text: function

text: that

text: gov-

text: erns

text: the

text: dynamics

text: of

text: the

text: acoustic

text: wave

text: equation

text: and

text: calculated

text: the

text: lowest

text: order

text: eﬀects

text: of

text: the

text: perturbation

text: operator

text: in

text: the

text: integral

text: formulation

text: of

text: the

text: acoustic

text: wave

text: equation.

text: For

text: the

text: boundary

text: vibrations

text: equations,

text: we

text: used

text: the

text: observation

text: that

text: the

text: relevant

text: operator

text: algebra

text: is

text: commutative

text: for

text: all

text: points

text: in

text: time

text: and

text: derived

text: a

text: closed

text: form

text: expression

text: for

text: the

text: operator

text: sine

text: function

text: that

text: governs

text: the

text: dynamics

text: of

text: the

text: solution

text: to

text: the

text: boundary

text: vibrations

text: equation.

text: We

text: have

text: checked

text: that

text: this

text: expression

text: reduces

text: to

text: the

text: expression

text: we

text: have

text: found

text: in

text: our

text: previous

text: papers

text: concentrating

text: exclusively

text: on

text: the

text: treatment

text: of

text: the

text: internally

text: coupled

text: ears

text: (ICE)

text: model.

text: These

text: equations

text: can

text: be

text: used

text: in

text: the

text: iteration

text: scheme

text: presented

text: in

text: the

text: preceding

text: paragraph

text: to

text: the

text: explicit

text: calculations.

text: Examples

text: of

text: how

text: the

text: formalism

text: interweaves

text: in

text: a

text: more

text: concrete

text: setting

text: can

text: be

text: found

text: in

text: our

text: previous

text: papers

text: on

text: acoustic

text: boundary

text: conditions

text: dynamics

text: (ABCD).

text: The

text: ﬁnal

text: point

text: we

text: have

text: addressed

text: for

text: the

text: derivation

text: of

text: the

text: perturbation

text: theory

text: is

text: the

text: question

text: of

text: convergence

text: of

text: semigroups.

text: Since

text: partial

text: diﬀerential

text: operators

text: are

text: in

text: general

text: unbounded

text: operators

text: and

text: the

text: Laplace-Beltrami

text: operators

text: in

text: particular

text: are

text: unbounded

text: linear

text: operators,

text: we

text: needed

text: to

text: restrict

text: their

text: domain

text: such

text: that

text: they

text: turn

text: into

text: bounded

text: and

text: by

text: compactness

text: of

text: the

text: geometrical

text: domains

text: of

text: the

text: operators

text: even

text: compact

text: operators.

text: Otherwise,

text: we

text: could

text: not

text: have

text: used

text: the

text: already

text: existent

text: results

text: on

text: convergence

text: of

text: the

text: Magnus

text: expansion.

text: By

text: linearity

text: of

text: the

text: Laplace-Beltrami

text: operators

text: it

text: followed

text: from

text: boundedness

text: that

text: the

text: Laplace-Beltrami

text: operators

text: are

text: also

text: continuous

text: and

text: thus

text: closed

text: linear

text: operators.

text: Because

text: the

text: Laplace-Beltrami

text: operators

text: are

text: then

text: no

text: longer

text: self-adjoint

text: but

text: just

text: symmetric

text: operators,

text: we

text: needed

text: for

text: the

text: check

text: on

text: the

text: relevant

text: items

text: of

text: well-posedness

text: of

text: the

text: problem,

text: i.e.,

text: existence,

text: uniqueness,

text: and

text: continuous

text: dependence

text: on

text: the

text: source

text: term

text: of

text: the

text: solution,

text: at

text: least

text: essentially

text: self-adjoint

text: operators,

text: i.e.,

text: symmetric

text: diﬀerential

text: operators

text: that

text: admit

text: a

text: unique

text: (maximal)

text: self-adjoint

text: extension,

text: the

text: Friedrichs

text: extension.

text: We

text: have

text: used

text: the

text: notion

text: of

text: analytic

text: vectors

text: combined

text: with

text: the

text: Sobolev

text: spaces

text: embedding

text: theorems

text: to

text: perform

text: the

text: conceptually

text: instructive

text: exercise

text: that

text: the

text: Laplace-Beltrami

text: operators

text: admit

text: self-adjoint

text: extensions.

text: Investigating

text: the

text: regularity

text: properties

text: of

text: the

text: overall

text: source

text: term

text: to

text: the

text: partial

text: diﬀerential

text: equation

text: system,

text: we

text: could

text: check

text: convergence

text: of

text: the

text: iteration

text: scheme

text: with

text: a

text: positive

text: result.

text: This

text: agrees

text: with

text: a

text: literature

text: result

text: on

text: convergence

text: properties

text: of

text: the

text: Magnus

text: expansion.

text: After

text: having

text: ﬁnished

text: the

text: analysis

text: of

text: out

text: geometric

text: perturbation

text: theory,

text: we

text: turned

text: to

text: the

text: investigation

text: of

text: an

text: applied

text: question.

text: Namely,

text: one

text: might

text: want

text: to

text: simplify

text: the

text: solutions

text: of

text: the

text: equation

text: by

text: neglecting

text: the

text: local

text: dependencies

text: of

text: the

text: boundary

text: vibrations

text: in

text: the

text: solution

text: to

text: the

text: acoustic

text: wave

text: equation.

text: This

text: corresponds

text: geometrically

text: with

text: replacing

text: the

text: boundary

text: vibrations

text: with

text: pistons

text: and

text: is

text: formalized

text: in

text: the

text: notation

text: of

text: piston

text: bundles

text: associated

text: to

text: a

text: perturbation

text: bundle.

text: The

text: pistons

text: are

text: displaced

text: from

text: their

text: equilibrium

text: position

text: by

text: the

text: geometrical

text: mean

text: of

text: the

text: boundary

text: vibrations.

text: While

text: it

text: is

text: clear

text: that

text: such

text: an

text: approximation

text: can

text: be

text: performed,

text: we

text: asked

text: what

text: the

text: approxima-

text: tion

text: means

text: geometrically.

text: We

text: have

text: found

text: that

text: the

text: approximation

text: corresponds

text: to

text: the

text: mean

text: curvature

text: of

text: the

text: boundaries

text: of

text: the

text: perturbed

text: ﬁbers

text: diﬀering

text: from

text: a

text: constant

text: mean

text: curvature

text: only

text: by

text: a

text: small

text: additional

text: perturbation

text: operator.

text: From

text: the

text: experimental

text: viewpoint,

text: the

text: local

text: mean

text: curvature

text: can

text: be

text: assessed

text: easily.

text: One

text: simply

text: looks

text: whether

text: the

text: mean

text: curvature

text: is

text: approximately

text: constant,

text: i.e.,

text: the

text: boundary

text: vibrations

text: (e.g.

text: plates

text: or

text: mem-

text: branes

text: in

text: the

text: physical

text: setting)

text: only

text: vibrate

text: very

text: mildly.

text: More

text: precisely,

text: the

text: local

text: mean

text: curvature

text: should

text: only

text: diﬀer

text: from

text: the

text: local

text: mean

text: curvature

text: constant

text: by

text: a

text: quantity

text: in

text: the

text: order

text: of

text: the

text: perturbation

text: strength.

text: Under

text: this

text: circumstances,

text: it

text: is

text: consistent

text: with

text: the

text: perturbation

text: theory

text: developed

text: before

text: to

text: replace

text: the

text: boundary

text: vibrations

text: by

text: piston

text: boundary

text: vibrations

text: or,

text: in

text: geometrical

text: language,

text: one

text: can

text: associate

text: to

text: the

text: original

text: perturbation

text: bundle

text: a

text: piston

text: perturbation

text: bundle

text: of

text: the

text: same

text: perturbation

text: strength

text: such

text: that

text: one

text: loses

text: spatial

text: dependencies

text: of

text: the

text: perturbations

text: of

text: the

text: ﬁbers

text: around

text: the

text: unperturbed

text: ﬁber

text: without


text: B R → R ⊂ R Lebesgue-Borel null-set w.r.t. the Lebesgue-Borel measure Vol : ( n) + restricted to Bn(0) n n | → 1 such that this does not hinder our further progress. Now, φ = ψ : ∂Ω Sn(0) induces an analogous 0 0 ∂Ω0 0 1 ﬁbration on Ω if 0 × → \ { } σ :[0, 1) ∂Ω Ω 0 ; (7) 0 0 0 (cid:55)→ − (cid:107) (cid:107) × ∗ (s, y ) σ (s, y ) = y snˆ (y ) y = (id φ ) σ (s, y). (8) 0 0 0 0 ∂Ω0 0 0 2 [0,1) 0 ref \ { } In the previous equation, a point on the boundary is denoted y and a point in Ω 0 by x . → 0 0 0 Analogously, φ : ∂Ω Sn(0) induces a ﬁbration on Ω is required by the deﬁnition if t t 1 t × → \ { } σ :[0, 1) ∂Ω Ω 0 ; (9) t t t (cid:55)→ − (cid:107) (cid:107) × ∗ (s, y ) σ (s, y ) = y snˆ (y ) y = (id φ ) σ (s, y), (10) 0 t t t ∂Ωt t t 2 [0,1) t ref ∈ \ { } where y ∂Ω and x denotes a generic point in Ω 0 . Denoting symbolically σ (s, ∂Ω ) := − · t t t − · t 0 0 (1 s) ∂Ω and similarly σ (s, ∂Ω ) := ((cid:93)1 s) ∂Ω , we can w(cid:93)rite 0 t t t \ { } ≡ − · ≡ (cid:93) (cid:93) Ω 0 (1 s) ∂Ω σ (s, ∂Ω ) , 0 0 0 0 ∈ ∈ s [0,1) s [0,1) \ { } ≡ − · ≡ (11) Ω 0 (1 s) ∂Ω σ (s, ∂Ω ) . t t 0 t ∈ ∈ s [0,1) s [0,1) The geometric intuition stored in the deﬁnitions of σ , σ (and σ ) is illustrated in the subsequent 0 t ref ﬁgure (Fig. 6). Physical perturbation bundles We call a proper perturbation bundle physical if there is a (dimen- (cid:111) M | sional) constant C satisfying C/[C] > 0 and C/c2 = ((cid:15)2) such that in the metric G continued to (cid:107)∇ − (cid:107) ≤ (cid:107)∇ − (cid:107) 0 n+1 (φ → id ) 2 C (φ → id ) 2 , conservative if the above inequality is an equality and ∂t 0 t Ω0 g0 ∂i 0 t Ω0 g0 dissipative if the above inequality is strict. Explanation: The property of a perturbation bundle to be physical states that the internal energy density U of the perturbations is non-increasing. We consider the boundary ∂Ω to be a membrane. In t the physical picture, membranes can be thought of as a continuum of harmonic oscillators coupled to each V ∇ ∇ other. This gives rise to the potential energy density = T gij u u where T is a physical constant, the 0 (0) i j 0 membrane tension. The vibration of the membranes on the other hand gives rise to a kinetic energy density T − = σ Gtt∂ u∂ u noting that Gtt < 0 if G is brought into diagonal form. σ denotes a physical constant, m 0 t t 0 0 m the surface mass density of the membrane. The property “conservative” states that each oscillator can convert the entire potential energy into kinetic energy. The property “dissipative” expresses the physical reality that there are heat losses due to internal friction eﬀects between the individual oscillators and external friction due to the membrane interacting with its environment which hinder the conversion of all V T T V potential energy into kinetic energy on a local level, i.e., for each oscillator. I.e., < . Equating ≡ (cid:28) (cid:111) C c2 = T /σ we recover the deﬁnition. The condition C c2 now turns into c /c = ((cid:15)). For the m 0 m (cid:28) m values given in [98] it can be fulﬁlled by choosing a suitable (cid:15) 1. 15

text: B

label: paragraph

text: R

text: R

text: R

text: Lebesgue-Borel

text: null-set

text: w.r.t.

text: the

text: Lebesgue-Borel

text: measure

text: Vol

text: n)

text: restricted

text: to

text: n

text: n

text: such

text: that

text: this

text: does

text: not

text: hinder

text: our

text: further

text: progress.

text: Now,

text: φ

text: ψ

text: ∂Ω

text: induces

text: an

text: analogous

text: ∂Ω0

text: ﬁbration

text: on

text: Ω

text: if

label: equation

text: σ

text: ∂Ω

text: Ω

text: (s,

text: y

text: σ

text: (s,

text: y

text: y

text: snˆ

text: (y

text: y

text: (id

text: φ

text: σ

text: (s,

text: y).

text: ∂Ω0

text: ref

text: In

text: the

text: previous

text: equation,

text: a

text: point

text: on

text: the

text: boundary

text: is

text: denoted

text: y

text: and

text: a

text: point

text: in

text: Ω

text: by

text: x

text: Analogously,

text: φ

text: ∂Ω

text: induces

text: a

text: ﬁbration

text: on

text: Ω

text: is

text: required

text: by

text: the

text: deﬁnition

text: if

text: t

text: t

text: t

text: σ

text: ∂Ω

text: Ω

text: t

text: t

text: t

text: (s,

text: y

text: σ

text: (s,

text: y

text: y

text: snˆ

text: (y

text: y

text: (id

text: φ

text: σ

text: (s,

text: y),

text: t

text: t

text: t

text: ∂Ωt

text: t

text: t

text: t

text: ref

text: where

text: y

text: ∂Ω

text: and

text: x

text: denotes

text: a

text: generic

text: point

text: in

text: Ω

text: Denoting

text: symbolically

text: σ

text: (s,

text: ∂Ω

text: t

text: t

text: t

text: t

text: s)

text: ∂Ω

text: and

text: similarly

text: σ

text: (s,

text: ∂Ω

text: s)

text: ∂Ω

text: we

text: can

text: t

text: t

text: t

text: Ω

text: s)

text: ∂Ω

text: σ

text: (s,

text: ∂Ω

text: s

text: s

text: Ω

text: s)

text: ∂Ω

text: σ

text: (s,

text: ∂Ω

text: t

text: t

text: t

text: s

text: s

text: The

text: geometric

text: intuition

text: stored

text: in

text: the

text: deﬁnitions

text: of

text: σ

text: σ

text: (and

text: σ

text: is

text: illustrated

text: in

text: the

text: subsequent

text: t

text: ref

text: ﬁgure

text: (Fig.

text: Physical

text: perturbation

text: bundles

text: We

text: call

text: a

text: proper

text: perturbation

text: bundle

text: physical

text: if

text: there

text: is

text: a

text: (dimen-

text: M

text: sional)

text: constant

text: C

text: satisfying

text: C/[C]

text: and

text: such

text: that

text: in

text: the

text: metric

text: G

text: continued

text: to

text: (φ

text: id

text: C

text: (φ

text: id

text: conservative

text: if

text: the

text: above

text: inequality

text: is

text: an

text: equality

text: and

text: ∂t

text: t

text: Ω0

text: ∂i

text: t

text: Ω0

text: dissipative

text: if

text: the

text: above

text: inequality

text: is

text: strict.

text: Explanation:

text: The

text: property

text: of

text: a

text: perturbation

text: bundle

text: to

text: be

text: physical

text: states

text: that

text: the

text: internal

text: energy

text: density

text: U

text: of

text: the

text: perturbations

text: is

text: non-increasing.

text: We

text: consider

text: the

text: boundary

text: ∂Ω

text: to

text: be

text: a

text: membrane.

text: In

text: t

text: the

text: physical

text: picture,

text: membranes

text: can

text: be

text: thought

text: of

text: as

text: a

text: continuum

text: of

text: harmonic

text: oscillators

text: coupled

text: to

text: each

text: V

text: other.

text: This

text: gives

text: rise

text: to

text: the

text: potential

text: energy

text: density

text: T

text: gij

text: u

text: u

text: where

text: T

text: is

text: a

text: physical

text: constant,

text: the

text: i

text: j

text: membrane

text: tension.

text: The

text: vibration

text: of

text: the

text: membranes

text: on

text: the

text: other

text: hand

text: gives

text: rise

text: to

text: a

text: kinetic

text: energy

text: density

text: T

text: σ

text: Gtt∂

text: u∂

text: u

text: noting

text: that

text: Gtt

text: if

text: G

text: is

text: brought

text: into

text: diagonal

text: form.

text: σ

text: denotes

text: a

text: physical

text: constant,

text: m

text: t

text: t

text: m

text: the

text: surface

text: mass

text: density

text: of

text: the

text: membrane.

text: The

text: property

text: “conservative”

text: states

text: that

text: each

text: oscillator

text: can

text: convert

text: the

text: entire

text: potential

text: energy

text: into

text: kinetic

text: energy.

text: The

text: property

text: “dissipative”

text: expresses

text: the

text: physical

text: reality

text: that

text: there

text: are

text: heat

text: losses

text: due

text: to

text: internal

text: friction

text: eﬀects

text: between

text: the

text: individual

text: oscillators

text: and

text: external

text: friction

text: due

text: to

text: the

text: membrane

text: interacting

text: with

text: its

text: environment

text: which

text: hinder

text: the

text: conversion

text: of

text: all

text: V

text: T

text: T

text: V

text: potential

text: energy

text: into

text: kinetic

text: energy

text: on

text: a

text: local

text: level,

text: i.e.,

text: for

text: each

text: oscillator.

text: I.e.,

text: Equating

text: C

text: T

text: /σ

text: we

text: recover

text: the

text: deﬁnition.

text: The

text: condition

text: C

text: now

text: turns

text: into

text: c

text: /c

text: For

text: the

text: m

text: m

text: m

text: values

text: given

text: in

text: it

text: can

text: be

text: fulﬁlled

text: by

text: choosing

text: a

text: suitable


text: its boundaries? Second, how can we quantify the smallness of those perturbations and relate them to solving the model equations in the unperturbed state of the system? Third, how can we account for time dependency of the perturbations in an operator-theoretic perturbation formalism. Again from the viewpoint of applications in the sciences, it is desirable to ﬁnd a geometrical answer to the question of how we may handle weakly curved deformations of the boundaries. Namely, we ask under what conditions is it appropriate to neglect the local curvature properties of the vibrating boundaries with respect to the equilibrium boundaries and simply consider their normal displacement from the unperturbed state? Strategy Our strategy consist of three parts. A geometrical one for the geometrical setup, a physical one for ﬁnding the appropriate model, and an analytical one for ﬁnding a perturbation theory to solve the model equations. • Geometry The ﬁrst part is mostly of geometrical ﬂavor spiced with a bit of topology. It concerns the question of what the appropriate formalism for the geometry of the problem is. This question has not been addressed in the vibrational-acoustics literature so far. We start from considering sub-manifolds of a surrounding Euclidean space as models for the volume under consideration. Speciﬁcally, we model the time-dependency of the problem by using smooth 1-parameter families of such manifolds. → One family is just the constant family t Ω mapping each point in time to he unperturbed volume. → 0 The other family is t Ω which maps the time t to the deforming volume. The reason why we start t → from this geometrical construction is that the time-evolution of the manifolds, i.e., t Ω can be t observed in experiments most easily. We then ask where a wave equation for the acoustic pressure lives geometrically. In the textbook literature on partial diﬀerential equations, the acoustic wave R × R equation is usually considered to live on a product manifold Ω consisting of a base manifold 0 R as a model for the time coordinate t and a manifold Ω embedded in n which is the living space of 0 the the spatial dependencies of the acoustic pressure, i.e., the solution to the acoustic wave equation. (cid:83) In a geometrical language, the living space of the solution is a product manifold which is an example of → a ﬁber bundle. Likewise, we can endow under suitable constraints on the topology of t Ω ,namely M◦ { } × t that it is unaﬀected by the perturbations, the object := t Ω with a ﬁber bundle t>0 t structure. Intuitively, a ﬁber bundle is just a manifold which admits locally a product structure of a R { } ⊂ R base manifold, the time axis + in our setup and a ﬁber manifold Ω glued to each point t + t of the time axis. M R × Geometrically, we then only need to relate := + Ω , the stationary ﬁber bundle to the M 0 0 M M perturbation bundle . The requirement to be made is that the topology of and and of the → → 0 ﬁbers t Ω and t Ω are left invariant by the perturbations aids at relating by means of bundle 0 t M (cid:39) M (cid:39) and manifold diﬀeomorphisms and Ω Ω globally. 0 0 t Using the imbedding space for the bundles, we give using the topological constraints and the therefore → → global Gauss map for the ﬁbers t Ω and t Ω the Lorentzian metric originating from restriction 0 t M of the Lorentzian metric of the imbedding space n+1, the (n + 1)-dimensional Minkowskian space. Combing the metric and the diﬀeomorphism, we are able to reduce the problem to comparing the M M reference bundle ( , G ) with time-independent metric to the pull-back bundle ( , G = G(t)) 0 0 0 with time dependent metric. This will be the starting point for the derivation of the perturbation operator in the model equations. • Physics Physically, the theories relevant to our present modeling belong to continuum ﬁeld theory. On the one hand, the acoustic pressure is quantity of the acoustic limit of ﬂuid dynamics assuming 4

text: its

text: boundaries?

text: Second,

text: how

text: can

text: we

text: quantify

text: the

text: smallness

text: of

text: those

text: perturbations

text: and

text: relate

text: them

text: to

text: solving

text: the

text: model

text: equations

text: in

text: the

text: unperturbed

text: state

text: of

text: the

text: system?

text: Third,

text: how

text: can

text: we

text: account

text: for

text: time

text: dependency

text: of

text: the

text: perturbations

text: in

text: an

text: operator-theoretic

text: perturbation

text: formalism.

text: Again

text: from

text: the

text: viewpoint

text: of

text: applications

text: in

text: the

text: sciences,

text: it

text: is

text: desirable

text: to

text: ﬁnd

text: a

text: geometrical

text: answer

text: to

text: the

text: question

text: of

text: how

text: we

text: may

text: handle

text: weakly

text: curved

text: deformations

text: of

text: the

text: boundaries.

text: Namely,

text: we

text: ask

text: under

text: what

text: conditions

text: is

text: it

text: appropriate

text: to

text: neglect

text: the

text: local

text: curvature

text: properties

text: of

text: the

text: vibrating

text: boundaries

text: with

text: respect

text: to

text: the

text: equilibrium

text: boundaries

text: and

text: simply

text: consider

text: their

text: normal

text: displacement

text: from

text: the

text: unperturbed

text: state?

text: Strategy

text: Our

text: strategy

text: consist

text: of

text: three

text: parts.

text: A

text: geometrical

text: one

text: for

text: the

text: geometrical

text: setup,

text: a

text: physical

text: one

text: for

text: ﬁnding

text: the

text: appropriate

text: model,

text: and

text: an

text: analytical

text: one

text: for

text: ﬁnding

text: a

text: perturbation

text: theory

text: to

text: solve

text: the

text: model

text: equations.

text: Geometry

text: The

text: ﬁrst

text: part

text: is

text: mostly

text: of

text: geometrical

text: ﬂavor

text: spiced

text: with

text: a

text: bit

text: of

text: topology.

text: It

text: concerns

text: the

text: question

text: of

text: what

text: the

text: appropriate

text: formalism

text: for

text: the

text: geometry

text: of

text: the

text: problem

text: is.

text: This

text: question

text: has

text: not

text: been

text: addressed

text: in

text: the

text: vibrational-acoustics

text: literature

text: so

text: far.

text: We

text: start

text: from

text: considering

text: sub-manifolds

text: of

text: a

text: surrounding

text: Euclidean

text: space

text: as

text: models

text: for

text: the

text: volume

text: under

text: consideration.

text: Speciﬁcally,

text: we

text: model

text: the

text: time-dependency

text: of

text: the

text: problem

text: by

text: using

text: smooth

text: families

text: of

text: such

text: manifolds.

text: One

text: family

text: is

text: just

text: the

text: constant

text: family

text: t

text: Ω

text: mapping

text: each

text: point

text: in

text: time

text: to

text: he

text: unperturbed

text: volume.

text: The

text: other

text: family

text: is

text: t

text: Ω

text: which

text: maps

text: the

text: time

text: t

text: to

text: the

text: deforming

text: volume.

text: The

text: reason

text: why

text: we

text: start

text: t

text: from

text: this

text: geometrical

text: construction

text: is

text: that

text: the

text: time-evolution

text: of

text: the

text: manifolds,

text: i.e.,

text: t

text: Ω

text: can

text: be

text: t

text: observed

text: in

text: experiments

text: most

text: easily.

text: We

text: then

text: ask

text: where

text: a

text: wave

text: equation

text: for

text: the

text: acoustic

text: pressure

text: lives

text: geometrically.

text: In

text: the

text: textbook

text: literature

text: on

text: partial

text: diﬀerential

text: equations,

text: the

text: acoustic

text: wave

text: R

text: R

text: equation

text: is

text: usually

text: considered

text: to

text: live

text: on

text: a

text: product

text: manifold

text: Ω

text: consisting

text: of

text: a

text: base

text: manifold

text: R

text: as

text: a

text: model

text: for

text: the

text: time

text: coordinate

text: t

text: and

text: a

text: manifold

text: Ω

text: embedded

text: in

text: n

text: which

text: is

text: the

text: living

text: space

text: of

text: the

text: the

text: spatial

text: dependencies

text: of

text: the

text: acoustic

text: pressure,

text: i.e.,

text: the

text: solution

text: to

text: the

text: acoustic

text: wave

text: equation.

text: In

text: a

text: geometrical

text: language,

text: the

text: living

text: space

text: of

text: the

text: solution

text: is

text: a

text: product

text: manifold

text: which

text: is

text: an

text: example

text: of

text: a

text: ﬁber

text: bundle.

text: Likewise,

text: we

text: can

text: endow

text: under

text: suitable

text: constraints

text: on

text: the

text: topology

text: of

text: t

text: Ω

text: ,namely

text: M◦

text: t

text: that

text: it

text: is

text: unaﬀected

text: by

text: the

text: perturbations,

text: the

text: object

text: t

text: Ω

text: with

text: a

text: ﬁber

text: bundle

text: t

text: structure.

text: Intuitively,

text: a

text: ﬁber

text: bundle

text: is

text: just

text: a

text: manifold

text: which

text: admits

text: locally

text: a

text: product

text: structure

text: of

text: a

text: R

text: R

text: base

text: manifold,

text: the

text: time

text: axis

text: in

text: our

text: setup

text: and

text: a

text: ﬁber

text: manifold

text: Ω

text: glued

text: to

text: each

text: point

text: t

text: t

text: of

text: the

text: time

text: axis.

text: M

text: R

text: Geometrically,

text: we

text: then

text: only

text: need

text: to

text: relate

text: Ω

text: the

text: stationary

text: ﬁber

text: bundle

text: to

text: the

text: M

text: M

text: M

text: perturbation

text: bundle

text: The

text: requirement

text: to

text: be

text: made

text: is

text: that

text: the

text: topology

text: of

text: and

text: and

text: of

text: the

text: ﬁbers

text: t

text: Ω

text: and

text: t

text: Ω

text: are

text: left

text: invariant

text: by

text: the

text: perturbations

text: aids

text: at

text: relating

text: by

text: means

text: of

text: bundle

text: t

text: M

text: M

text: and

text: manifold

text: diﬀeomorphisms

text: and

text: Ω

text: Ω

text: globally.

text: t

text: Using

text: the

text: imbedding

text: space

text: for

text: the

text: bundles,

text: we

text: give

text: using

text: the

text: topological

text: constraints

text: and

text: the

text: therefore

text: global

text: Gauss

text: map

text: for

text: the

text: ﬁbers

text: t

text: Ω

text: and

text: t

text: Ω

text: the

text: Lorentzian

text: metric

text: originating

text: from

text: restriction

text: t

text: M

text: of

text: the

text: Lorentzian

text: metric

text: of

text: the

text: imbedding

text: space

text: the

text: (n

text: Minkowskian

text: space.

text: Combing

text: the

text: metric

text: and

text: the

text: diﬀeomorphism,

text: we

text: are

text: able

text: to

text: reduce

text: the

text: problem

text: to

text: comparing

text: the

text: M

text: M

text: reference

text: bundle

text: G

text: with

text: time-independent

text: metric

text: to

text: the

text: pull-back

text: bundle

text: G

text: G(t))

text: with

text: time

text: dependent

text: metric.

text: This

text: will

text: be

text: the

text: starting

text: point

text: for

text: the

text: derivation

text: of

text: the

text: perturbation

text: operator

text: in

text: the

text: model

text: equations.

text: Physics

text: Physically,

text: the

text: theories

text: relevant

text: to

text: our

text: present

text: modeling

text: belong

text: to

text: continuum

text: ﬁeld

text: theory.

text: On

text: the

text: one

text: hand,

text: the

text: acoustic

text: pressure

text: is

text: quantity

text: of

text: the

text: acoustic

text: limit

text: of

text: ﬂuid

text: dynamics

text: assuming


text: further have (cid:90) (cid:112) (cid:104) | | (cid:105) λ (t) = Ψ (t) ∆ Ψ (t) n n g n g (cid:90) (cid:112)| | = dnx g Ψ¯ (t)∆ Ψ (t) n g n Ω0 (cid:90) = dnx(cid:112)|g |Ψ¯ ∆(cid:0) Ψ +(cid:0)O((cid:15)) (cid:1)(cid:1) 0 n g n Ω0 (cid:90) (cid:90) (cid:112)| | − (cid:112) = dnx g Ψ¯ ∆ + ∆ ∆ Ψ 0 n g0 g g(0) n Ω0 (cid:90) | |(cid:112) | | = dnx g Ψ¯ ∆ Ψ + dnx g Ψ¯ VΨ 0 n g(0) n n n Ω0 Ω0 | | = λ + dnx g Ψ¯ VΨ n 0 n n Ω0 Rearranging the equation, we ﬁnd (cid:90) (cid:112) ≡ − | | (cid:104) | | (cid:105) δλ λ (t) λ = dnx g Ψ¯ VΨ . = Ψ V Ψ . (68) n n n 0 n n n n g(0) Ω0 (cid:104) | | (cid:105) ∼ O If we manage to show that Ψ V Ψ ((cid:15)) (or better), we have not only found a perturbation n n g(0) theory, but also shown that Small changes in the ﬁbers Ω from the reference ﬁber Ω lead to small t 0 changes in the spectra of the Neumann-Laplacians ∆ on Ω and ∆ on Ω . A similar result has g 0 g(0) 0 already been sketched in a less general setting than ours in [DengLi]. M We turn to the investigation of ∆ , the Robin-Laplacian on . G0 M Self-adjointness of ∆ : The Robin-Laplacian ∆ satisﬁes self-adjointness of H1,2( ). Since we G0 G0 (cid:39) have already proven that ∆0 is self-adjoint on Ω by diﬀeometric equivalence (∆ , Ω ) (∆ , Ω ), we can g − t g 0 g0 t simply take the expression ∂2 ∆ absorbing again the c2 factor into ∆ and construct the Sobolev- M − Mt g0 g0 spaces H1,2( ) and H 1,2( ) just as we did before. As we have seen before, the boundary conditions 0 M to ∆ are equivalent to the boundary and initial conditions chosen for the wave equation on . Since G0 the self-adjointness is veriﬁed by considering only the homogeneous initial and boundary conditions, we ∈ M ∈ − M (cid:39) M can use integration by parts freely. We let F H1,2( ), F H 1,2( ) H1,2( ) with the antilinear (cid:39) (cid:90) 1 2 M → C isometric isomorphy valid again because of t(cid:112)he non-degeneracy of the L2 inner product on L2( ) G and calculate (cid:90) (cid:104) | (cid:105) (cid:112)| | (cid:112) (cid:112) F ∆ F = dnx G F¯ ∆ F 1 G0 2 L2G0 M 0 1 G0 2 = (cid:90)dnx (cid:112)|G |F¯ (cid:112)|G |−1∂ ((cid:112)|G |GIJ∂ f ) M 0 1 0 I 0 0 J 2 i.b=.p(cid:90)2 dn(cid:112)x |G(cid:112)| |G |−1∂(cid:112)( |G |GIJ∂ F¯ )F 0 0 I 0 0 J 1 2 Ωt | | | |− | | 1 = dnx G G ∂ ( G GIJ∂ F )F M 0 0 I 0 0 J 1 2 (cid:104) | (cid:105) = ∆ F F (cid:104) G∗ 0 1| 2(cid:105)L2G0 = ∆ F F . G0 1 2 L2G0 31 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: further

label: list

text: have

text: λ

text: (t)

text: Ψ

text: (t)

text: Ψ

text: (t)

text: n

text: n

text: g

text: n

text: g

text: dnx

text: g

text: Ψ¯

text: (t)∆

text: Ψ

text: (t)

text: n

text: g

text: n

text: Ω0

text: |Ψ¯

text: Ψ

text: n

text: g

text: n

text: Ω0

text: dnx

text: g

text: Ψ¯

text: Ψ

text: n

text: g

text: n

text: Ω0

text: dnx

text: g

text: Ψ¯

text: Ψ

text: dnx

text: g

text: Ψ¯

text: VΨ

text: n

text: n

text: n

text: n

text: Ω0

text: Ω0

text: λ

text: dnx

text: g

text: Ψ¯

text: VΨ

text: n

text: n

text: n

text: Ω0

text: Rearranging

text: the

text: equation,

text: we

text: ﬁnd

text: δλ

text: λ

text: (t)

text: λ

text: dnx

text: g

text: Ψ¯

text: VΨ

text: Ψ

text: V

text: Ψ

text: n

text: n

text: n

text: n

text: n

text: n

text: n

text: Ω0

text: O

text: If

text: we

text: manage

text: to

text: show

text: that

text: Ψ

text: V

text: Ψ

text: (or

text: better),

text: we

text: have

text: not

text: only

text: found

text: a

text: perturbation

text: n

text: n

text: theory,

text: but

text: also

text: shown

text: that

text: Small

text: changes

text: in

text: the

text: ﬁbers

text: Ω

text: from

text: the

text: reference

text: ﬁber

text: Ω

text: lead

text: to

text: small

text: t

text: changes

text: in

text: the

text: spectra

text: of

text: the

text: Neumann-Laplacians

text: on

text: Ω

text: and

text: on

text: Ω

text: A

text: similar

text: result

text: has

text: g

text: already

text: been

text: sketched

text: in

text: a

text: less

text: general

text: setting

text: than

text: ours

text: in

text: [DengLi].

text: M

text: We

text: turn

text: to

text: the

text: investigation

text: of

text: the

text: Robin-Laplacian

text: on

text: M

text: Self-adjointness

text: of

text: The

text: Robin-Laplacian

text: satisﬁes

text: self-adjointness

text: of

text: Since

text: we

text: have

text: already

text: proven

text: that

text: ∆0

text: is

text: self-adjoint

text: on

text: Ω

text: by

text: diﬀeometric

text: equivalence

text: Ω

text: Ω

text: we

text: can

text: g

text: t

text: g

text: t

text: simply

text: take

text: the

text: expression

text: ∂2

text: absorbing

text: again

text: the

text: factor

text: into

text: and

text: construct

text: the

text: Sobolev-

text: M

text: Mt

text: spaces

text: and

text: H

text: just

text: as

text: we

text: did

text: before.

text: As

text: we

text: have

text: seen

text: before,

text: the

text: boundary

text: conditions

text: M

text: to

text: are

text: equivalent

text: to

text: the

text: boundary

text: and

text: initial

text: conditions

text: chosen

text: for

text: the

text: wave

text: equation

text: on

text: Since

text: the

text: self-adjointness

text: is

text: veriﬁed

text: by

text: considering

text: only

text: the

text: homogeneous

text: initial

text: and

text: boundary

text: conditions,

text: we

text: M

text: M

text: M

text: can

text: use

text: integration

text: by

text: parts

text: freely.

text: We

text: let

text: F

text: F

text: H

text: with

text: the

text: antilinear

text: M

text: C

text: isometric

text: isomorphy

text: valid

text: again

text: because

text: of

text: non-degeneracy

text: of

text: the

text: inner

text: product

text: on

text: G

text: and

text: calculate

text: F

text: F

text: dnx

text: G

text: F¯

text: F

text: M

text: |F¯

text: |−1∂

text: |GIJ∂

text: f

text: M

text: I

text: J

text: |G

text: |G

text: |GIJ∂

text: F¯

text: )F

text: I

text: J

text: Ωt

text: dnx

text: G

text: G

text: G

text: GIJ∂

text: F

text: )F

text: M

text: I

text: J

text: F

text: F

text: G∗

text: F

text: F

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: the coordinate expressions in terms of a local orthonormal coordinate system w.r.t. G , we have for ∈ M 0 an F H1,2( ) 0 0 (cid:112) (cid:112) (cid:112) (cid:112) −| | −| | − −| | −| | W[F ] = G ∂ ( G GIJ∂ F ) G ∂ ( G GIJ∂ F ). (90) I J 0 I 0 0 J We recall that at the end of the previous section, we obtained after some work that the pull-back of M M the metric G on is the bundle metric G on given by G = G + δG. Since we only want to 0 0 stay in lowest order in δG, we can use the determinant expansion formulas given at the end of the − previous section as well. Last, we notice GIJ = GIJ δGIJ in ﬁrst order in δG, since δI = GIKG = 0 − O J KJ (GIK + δGIK)(G + δG ) = GIKG + GIKδG δGIKG + ((δG)2) = GIKG0 = δI 0 0,KJ KJ 0 0,KJ 0 KJ 0,KJ 0 KJ J because of symmetry of δG, δG = δG . The derivation is completely analogous to the derivation IJ JI of V. Hitting in the same vein, we deﬁne (cid:88) ∞ W = W , (91) k k=1 ∼ where W contains only the contributions of δG in the determinant expansion which scale as W k O k (δG≡)k. Since(cid:112)we are on(cid:112)ly interested in the lowest order, we have W = W1 + ((δG)2). Identifying W W as we did before for V we can write at ﬁrst 1 (cid:112) (cid:112) (cid:3) −| | −| | F = G ∂ ( G GIJ∂ F ) G (cid:112) I (cid:112) J (cid:112) (cid:112) −| | −| | − = G + δG ∂ ( G + δG (GIJ δGIJ)∂ F ) 0 I 0 0 J (cid:112)−| | −(cid:112)| | − −| |1 (cid:112)− −|(cid:112) | = G ∂ ( G GIJ∂ F ) G Tr(G 1δG)∂ ( G GIJ∂ F ) 0 I 0 0 J 0 2 0 I 0 0 J 1(cid:112) −| |−1(cid:112) −| | − (cid:112) − −| (cid:112)|−1 −| | + G ∂ ( G Tr(G 1δG)GIJ∂ F ) G ∂ ( G δGIJ∂ F ) 2 0 I 0 0 0 J 0 I 0 J (cid:112)−| | (cid:112)−| | − | |1 (cid:112) − (cid:112) −| | = G ∂ ( G GIJ∂ F ) g Tr(G 1δG)∂ ( g GIJ∂ F ) 0 I 0 0 J 0 2 0 I 0 0 J 1(cid:112) | |−1 (cid:112) | | − − | |−1 | |(cid:112) (cid:112) + g ∂ ( g Tr(G 1δG)GIJ∂ F ) g ∂ ( g δGIJ∂ F ) 2 0 I 0 0 0 J 0 I 0 J −| | −| | 1 − − | |−1 | | = G ∂ ( G GIJ∂ F ) + ∂ Tr(G 1δG)GIJ∂ F g ∂ ( g δGIJ∂ F ) 0 I 0 0 J 2 I 0 0 J 0 I 0 J (cid:3) = [F ] + W[F ]. G0 | | −| | ∈ { } During the fourth step we have used g = G by deﬁnition of G . Notice that if I, J 1, 2, ..., n , 0 0 0 the last three contributions are precisely the prior perturbation operator V. We deﬁne a third operator T which contains all temporal derivatives. T is deﬁned in all orders in δG ≡ − T W V, (92) 38 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: the

text: coordinate

text: expressions

text: in

text: terms

text: of

text: a

text: local

text: orthonormal

text: coordinate

text: system

text: w.r.t.

text: G

text: we

text: have

text: for

text: M

text: an

text: F

text: W[F

text: G

text: G

text: GIJ∂

text: F

text: G

text: G

text: GIJ∂

text: F

text: I

text: J

text: I

text: J

text: We

text: recall

text: that

text: at

text: the

text: end

text: of

text: the

text: previous

text: section,

text: we

text: obtained

text: after

text: some

text: work

text: that

text: the

text: pull-back

text: of

text: M

text: M

text: the

text: metric

text: G

text: on

text: is

text: the

text: bundle

text: metric

text: G

text: on

text: given

text: by

text: G

text: G

text: δG.

text: Since

text: we

text: only

text: want

text: to

text: stay

text: in

text: lowest

text: order

text: in

text: δG,

text: we

text: can

text: use

text: the

text: determinant

text: expansion

text: formulas

text: given

text: at

text: the

text: end

text: of

text: the

text: previous

text: section

text: as

text: well.

text: Last,

text: we

text: notice

text: GIJ

text: GIJ

text: δGIJ

text: in

text: ﬁrst

text: order

text: in

text: δG,

text: since

text: δI

text: GIKG

text: O

text: J

text: KJ

text: (GIK

text: δGIK)(G

text: δG

text: GIKG

text: GIKδG

text: δGIKG

text: δI

text: KJ

text: KJ

text: KJ

text: J

text: because

text: of

text: symmetry

text: of

text: δG,

text: δG

text: δG

text: The

text: derivation

text: is

text: completely

text: analogous

text: to

text: the

text: derivation

text: IJ

text: JI

text: of

text: V.

text: Hitting

text: in

text: the

text: same

text: vein,

text: we

text: deﬁne

text: W

text: W

text: k

text: where

text: W

text: contains

text: only

text: the

text: contributions

text: of

text: δG

text: in

text: the

text: determinant

text: expansion

text: which

text: scale

text: as

text: W

text: k

text: O

text: k

text: (δG≡)k.

text: are

text: interested

text: in

text: the

text: lowest

text: order,

text: we

text: have

text: W

text: Identifying

text: W

text: W

text: as

text: we

text: did

text: before

text: for

text: V

text: we

text: can

text: write

text: at

text: ﬁrst

text: F

text: G

text: G

text: GIJ∂

text: F

text: G

text: I

text: J

text: G

text: δG

text: G

text: δG

text: (GIJ

text: δGIJ)∂

text: F

text: I

text: J

text: G

text: G

text: GIJ∂

text: F

text: G

text: Tr(G

text: G

text: GIJ∂

text: F

text: I

text: J

text: I

text: J

text: G

text: G

text: Tr(G

text: F

text: G

text: G

text: δGIJ∂

text: F

text: I

text: J

text: I

text: J

text: G

text: G

text: GIJ∂

text: F

text: g

text: Tr(G

text: g

text: GIJ∂

text: F

text: I

text: J

text: I

text: J

text: g

text: g

text: Tr(G

text: F

text: g

text: g

text: δGIJ∂

text: F

text: I

text: J

text: I

text: J

text: G

text: G

text: GIJ∂

text: F

text: Tr(G

text: F

text: g

text: g

text: δGIJ∂

text: F

text: I

text: J

text: I

text: J

text: I

text: J

text: [F

text: W[F

text: During

text: the

text: fourth

text: step

text: we

text: have

text: used

text: g

text: G

text: by

text: deﬁnition

text: of

text: G

text: Notice

text: that

text: if

text: I,

text: J

text: n

text: the

text: last

text: three

text: contributions

text: are

text: precisely

text: the

text: prior

text: perturbation

text: operator

text: V.

text: We

text: deﬁne

text: a

text: third

text: operator

text: T

text: which

text: contains

text: all

text: temporal

text: derivatives.

text: T

text: is

text: deﬁned

text: in

text: all

text: orders

text: in

text: δG

text: T

text: W

text: V,

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: irrotational, isentropic and inviscid ﬂuid ﬂow. We start by a modiﬁcation of a literature action functional of this theory of ﬂuid dynamics to curved space-time. Using the theory of diﬀerential forms, we can derive the Euler equations and by acoustic linearization we recover a scalar wave equation in curved space-time - the acoustic wave equation. The dynamics of the boundaries, i.e., the boundary vibrations, can be well described by the motion of a massive membrane-like structure comparable to bio-membranes which we have in mind as prototypical applications of our theory. Our starting point is an action functional including the variation of area of the boundaries and including by an approach inspired by Chern-Weyl theory or, more physically, (Dirac-)Born-Infeld electrodynamics, curvature eﬀects, we arrive at a diﬀerential equation which is of second order in time and fourth order in spatial variables. The equation reproduces in suitable limits the bending membrane equation derived before and the conservative ﬂexible membrane equation. To include dissipation, we transfer the concept of time-lapsing from general relativity to continuum ﬁeld theory and modify the derivation of the boundary vibrations equation accordingly. In the end, we discuss how to include boundary and initial conditions as well as source terms to our model equations. A Cauchy-Kovalevskaya argument shows that we can convert boundary conditions in source terms and vice versa. We answer the question of how to model the practically more relevant case of localized boundary vibrations: Localized boundary vibrations are boundary vibrations which are non-zero only on one or more mutually dis-connected sub-manifolds of the boundary of the unperturbed ﬁber Ω . 0 • Analysis The analytical part is the bridge between the geometrical and the physical part. Physically, we are interested in solving the model equations and a convenient tools is perturbation theory as developed for quantum mechanics by Dirac [22, 23], Dyson [24, 25] (see also Reed and Simon [81, Section X.12]), and a decade later later by Magnus [58, 8], whose method has been explained nicely by Blanes et al. [7] and has been reﬁned recently by Fern´andez [29]. We will present a detailed derivation of a time-dependent perturbation theory in the spirit of Magnus, which then is applied by using Duhamel’s principle and Banach’s ﬁxed-point theorem in the spirit of Dirac’s perturbation ansatz. The relevant mathematical theory is the theory of semigroups of operators, operator evolution equations, and perturbation theory for linear operators applied to “classical” partial diﬀerential equations. As a check of our formalism, we compare our results with the literature. Particularly, for the conceptual basis of the analytical perturbation theory, with the theories of Brillouin [9], Fro¨hlich [33], and Cabrera [11, 12] as well as Feshbach [30], which partially date back to the thirties of last century. From the result-oriented point of view, we compare our approach with the perturbation theory developed by Deng and Li [56] more recently using the seminal work of Beale, which has since then undergone reﬁnement by Casarino et al. [15], Gal et al. [34], once more Casarino et al. [16], and many more who are to remain unnamed here. In our perturbation theory, we use a decoupling argument that implicitly assumes the smallness of a certain parameter. Upon formulating our model, we will see that this is the case in a large class of vibrational acoustics models; particularly, those used in bio-acoustics and (classical) vibrational acoustics. The ﬁnal consistency check will be performed by comparing the detailed results we are going to obtain in the general formalism with those that have been achieved in our previous papers [101, 98, 99]. In doing so, we will verify as to whether the semigroups obtained in this paper agree with the semigroups we found before when specializing to the model of internally coupled ears, ICE for short. 5

text: irrotational,

text: isentropic

text: and

text: inviscid

text: ﬂuid

text: ﬂow.

text: We

text: start

text: by

text: a

text: modiﬁcation

text: of

text: a

text: literature

text: action

text: functional

text: of

text: this

text: theory

text: of

text: ﬂuid

text: dynamics

text: to

text: curved

text: space-time.

text: Using

text: the

text: theory

text: of

text: diﬀerential

text: forms,

text: we

text: can

text: derive

text: the

text: Euler

text: equations

text: and

text: by

text: acoustic

text: linearization

text: we

text: recover

text: a

text: scalar

text: wave

text: equation

text: in

text: curved

text: space-time

text: the

text: acoustic

text: wave

text: equation.

text: The

text: dynamics

text: of

text: the

text: boundaries,

text: i.e.,

text: the

text: boundary

text: vibrations,

text: can

text: be

text: well

text: described

text: by

text: the

text: motion

text: of

text: a

text: massive

text: membrane-like

text: structure

text: comparable

text: to

text: bio-membranes

text: which

text: we

text: have

text: in

text: mind

text: as

text: prototypical

text: applications

text: of

text: our

text: theory.

text: Our

text: starting

text: point

text: is

text: an

text: action

text: functional

text: including

text: the

text: variation

text: of

text: area

text: of

text: the

text: boundaries

text: and

text: including

text: by

text: an

text: approach

text: inspired

text: by

text: Chern-Weyl

text: theory

text: or,

text: more

text: physically,

text: (Dirac-)Born-Infeld

text: electrodynamics,

text: curvature

text: eﬀects,

text: we

text: arrive

text: at

text: a

text: diﬀerential

text: equation

text: which

text: is

text: of

text: second

text: order

text: in

text: time

text: and

text: fourth

text: order

text: in

text: spatial

text: variables.

text: The

text: equation

text: reproduces

text: in

text: suitable

text: limits

text: the

text: bending

text: membrane

text: equation

text: derived

text: before

text: and

text: the

text: conservative

text: ﬂexible

text: membrane

text: equation.

text: To

text: include

text: dissipation,

text: we

text: transfer

text: the

text: concept

text: of

text: time-lapsing

text: from

text: general

text: relativity

text: to

text: continuum

text: ﬁeld

text: theory

text: and

text: modify

text: the

text: derivation

text: of

text: the

text: boundary

text: vibrations

text: equation

text: accordingly.

text: In

text: the

text: end,

text: we

text: discuss

text: how

text: to

text: include

text: boundary

text: and

text: initial

text: conditions

text: as

text: well

text: as

text: source

text: terms

text: to

text: our

text: model

text: equations.

text: A

text: Cauchy-Kovalevskaya

text: argument

text: shows

text: that

text: we

text: can

text: convert

text: boundary

text: conditions

text: in

text: source

text: terms

text: and

text: vice

text: versa.

text: We

text: answer

text: the

text: question

text: of

text: how

text: to

text: model

text: the

text: practically

text: more

text: relevant

text: case

text: of

text: localized

text: boundary

text: vibrations:

text: Localized

text: boundary

text: vibrations

text: are

text: boundary

text: vibrations

text: which

text: are

text: non-zero

text: only

text: on

text: one

text: or

text: more

text: mutually

text: dis-connected

text: sub-manifolds

text: of

text: the

text: boundary

text: of

text: the

text: unperturbed

text: ﬁber

text: Ω

text: Analysis

text: The

text: analytical

text: part

text: is

text: the

text: bridge

text: between

text: the

text: geometrical

text: and

text: the

text: physical

text: part.

text: Physically,

text: we

text: are

text: interested

text: in

text: solving

text: the

text: model

text: equations

text: and

text: a

text: convenient

text: tools

text: is

text: perturbation

text: theory

text: as

text: developed

text: for

text: quantum

text: mechanics

text: by

text: Dirac

text: Dyson

text: (see

text: also

text: Reed

text: and

text: Simon

text: Section

text: and

text: a

text: decade

text: later

text: later

text: by

text: Magnus

text: whose

text: method

text: has

text: been

text: explained

text: nicely

text: by

text: Blanes

text: et

text: al.

text: and

text: has

text: been

text: reﬁned

text: recently

text: by

text: Fern´andez

text: We

text: will

text: present

text: a

text: detailed

text: derivation

text: of

text: a

text: time-dependent

text: perturbation

text: theory

text: in

text: the

text: spirit

text: of

text: Magnus,

text: which

text: then

text: is

text: applied

text: by

text: using

text: Duhamel’s

text: principle

text: and

text: Banach’s

text: ﬁxed-point

text: theorem

text: in

text: the

text: spirit

text: of

text: Dirac’s

text: perturbation

text: ansatz.

text: The

text: relevant

text: mathematical

text: theory

text: is

text: the

text: theory

text: of

text: semigroups

text: of

text: operators,

text: operator

text: evolution

text: equations,

text: and

text: perturbation

text: theory

text: for

text: linear

text: operators

text: applied

text: to

text: “classical”

text: partial

text: diﬀerential

text: equations.

text: As

text: a

text: check

text: of

text: our

text: formalism,

text: we

text: compare

text: our

text: results

text: with

text: the

text: literature.

text: Particularly,

text: for

text: the

text: conceptual

text: basis

text: of

text: the

text: analytical

text: perturbation

text: theory,

text: with

text: the

text: theories

text: of

text: Brillouin

text: Fro¨hlich

text: and

text: Cabrera

text: as

text: well

text: as

text: Feshbach

text: which

text: partially

text: date

text: back

text: to

text: the

text: thirties

text: of

text: last

text: century.

text: From

text: the

text: result-oriented

text: point

text: of

text: view,

text: we

text: compare

text: our

text: approach

text: with

text: the

text: perturbation

text: theory

text: developed

text: by

text: Deng

text: and

text: Li

text: more

text: recently

text: using

text: the

text: seminal

text: work

text: of

text: Beale,

text: which

text: has

text: since

text: then

text: undergone

text: reﬁnement

text: by

text: Casarino

text: et

text: al.

text: Gal

text: et

text: al.

text: once

text: more

text: Casarino

text: et

text: al.

text: and

text: many

text: more

text: who

text: are

text: to

text: remain

text: unnamed

text: here.

text: In

text: our

text: perturbation

text: theory,

text: we

text: use

text: a

text: decoupling

text: argument

text: that

text: implicitly

text: assumes

text: the

text: smallness

text: of

text: a

text: certain

text: parameter.

text: Upon

text: formulating

text: our

text: model,

text: we

text: will

text: see

text: that

text: this

text: is

text: the

text: case

text: in

text: a

text: large

text: class

text: of

text: vibrational

text: acoustics

text: models;

text: particularly,

text: those

text: used

text: in

text: bio-acoustics

text: and

text: (classical)

text: vibrational

text: acoustics.

text: The

text: ﬁnal

text: consistency

text: check

text: will

text: be

text: performed

text: by

text: comparing

text: the

text: detailed

text: results

text: we

text: are

text: going

text: to

text: obtain

text: in

text: the

text: general

text: formalism

text: with

text: those

text: that

text: have

text: been

text: achieved

text: in

text: our

text: previous

text: papers

text: In

text: doing

text: so,

text: we

text: will

text: verify

text: as

text: to

text: whether

text: the

text: semigroups

text: obtained

text: in

text: this

text: paper

text: agree

text: with

text: the

text: semigroups

text: we

text: found

text: before

text: when

text: specializing

text: to

text: the

text: model

text: of

text: internally

text: coupled

text: ears,

text: ICE

text: for

text: short.


text: 4 Derivation of the Perturbation Equations Convention: We use big Latin indices I, J, K, L, ... to denote local orthornormal coordinates stemming M M M from on and , small Greek indices µ, ν, κ, λ, ... to denote the local orthornormal coordinates ref 0 on the second component of the ﬁber bundles, Ω and Ω stemming from Bn(0) and small Latin indices t 0 1 i, j, k, l, ... to denote local orthornormal coordinates on ∂Ω and ∂Ω stemming from Sn(0). Further, we M t 0 M 1 will distinguish the perturbation bundle from the unperturbed bundle by ¨priming¨the indices for M 0 , Ω , ∂Ω if ambiguities arise. 0 0 0 Modelling: We will derive an abstract ICE model from an action functional S consisting of two ICE parts. A ﬂuid dynamical action S and a geometric action S . fluid geom Fluid action: We will use a simpliﬁed version of the Bateman Lagrangian [?, ?] for irrotational, isentropic M non-viscous ﬂow, but put the action functional in the metric background on the perturbation bundle given by G. We need several steps. • ∈ M Fluid velocity normalization: Recall that V Γ(T ). Since V 0 = V t as the ”time”-component − of the ﬂuid velocity ﬁeld bears no physical information, we use the normalization V 0V = 4, i.e., 0 V 0 = 2, which is conceptually familiar from Einsteinian gravitation theory. That we normalize V 0 = 2 instead of V 0 = 1 has aesthetic reasons that will become clear below. On the other hand, ∈ V M M → V ( ). By deﬁnition of the perturbation bundle is compact and the mapping t Ω is (cid:55)→ t smooth in t. We now levy this deﬁnition up to deﬁne the map V V = V µ∂ , where the index t t t µ ∈ V indicates that we now have a smoothly parameterized family of vector ﬁelds such that V (Ω ) t t for all positive t. • Hodge decomposition theorem: Since the dimensions of our setups are not necessarily n = 3 although we have that as the main application in mind, we can’t use the Helmholtz decomposition theorem straight away but need a digression in cohomology theory on Riemannian manifolds. More precisely, we give a minimal account of the Hodge decomposition theorem. Since Ω is compact, t ∗ the deRham cohomology exact sequence obtained by acting with the cohomology functor Ω on Ω . Z →t For that regard the -graded exterior algebra Ω(Ω ) with the exterior derivative d : Ωk(Ω ) ≤ n≤+1 − t k t Ωk+1(Ω ) for all 0 k n 1, t {0} (cid:44)→ Ω0(Ω ) →d0 Ω1(Ω ) →d1 Ω2(Ω ) · · · Ωn−1(Ω ) d→n−1 Ωn(Ω ) (cid:44)→ {0}, (98) t t t t t ◦ ≤ ≤ − Treating the spaces Ωk(Ω ) as vector spaces and using nilpotency of d d for all 0 k n 2, t k+1 k we can deﬁne the deRham complex { } → → → · · · − → → { } 0 (cid:44) H0(Ω ) d H1(Ω ) d H2(Ω ) Hn 1(Ω ) d Hn(Ω ) (cid:44) 0 , (99) t t t t t (cid:39) where Hk(Ω ) ker d /imd − . By deﬁnition of the deRham groups Hk(Ω ), d − can be inverted in t k k 1 → t k 1 the sense that δ : Hk+1(Ω ) Hk(Ω ) is the adjoint operator w.r.t. the L2 inner product deﬁned k+1 t t on Ωk(Ω ) by means of the metric tensor g on Ω . The resultant complex is t t 41

label: section

text: Derivation

text: of

text: the

text: Perturbation

text: Equations

text: Convention:

text: We

text: use

text: big

text: Latin

text: indices

text: I,

text: J,

text: K,

text: L,

text: to

text: denote

text: local

text: orthornormal

text: coordinates

text: stemming

text: M

text: M

text: M

text: from

text: on

text: and

text: small

text: Greek

text: indices

text: µ,

text: ν,

text: κ,

text: λ,

text: to

text: denote

text: the

text: local

text: orthornormal

text: coordinates

text: ref

text: on

text: the

text: second

text: component

text: of

text: the

text: ﬁber

text: bundles,

text: Ω

text: and

text: Ω

text: stemming

text: from

text: and

text: small

text: Latin

text: indices

text: t

text: i,

text: j,

text: k,

text: l,

text: to

text: denote

text: local

text: orthornormal

text: coordinates

text: on

text: ∂Ω

text: and

text: ∂Ω

text: stemming

text: from

text: Further,

text: we

text: M

text: t

text: M

text: will

text: distinguish

text: the

text: perturbation

text: bundle

text: from

text: the

text: unperturbed

text: bundle

text: by

text: ¨priming¨the

text: indices

text: for

text: M

text: Ω

text: ∂Ω

text: if

text: ambiguities

text: arise.

text: Modelling:

text: We

text: will

text: derive

text: an

text: abstract

text: ICE

text: model

text: from

text: an

text: action

text: functional

text: S

text: consisting

text: of

text: two

text: ICE

text: parts.

text: A

text: ﬂuid

text: dynamical

text: action

text: S

text: and

text: a

text: geometric

text: action

text: S

text: fluid

text: geom

text: Fluid

text: action:

text: We

text: will

text: use

text: a

text: simpliﬁed

text: version

text: of

text: the

text: Bateman

text: Lagrangian

text: for

text: irrotational,

text: isentropic

text: M

text: non-viscous

text: ﬂow,

text: but

text: put

text: the

text: action

text: functional

text: in

text: the

text: metric

text: background

text: on

text: the

text: perturbation

text: bundle

text: given

text: by

text: G.

text: We

text: need

text: several

text: steps.

text: M

text: Fluid

text: velocity

text: normalization:

text: Recall

text: that

text: V

text: Γ(T

text: Since

text: V

text: V

text: t

text: as

text: the

text: ”time”-component

text: of

text: the

text: ﬂuid

text: velocity

text: ﬁeld

text: bears

text: no

text: physical

text: information,

text: we

text: use

text: the

text: normalization

text: V

text: i.e.,

text: V

text: which

text: is

text: conceptually

text: familiar

text: from

text: Einsteinian

text: gravitation

text: theory.

text: That

text: we

text: normalize

text: V

text: instead

text: of

text: V

text: has

text: aesthetic

text: reasons

text: that

text: will

text: become

text: clear

text: below.

text: On

text: the

text: other

text: hand,

text: V

text: M

text: M

text: V

text: By

text: deﬁnition

text: of

text: the

text: perturbation

text: bundle

text: is

text: compact

text: and

text: the

text: mapping

text: t

text: Ω

text: is

text: t

text: smooth

text: in

text: t.

text: We

text: now

text: levy

text: this

text: deﬁnition

text: up

text: to

text: deﬁne

text: the

text: map

text: V

text: V

text: V

text: µ∂

text: where

text: the

text: index

text: t

text: t

text: t

text: µ

text: V

text: indicates

text: that

text: we

text: now

text: have

text: a

text: smoothly

text: parameterized

text: family

text: of

text: vector

text: ﬁelds

text: such

text: that

text: V

text: (Ω

text: t

text: t

text: for

text: all

text: positive

text: t.

text: Hodge

text: decomposition

text: theorem:

text: Since

text: the

text: dimensions

text: of

text: our

text: setups

text: are

text: not

text: necessarily

text: n

text: although

text: we

text: have

text: that

text: as

text: the

text: main

text: application

text: in

text: mind,

text: we

text: can’t

text: use

text: the

text: Helmholtz

text: decomposition

text: theorem

text: straight

text: away

text: but

text: need

text: a

text: digression

text: in

text: cohomology

text: theory

text: on

text: Riemannian

text: manifolds.

text: More

text: precisely,

text: we

text: give

text: a

text: minimal

text: account

text: of

text: the

text: Hodge

text: decomposition

text: theorem.

text: Since

text: Ω

text: is

text: compact,

text: t

text: the

text: deRham

text: cohomology

text: exact

text: sequence

text: obtained

text: by

text: acting

text: with

text: the

text: cohomology

text: functor

text: Ω

text: on

text: Ω

text: Z

text: →t

text: For

text: that

text: regard

text: the

text: -graded

text: exterior

text: algebra

text: Ω(Ω

text: with

text: the

text: exterior

text: derivative

text: d

text: Ωk(Ω

text: t

text: k

text: t

text: for

text: all

text: k

text: n

text: t

text: Ω0(Ω

text: Ω1(Ω

text: Ω2(Ω

text: Ωn−1(Ω

text: Ωn(Ω

text: t

text: t

text: t

text: t

text: t

text: Treating

text: the

text: spaces

text: Ωk(Ω

text: as

text: vector

text: spaces

text: and

text: using

text: nilpotency

text: of

text: d

text: d

text: for

text: all

text: k

text: n

text: t

text: k

text: we

text: can

text: deﬁne

text: the

text: deRham

text: complex

text: d

text: d

text: Hn

text: d

text: Hn(Ω

text: t

text: t

text: t

text: t

text: t

text: where

text: Hk(Ω

text: ker

text: d

text: /imd

text: By

text: deﬁnition

text: of

text: the

text: deRham

text: groups

text: Hk(Ω

text: d

text: can

text: be

text: inverted

text: in

text: t

text: k

text: k

text: t

text: k

text: the

text: sense

text: that

text: δ

text: Hk(Ω

text: is

text: the

text: adjoint

text: operator

text: w.r.t.

text: the

text: inner

text: product

text: deﬁned

text: t

text: t

text: on

text: Ωk(Ω

text: by

text: means

text: of

text: the

text: metric

text: tensor

text: g

text: on

text: Ω

text: The

text: resultant

text: complex

text: is

text: t

text: t


text: Figure 9: Diﬀeomorphy of Ω , Ω and Ω (cid:48) induces an isomorphism between the de Rham cohomology t 0 t complex of Ω , Ω and Ω (cid:48) t 0 t M M M Figure 10: Diﬀeomorphy of , and induces an isomorphism between the de Rham cohomology M M M 0 ref complex of , and 0 ref M M − ◦ M → M a global bundle diﬀeomorphism between and , Φ 1 Φ : . This can be used to obtain 0 0 0 the correspondences depicted in Fig. 10 following the previous logic. The only thing left is to relate M R the deRham cohomology complex of to the cohomology complex of the base space + and Ω . By M M M t triviality of the bundles , , , the Serre spectral sequence reduces to the Ku¨nneth formula for ref the cohomology of product manifolds. Since the deRham cohomology is a cohomology sequence over the R group , the torsion groups in the general Ku¨nneth formula become trivial and can thus be ignored up to isomorphy / diﬀeomorphy. Since the cohomology complexes of the (Ω ) are isomorphic, we have t t>0 (cid:77) M (cid:39) k ⊗ − R Hk( ) (cid:77) Hl(Ω ) Hk 1( +) (143) t l=0 (cid:39) k ⊗ − R Hl(Ω ) Hk 1( +) (144) 0 l=0 (cid:39) − ⊗ R ⊕ ⊗ R (Hk 1(Ω ) H1( +)) (Hk(Ω ) H0( +)) (145) 0 0 R R → R R The group H0( +) can be thought of as the space of functions f : + and H1( +) is just the R → R M space of forms ω dt where ω is again a function ω : + . Since we are only interested in Ωk ( ), the t t t M (cid:39) ⊗ Rsp ﬁrst contribution in the above decomposition is the relevant one, i.e., Hk ( ) Hk(Ω ) H0( +). By the M sp 0 ∈ M Hodge decomposition theorem, we can obtain from Hk ( ) the decomposition of a k-form ω Ωk ( ) sp sp where we have to ignore any contributions including a 1-form basis vector dt. But this reduces by the 49 ##LTFigure## ##LTFigure##

text: Figure

text: Diﬀeomorphy

label: caption

text: of

text: Ω

text: Ω

text: and

text: Ω

text: induces

text: an

text: isomorphism

text: between

text: the

text: de

text: Rham

text: cohomology

text: t

text: t

text: complex

text: of

text: Ω

text: Ω

text: and

text: Ω

text: t

text: t

text: M

text: M

text: M

text: Figure

text: Diﬀeomorphy

text: of

text: and

text: induces

text: an

text: isomorphism

text: between

text: the

text: de

text: Rham

text: cohomology

text: M

text: M

text: M

text: ref

text: complex

text: of

text: and

text: ref

text: M

text: M

text: M

text: M

text: a

text: global

text: bundle

text: diﬀeomorphism

text: between

text: and

text: Φ

text: Φ

text: This

text: can

text: be

text: used

text: to

text: obtain

text: the

text: correspondences

text: depicted

text: in

text: Fig.

text: following

text: the

text: previous

text: logic.

text: The

text: only

text: thing

text: left

text: is

text: to

text: relate

text: M

text: R

text: the

text: deRham

text: cohomology

text: complex

text: of

text: to

text: the

text: cohomology

text: complex

text: of

text: the

text: base

text: space

text: and

text: Ω

text: By

text: M

text: M

text: M

text: t

text: triviality

text: of

text: the

text: bundles

text: the

text: Serre

text: spectral

text: sequence

text: reduces

text: to

text: the

text: Ku¨nneth

text: formula

text: for

text: ref

text: the

text: cohomology

text: of

text: product

text: manifolds.

text: Since

text: the

text: deRham

text: cohomology

text: is

text: a

text: cohomology

text: sequence

text: over

text: the

text: R

text: group

text: the

text: torsion

text: groups

text: in

text: the

text: general

text: Ku¨nneth

text: formula

text: become

text: trivial

text: and

text: can

text: thus

text: be

text: ignored

text: up

text: to

text: isomorphy

text: diﬀeomorphy.

text: Since

text: the

text: cohomology

text: complexes

text: of

text: the

text: (Ω

text: are

text: isomorphic,

text: we

text: have

text: t

text: M

text: k

text: R

text: Hk(

text: Hl(Ω

text: Hk

text: t

text: k

text: R

text: Hl(Ω

text: Hk

text: R

text: R

text: (Hk

text: (Hk(Ω

text: R

text: R

text: R

text: R

text: The

text: group

text: can

text: be

text: thought

text: of

text: as

text: the

text: space

text: of

text: functions

text: f

text: and

text: is

text: just

text: the

text: R

text: R

text: M

text: space

text: of

text: forms

text: ω

text: dt

text: where

text: ω

text: is

text: again

text: a

text: function

text: ω

text: Since

text: we

text: are

text: only

text: interested

text: in

text: Ωk

text: the

text: t

text: t

text: t

text: M

text: Rsp

text: ﬁrst

text: contribution

text: in

text: the

text: above

text: decomposition

text: is

text: the

text: relevant

text: one,

text: i.e.,

text: Hk

text: Hk(Ω

text: By

text: the

text: M

text: sp

text: M

text: Hodge

text: decomposition

text: theorem,

text: we

text: can

text: obtain

text: from

text: Hk

text: the

text: decomposition

text: of

text: a

text: k-form

text: ω

text: Ωk

text: sp

text: sp

text: where

text: we

text: have

text: to

text: ignore

text: any

text: contributions

text: including

text: a

text: basis

text: vector

text: dt.

text: But

text: this

text: reduces

text: by

text: the

text: ##LTFigure##

text: ##LTFigure##


text: → Using that φ → : ∂Ω ∂Ω paramterizes ∂Ω in terms of coordinates on ∂Ω and that ∂Ω = 0 t 0 t t 0 t graph (u(t, y)), we obtain the easier expression, y (cid:112) ∇ ∇ − Tr [ ∂,i ∂,ju] ∇ ∇ O (n 1)H(u) = g0 ∇ ∇ = Tr [ ∂,i ∂,ju] + ((cid:15)3) = ∆∂ u (149) 1 + ∂u ∂,iu g0 g0 i ∇ Here, ∂ denotes the covariant derivative w.r.t. to the basis vectors induces from φ → as a parme- 0 t ∇ terization. Expanding further, we can neglect and non-linear contributions in u and identify ∂ with the Levi-Civita-connection on ∂Ω , ignoring contributions containing u and derivatives thereof t at least quadratically. ∆∂ is the Laplace-Beltrami operator on ∂Ω . Last, we need a matrix J which −g0 0 − is anti-symmetric, J = J and that satisﬁes the normalization condition Tr((g 1J)2) = 1. If g is ij ji × − 0 0 the Euclidean metric and n = 3, we may take e.g. the 2 2 matrix with 1 on the upper to the diagonal and 1 on the lower to the diagonal. Then we deﬁne the anti-symmetric matrix Tr [II]J = Λ g0 and call the object mean curvature form on ∂Ω . We can put the stationary contribution to the t geometrical action together. Introduce a dimensionful parameter µ and set (cid:112) H (cid:112) | ≡ det (g + 2µ Λ) V T 0 ∂Ωt | H . (150) 0 det g 0 ∂Ωt Taylor-expansion around g up to quadratic order in µ gives us, using the deﬁnition of J, 0 (cid:18) H(cid:19) ◦ µ2 µ V φ → = T 1 + H (Tr (II))2 = T + (∆ u)2, (151) 0 t 0 2 g0 0 2 g0 on ∂Ω instead of ∂Ω . The stationary contribution to the action functional is then given by 0 t (cid:90) (cid:112) (cid:112) (cid:112) | S = T (cid:90) dn−1y (cid:112)|g | (cid:32)det (g0 ∂Ωt +| 2µHΛ) (cid:33) (152) geom,stat 0 t 0 det g ∂Ωt 0 ∂Ωt − | | gij ∇ ∇ µ = T dtdn 1y g 1 + 0 ∂u ∂u + (∆∂ u)2 , (153) 0 0 2 i j 2 g0 ∂Ω0 where we have pulled back ∂Ω to ∂Ω by means of φ → and expanded the pullback metric g = g +δg. t 0 0 t 0 Contributions of order (cid:15)3 have been ignored. The base coordinate t has not been pulled back because we have to treat the kinetic energy separately because of a diﬀerent pre-factor σ = ρ d. Let us m m integrate the stationary contribution w.r.t. the base space coordinate t of the reference ﬁber bundle M 0 . The ﬁrst contribution is a constant volume term which has no physical information and is 0 − M discarded. Noting that G is always in block-diagonal form, with Gtt = 1 on ∂ , we obtain the 0 0 0 following formulation for the stationary contribution to the geometrical action functional. 51 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Using

text: that

text: φ

text: ∂Ω

text: ∂Ω

text: paramterizes

text: ∂Ω

text: in

text: terms

text: of

text: coordinates

text: on

text: ∂Ω

text: and

text: that

text: ∂Ω

text: t

text: t

text: t

text: t

text: graph

text: (u(t,

text: y)),

text: we

text: obtain

text: the

text: easier

text: expression,

text: y

text: Tr

text: ∂,i

text: ∂,ju]

text: O

text: (n

text: Tr

text: ∂,i

text: ∂,ju]

text: u

text: ∂u

text: ∂,iu

text: i

text: Here,

text: denotes

text: the

text: covariant

text: derivative

text: w.r.t.

text: to

text: the

text: basis

text: vectors

text: induces

text: from

text: φ

text: as

text: a

text: parme-

text: t

text: terization.

text: Expanding

text: further,

text: we

text: can

text: neglect

text: and

text: non-linear

text: contributions

text: in

text: u

text: and

text: identify

text: with

text: the

text: Levi-Civita-connection

text: on

text: ∂Ω

text: ignoring

text: contributions

text: containing

text: u

text: and

text: derivatives

text: thereof

text: t

text: at

text: least

text: quadratically.

text: is

text: the

text: Laplace-Beltrami

text: operator

text: on

text: ∂Ω

text: Last,

text: we

text: need

text: a

text: matrix

text: J

text: which

text: is

text: anti-symmetric,

text: J

text: J

text: and

text: that

text: satisﬁes

text: the

text: normalization

text: condition

text: Tr((g

text: If

text: g

text: is

text: ij

text: ji

text: the

text: Euclidean

text: metric

text: and

text: n

text: we

text: may

text: take

text: e.g.

text: the

text: matrix

text: with

text: on

text: the

text: upper

text: to

text: the

text: diagonal

text: and

text: on

text: the

text: lower

text: to

text: the

text: diagonal.

text: Then

text: we

text: deﬁne

text: the

text: anti-symmetric

text: matrix

text: Tr

text: [II]J

text: Λ

text: and

text: call

text: the

text: object

text: mean

text: curvature

text: form

text: on

text: ∂Ω

text: We

text: can

text: put

text: the

text: stationary

text: contribution

text: to

text: the

text: t

text: geometrical

text: action

text: together.

text: Introduce

text: a

text: dimensionful

text: parameter

text: µ

text: and

text: set

text: H

text: det

text: (g

text: 2µ

text: Λ)

text: V

text: T

text: ∂Ωt

text: H

text: det

text: g

text: ∂Ωt

text: Taylor-expansion

text: around

text: g

text: up

text: to

text: quadratic

text: order

text: in

text: µ

text: gives

text: us,

text: using

text: the

text: deﬁnition

text: of

text: J,

text: µ2

text: µ

text: V

text: φ

text: T

text: H

text: (Tr

text: T

text: t

text: on

text: ∂Ω

text: instead

text: of

text: ∂Ω

text: The

text: stationary

text: contribution

text: to

text: the

text: action

text: functional

text: is

text: then

text: given

text: by

text: t

text: S

text: T

text: ∂Ωt

text: geom,stat

text: t

text: det

text: g

text: ∂Ωt

text: ∂Ωt

text: gij

text: µ

text: T

text: dtdn

text: g

text: ∂u

text: ∂u

text: i

text: j

text: ∂Ω0

text: where

text: we

text: have

text: pulled

text: back

text: ∂Ω

text: to

text: ∂Ω

text: by

text: means

text: of

text: φ

text: and

text: expanded

text: the

text: pullback

text: metric

text: g

text: g

text: +δg.

text: t

text: t

text: Contributions

text: of

text: order

text: have

text: been

text: ignored.

text: The

text: base

text: coordinate

text: t

text: has

text: not

text: been

text: pulled

text: back

text: because

text: we

text: have

text: to

text: treat

text: the

text: kinetic

text: energy

text: separately

text: because

text: of

text: a

text: diﬀerent

text: pre-factor

text: σ

text: ρ

text: d.

text: Let

text: us

text: m

text: m

text: integrate

text: the

text: stationary

text: contribution

text: w.r.t.

text: the

text: base

text: space

text: coordinate

text: t

text: of

text: the

text: reference

text: ﬁber

text: bundle

text: M

text: The

text: ﬁrst

text: contribution

text: is

text: a

text: constant

text: volume

text: term

text: which

text: has

text: no

text: physical

text: information

text: and

text: is

text: M

text: discarded.

text: Noting

text: that

text: G

text: is

text: always

text: in

text: block-diagonal

text: form,

text: with

text: Gtt

text: on

text: we

text: obtain

text: the

text: following

text: formulation

text: for

text: the

text: stationary

text: contribution

text: to

text: the

text: geometrical

text: action

text: functional.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: t = 0, we have φ → = id such that by the identiﬁcation of φ → with its diﬀerential by properness 0 t ∂Ω0 | 0 t of the perturbation bundle, we can conclude ∂ u = 0 as well. We note down the result of our t t=0 considerations ∈ u(t = 0, y) = 0 = ∂ (t = 0, y) for all y ∂Ω . (182) t 0 M Physically, this states that the reference bundle will not deform its ﬁbers Ω in time t by M 0 0 itself, but the formation of a perturbation bundle needs an external stimulation. In view of the principle of least action, this is reasonable from the physical viewpoint. We turn to summarizing our considerations to build up the geometrical action functional for the boundary vibrations and explaining the physical intuition and meaning of the contributions stored in the symbolic expression. M Denoting by GΣ the modiﬁed metric tensor on to account for dissipativity of the perturbation M 0 0 bundle , we have found in the previous sub-paragraphs, 0 (cid:90) (cid:113) (cid:0) (cid:1) S [u] geom.,diss. 1 −| | (183) = GΣ σ (GΣ)tt(∂ u)(∂ u) + T (GΣ)ij(∂ u)(∂ u) + d2T (∆∂ u)(∆∂ u) . 2 M 0 m 0 t t 0 0 i j curv g0 g0 ∂ 0 − The quantity σ = ρ d is the boundary vibrations’ mass density per (n 1)-dimensional area. m m It is expressed via the volume mass density of the boundary vibrations ρ and a characteristic m intrinsic length scale of the boundary vibrations u. In [99] this is the thickness of the tympanic membranes, denoted by d. Mathematically, we could have worked with the symbol σ straight m away, but experimentally ρ and d can be determined, whereas σ is a derived quantity not directly m m accessible to our experimental collaborators. The next constant is the tension of the boundary vibrations, T . The same letter denotes in [99] the membrane tension of the tympanic membranes. 0 It is a material constant that is needed to derive the phase velocity of the membrane’s ﬂexural waves. The next constant is new and called T . In [96], a n = 3-model for bio-membranes had curv been derived from a geometrical viewpoint starting form the equations of elasticity theory. We interpret T here as a curvature tension that describes how much the membranes resist to changes curv (cid:29) in mean curvature. If T /(T ) 1, the changes in curvature dominate over the ﬂexural membrane curv 0 vibrations and we could idealize the boundary vibrations as a damped Kirchhoﬀ-plate that gives bending waves predominantly. Eﬀectively, the boundary vibrations equation would then reduce to a damped higher-dimensional analogue of the bio-membrane equation of motion derived in [96]. If (cid:28) we have T /T 1 on the contrast, the bending waves propagate much slower than the ﬂexural curv 0 waves. This means that we can neglect the bending contribution already in the action functional S and consider only the ﬂexural vibrations of the boundary vibrations. We arrive at a higher- geom dimensional analog of the damped membrane equation considered in [101, 98, 99] that has been investigated as a special case in [42]. In the following sub-paragraph we will be concerned with three issues. Firstly, we want to localize the boundary vibrations. In biophysics or classical acoustics as the main physical motivation for the theory, one usually has not a vibration of the entire boundary { } of ∂Ω but only some, pathwisely unconnected parts, say Γ ≤ ≤ vibrate. In the biophysics of t i 1 i N animal hearing, this intuitively clear. If e.g. lizards want to localize the sounds a prey makes, not the entire interaural cavity in the lizards head starts to vibrate but only the tympanic membranes. 60 ##LTLine## ##LTLine##

text: t

text: we

text: have

text: φ

text: id

text: such

text: that

text: by

text: the

text: identiﬁcation

text: of

text: φ

text: with

text: its

text: diﬀerential

text: by

text: properness

text: t

text: ∂Ω0

text: t

text: of

text: the

text: perturbation

text: bundle,

text: we

text: can

text: conclude

text: u

text: as

text: well.

text: We

text: note

text: down

text: the

text: result

text: of

text: our

text: t

text: considerations

text: u(t

text: y)

text: (t

text: y)

text: for

text: all

text: y

text: ∂Ω

text: t

text: M

text: Physically,

text: this

text: states

text: that

text: the

text: reference

text: bundle

text: will

text: not

text: deform

text: its

text: ﬁbers

text: Ω

text: in

text: time

text: t

text: by

text: M

text: itself,

text: but

text: the

text: formation

text: of

text: a

text: perturbation

text: bundle

text: needs

text: an

text: external

text: stimulation.

text: In

text: view

text: of

text: the

text: principle

text: of

text: least

text: action,

text: this

text: is

text: reasonable

text: from

text: the

text: physical

text: viewpoint.

text: We

text: turn

text: to

text: summarizing

text: our

text: considerations

text: to

text: build

text: up

text: the

text: geometrical

text: action

text: functional

text: for

text: the

text: boundary

text: vibrations

text: and

text: explaining

text: the

text: physical

text: intuition

text: and

text: meaning

text: of

text: the

text: contributions

text: stored

text: in

text: the

text: symbolic

text: expression.

text: M

text: Denoting

text: by

text: GΣ

text: the

text: modiﬁed

text: metric

text: tensor

text: on

text: to

text: account

text: for

text: dissipativity

text: of

text: the

text: perturbation

text: M

text: bundle

text: we

text: have

text: found

text: in

text: the

text: previous

text: sub-paragraphs,

text: S

text: [u]

text: geom.,diss.

text: GΣ

text: σ

text: (GΣ)tt(∂

text: u)(∂

text: u)

text: T

text: (GΣ)ij(∂

text: u)(∂

text: u)

text: u)(∆∂

text: u)

text: M

text: m

text: t

text: t

text: i

text: j

text: curv

text: The

text: quantity

text: σ

text: ρ

text: d

text: is

text: the

text: boundary

text: vibrations’

text: mass

text: density

text: per

text: (n

text: area.

text: m

text: m

text: It

text: is

text: expressed

text: via

text: the

text: volume

text: mass

text: density

text: of

text: the

text: boundary

text: vibrations

text: ρ

text: and

text: a

text: characteristic

text: m

text: intrinsic

text: length

text: scale

text: of

text: the

text: boundary

text: vibrations

text: u.

text: In

text: this

text: is

text: the

text: thickness

text: of

text: the

text: tympanic

text: membranes,

text: denoted

text: by

text: d.

text: Mathematically,

text: we

text: could

text: have

text: worked

text: with

text: the

text: symbol

text: σ

text: straight

text: m

text: away,

text: but

text: experimentally

text: ρ

text: and

text: d

text: can

text: be

text: determined,

text: whereas

text: σ

text: is

text: a

text: derived

text: quantity

text: not

text: directly

text: m

text: m

text: accessible

text: to

text: our

text: experimental

text: collaborators.

text: The

text: next

text: constant

text: is

text: the

text: tension

text: of

text: the

text: boundary

text: vibrations,

text: T

text: The

text: same

text: letter

text: denotes

text: in

text: the

text: membrane

text: tension

text: of

text: the

text: tympanic

text: membranes.

text: It

text: is

text: a

text: material

text: constant

text: that

text: is

text: needed

text: to

text: derive

text: the

text: phase

text: velocity

text: of

text: the

text: membrane’s

text: ﬂexural

text: waves.

text: The

text: next

text: constant

text: is

text: new

text: and

text: called

text: T

text: In

text: a

text: n

text: for

text: bio-membranes

text: had

text: curv

text: been

text: derived

text: from

text: a

text: geometrical

text: viewpoint

text: starting

text: form

text: the

text: equations

text: of

text: elasticity

text: theory.

text: We

text: interpret

text: T

text: here

text: as

text: a

text: curvature

text: tension

text: that

text: describes

text: how

text: much

text: the

text: membranes

text: resist

text: to

text: changes

text: curv

text: in

text: mean

text: curvature.

text: If

text: T

text: /(T

text: the

text: changes

text: in

text: curvature

text: dominate

text: over

text: the

text: ﬂexural

text: membrane

text: curv

text: vibrations

text: and

text: we

text: could

text: idealize

text: the

text: boundary

text: vibrations

text: as

text: a

text: damped

text: Kirchhoﬀ-plate

text: that

text: gives

text: bending

text: waves

text: predominantly.

text: Eﬀectively,

text: the

text: boundary

text: vibrations

text: equation

text: would

text: then

text: reduce

text: to

text: a

text: damped

text: higher-dimensional

text: analogue

text: of

text: the

text: bio-membrane

text: equation

text: of

text: motion

text: derived

text: in

text: If

text: we

text: have

text: T

text: /T

text: on

text: the

text: contrast,

text: the

text: bending

text: waves

text: propagate

text: much

text: slower

text: than

text: the

text: ﬂexural

text: curv

text: waves.

text: This

text: means

text: that

text: we

text: can

text: neglect

text: the

text: bending

text: contribution

text: already

text: in

text: the

text: action

text: functional

text: S

text: and

text: consider

text: only

text: the

text: ﬂexural

text: vibrations

text: of

text: the

text: boundary

text: vibrations.

text: We

text: arrive

text: at

text: a

text: higher-

text: geom

text: dimensional

text: analog

text: of

text: the

text: damped

text: membrane

text: equation

text: considered

text: in

text: that

text: has

text: been

text: investigated

text: as

text: a

text: special

text: case

text: in

text: In

text: the

text: following

text: sub-paragraph

text: we

text: will

text: be

text: concerned

text: with

text: three

text: issues.

text: Firstly,

text: we

text: want

text: to

text: localize

text: the

text: boundary

text: vibrations.

text: In

text: biophysics

text: or

text: classical

text: acoustics

text: as

text: the

text: main

text: physical

text: motivation

text: for

text: the

text: theory,

text: one

text: usually

text: has

text: not

text: a

text: vibration

text: of

text: the

text: entire

text: boundary

text: of

text: ∂Ω

text: but

text: only

text: some,

text: pathwisely

text: unconnected

text: parts,

text: say

text: Γ

text: vibrate.

text: In

text: the

text: biophysics

text: of

text: t

text: i

text: i

text: N

text: animal

text: hearing,

text: this

text: intuitively

text: clear.

text: If

text: e.g.

text: lizards

text: want

text: to

text: localize

text: the

text: sounds

text: a

text: prey

text: makes,

text: not

text: the

text: entire

text: interaural

text: cavity

text: in

text: the

text: lizards

text: head

text: starts

text: to

text: vibrate

text: but

text: only

text: the

text: tympanic

text: membranes.

text: ##LTLine##

text: ##LTLine##