Harmonic generation and mode conversion in a rigid duct with a time-modulated diaphragm: A computational study

Abstract

We present a computational study of scattering of linear acoustic waves in a rigid waveguide by an infinitesimally thin vertical rigid baffle whose penetration into the cross-section varies with time. The stationary-baffle case is analysed first to provide a benchmark multimodal description of the scattered field together with the associated reflection and transmission behaviour. We then introduce a periodic modulation of the penetration height, which converts an incident monochromatic wave into a family of temporal harmonics. This interaction is captured through a Floquet-scattering formulation in which the harmonic amplitudes are coupled by interface conditions imposed at the baffle location and solved numerically by truncating the harmonic system. In the long-wavelength regime, we additionally derive a reduced one-dimensional model where the baffle is represented by jump relations that encode the effective kinematic and dynamic constraints induced by partial blockage. For sufficiently slow modulation, a quasistatic approximation is developed and compared against the full Floquet solution. All results are obtained from a self-contained computational framework whose convergence and accuracy are verified throughout. The results demonstrate how time-dependent internal elements redistribute acoustic energy across harmonics and modify reflection and transmission, providing practical modelling tools for waveguides with mechanically driven or actively controlled internal boundaries.

Keywords

duct acoustics acoustic scattering Floquet scattering harmonic generation quasistatic approximation

1. Introduction

Time-dependent wave systems, in which material parameters or boundary conditions are deliberately modulated in time, have attracted sustained attention because they enable responses that are inaccessible in stationary media, such as time-reversal concepts, frequency conversion and cascaded mixing, parametric and transient amplification, and temporal steering of wave energy (Apffel and Fort, 2022; Bacot et al., 2016; Galiffi et al., 2022; Hayran and Monticone, 2023; Kiorpelidis et al., 2024; Pacheco-Peña and Engheta, 2020; Pacheco-Peña et al., 2022; Ptitcyn et al., 2023). Foundational ideas on rapid phase-velocity modification (Morgenthaler, 1958), together with recent developments in time-varying metamaterials and dynamic-media platforms (Engheta, 2023), have reinforced the view that time modulation provides an additional design dimension for controlling wave scattering and transport. Broader links to waves in periodic media and the associated mathematical viewpoint are discussed in tutorial form in Gazalet et al. (2013), while a general mathematical framework for dynamic materials is presented in Lurie (2007).

From a modelling perspective, periodic time dependence naturally leads to linear problems with periodic coefficients (in the bulk or at boundaries), for which Floquet theory provides a systematic framework to compute generated harmonics and their scattering amplitudes (Yakubovich and Starzhinskii, 1975). Floquet scattering concepts have been applied in various physical settings, including scattering through time-periodic potentials (Li and Reichl, 1999; Martinez and Reichl, 2001) and scattering-matrix formulations in layered time-driven structures (Mei et al., 2005; Pantazopoulos and Stefanou, 2019). In duct acoustics, these ideas translate directly: a monochromatic incident field interacting with a periodically actuated boundary or insert generates temporal sidebands whose amplitudes are coupled through interface conditions.

A key requirement in duct scattering is a tractable multimodal model that enforces interface and boundary conditions accurately across coupled sub-regions. Mode-matching and Galerkin-type extensions reduce complex configurations to coupled linear systems for modal amplitudes and have been used to quantify how membranes and compliant inserts reshape reflection and transmission in lined or wave-bearing cavities (Afzal and Bilal, 2022; Afzal et al., 2023c; Safdar et al., 2023). These approaches also accommodate porous or complex-media loading effects on silencing performance (Bilal and Afzal, 2024; Bilal et al., 2023). Tailored Galerkin projections further incorporate higher-order interface conditions and geometric variations when classical continuity conditions are insufficient (Afzal et al., 2023a), which is particularly relevant for time-modulated inserts where harmonic coupling occurs through the same interfaces governing the static problem. More broadly, fluid–structure interaction formulations address mixed rigid–flexible duct interfaces (Afzal et al., 2024a), while flexible shells, absorbent linings, layered conduits, and branching (trifurcated) geometries highlight the sensitivity of scattering to impedance contrast and interface modelling (Afzal et al., 2023b, 2024b; Afzal and Nawaz, 2024; Afzal and Safdar, 2023; Alruwaili et al., 2024a, 2024b; Shafique et al., 2024). Related mode-matching formulations also appear in multi-region cylindrical waveguides involving plasma or dielectric media (Rizvi and Afzal, 2024a, 2024b, 2025a, 2025b), underscoring the generality of interface-based modal scattering models.

In this study, we investigate linear acoustic scattering in a rigid duct caused by an infinitesimally thin vertical rigid baffle whose penetration height varies in time.

The motivation is twofold. First, mechanically actuated or active in-duct elements (movable diaphragms, deployable baffles, controllable shutters, and piezoelectrically driven blades) are already used in HVAC silencers, exhaust manifolds, fluidic logic and active noise-control hardware, and they provide a controllable analogue of time-varying material parameters without modifying the bulk medium. Second, recent activity on time-varying media, non-reciprocal devices and low-frequency metamaterials (Cai et al., 2020; Chen et al., 2026; Jamil et al., 2022; Mei et al., 2012) has shown that strong frequency conversion, bandgap engineering and unidirectional transport can be achieved at acoustically interesting (and indeed sub-Hertz) scales when an internal boundary or impedance is modulated in time. Compact, geometrically simple time-modulated elements are therefore attractive both as building blocks for active sound-control devices and as canonical settings in which the underlying physics can be understood quantitatively.

We first analyse the reference configuration of a static baffle and formulate a two-dimensional boundary-value problem that admits a multimodal (mode-matching) description of the reflected and transmitted acoustic fields over the full frequency range. We then introduce a periodic modulation of the baffle penetration. In the low-frequency (long-wavelength) regime, we derive an effective reduced model in which the baffle acts as a compact element represented by jump conditions imposed at its axial location. For periodic oscillations, we develop a Floquet-scattering formulation to compute the reflected and transmitted harmonic components (Koukouraki et al., 2025; Li and Reichl, 1999; Martinez and Reichl, 2001; Pantazopoulos and Stefanou, 2019; Yakubovich and Starzhinskii, 1975). We also present a quasistatic (adiabatic) approximation appropriate when the baffle motion is slow compared with the acoustic period. Throughout, the actuation enters the formulation through three physically distinct control parameters whose roles are made explicit: a mean blockage that sets the time-averaged scattering strength of the baffle, a modulation amplitude that controls the depth of the actuation, and hence the strength of the energy transfer between the carrier wave and its newly generated harmonic sidebands, and a pumping frequency that sets the spectral spacing of those sidebands and the regime (slow, adiabatic vs fast, strongly non-adiabatic) of the response.

In parallel, the water-wave community has extensively studied how spatial variability and moving boundaries influence reflection and transmission, particularly through the role of bathymetry in scattering and long-wave approximations. Conformal-mapping and related transformations have been used to treat variable-depth configurations (Evans and Linton, 1994; Fitz-Gerald, 1976; Hamilton, 1977), while step-approximation approaches were developed to handle more general profiles (Evans and Linton, 1994; Porter and Porter, 2006). Scattering by vertical barriers and submerged plates has also been investigated in canonical configurations using complementary approximation strategies (Evans, 1970; Newman, 1974; Porter and Evans, 1995). More recent works consider structured beds and periodically arranged elements in shallow-water settings (Marangos and Porter, 2021; Maurel et al., 2017, 2019). Although much of this literature addresses static geometries, moving underwater barriers and space–time dependent profiles have also been analysed, including matched-asymptotic approaches for space–time dependent shallow-water problems (Tuck, 1977). These developments motivate analogous questions in duct acoustics, where internal baffles, partitions, and liners play a similar role in shaping scattering, and where time dependence can be introduced through actuation or controlled motion.

Here these issues are addressed in detail within a unified computational framework that combines mode-matching, Floquet scattering, and quasistatic reduction, together with explicit convergence and power-balance diagnostics. Section 2 introduces the governing acoustic model and the duct/baffle geometry, and derives the long-wavelength (plane-wave) jump description used later for time-periodic actuation. Section 3 treats the static-baffle benchmark problem, presenting the mode-matching formulation and quantifying the range of validity of the reduced jump model. Section 4 develops the periodically modulated baffle model, including the truncated Floquet system, harmonic generation results, and the quasistatic approximation, together with comparisons and diagnostics (including power/flux balance). Concluding remarks are given in the final section.

2. Governing equations

We consider a compressible, inviscid fluid in a two-dimensional rigid channel of height h, unbounded in the axial direction x. The Cartesian coordinates are (x, y), where x denotes the axial coordinate and y the transverse coordinate. The upper wall is located at y = 0 and the lower wall at y = −h, so that the fluid occupies $D = {(x, y) : x \in R, - h < y < 0} .$ The geometrical configuration can be seen in Figure 1.

Figure 1.

Physical configuration of the time-modulated diaphragm in a rigid duct.

An infinitesimally thin rigid vertical plate is located at x = 0 and mounted on the lower wall. At time t it penetrates the fluid over a height h_p(t) with 0 < h_p(t) < h. The blocked plate segment is therefore Γ_p(t) = {(0, y): − h < y < − (h − h_p(t))}, while the open region above the plate is Γ_o(t) = {(0, y): − (h − h_p(t)) < y < 0}. Accordingly, the time-dependent acoustic domain is $Ω (t) = D \ Γ_{p} (t) .$ Although h_p(t) varies in time, the plate remains vertical at x = 0, so the impermeability condition on its face involves the normal velocity and remains homogeneous. On Γ_p(t) the outward normal is parallel to the x-axis, and the boundary condition reduces to ∂_xΦ = 0. Within linear acoustics (see, e.g. (Afzal et al., 2024b; Alruwaili et al., 2024a)), the acoustic velocity potential Φ(x, y, t) is defined through u(x, y, t) = ∇Φ(x, y, t), where u is the acoustic particle-velocity vector. The associated acoustic pressure is

\begin{array}{l} p (x, y, t) = - ρ_{0} \frac{\partial Φ}{\partial t}, \end{array}

where ρ₀ is the mean fluid density. The governing boundary-value problem for Φ is

\begin{align} Δ Φ - \frac{1}{c^{2}} \frac{\partial^{2} Φ}{\partial t^{2}} & = 0, & (x, y) \in Ω (t), \end{align}

(1a)

\begin{align} \hat{n} \cdot \nabla Φ & = 0, & on Γ_{w}, \end{align}

(1b)

\begin{align} \hat{n} \cdot \nabla Φ & = 0, & on Γ_{p} (t), \end{align}

(1c)

where c is the constant sound speed and

\hat{n}

denotes the outward unit normal vector. The rigid channel walls are

Γ_{w} = {(x, - h) : x \in R} \cup {(x, 0) : x \in R} .

We decompose the total potential as Φ = Φ^inc + Φ^sc, where Φ^inc represents a prescribed incoming plane wave from x = −∞ and Φ^sc is the scattered field. The scattered field satisfies a duct radiation condition: as x → +∞ it contains only right-going duct modes, while as x → −∞ it contains only left-going modes, with evanescent contributions decaying exponentially with |x|. In the time-periodic setting, this radiation condition is imposed on each Floquet harmonic introduced later.

2.1. Plane-wave acoustic model with jump conditions

In the long-wavelength regime, the acoustic wavelength is large compared with the channel height h, so only the fundamental (plane) duct mode propagates and the field is approximately uniform in y. For a monochromatic incident wave of angular frequency ω, the acoustic wavenumber is k = ω/c, and the plane-wave regime corresponds to kh ≪ π (i.e. below the first transverse cut-off). We therefore introduce the one-dimensional velocity potential ϕ(x, t) representing the dominant mode amplitude.

For concreteness, ‘low frequency’ here refers to this long-wavelength regime kh ≪ π. For airborne sound this corresponds to frequencies up to a few hundred hertz for typical duct heights. The same long-wavelength window is the operating range of recent low-frequency acoustic metamaterials and inerter-based bandgap devices (Cai et al., 2020; Chen et al., 2026; Jamil et al., 2022; Mei et al., 2012).

The thin plate is modelled as a compact element at x = 0 that produces jump conditions for ϕ. Using the plane-wave pressure relation p(x, t) = −ρ₀ ∂_tϕ(x, t), the jump in potential implies the pressure jump

{[p]}_{0^{-}}^{0^{+}} = - ρ_{0} \partial_{t} ({[ϕ]}_{0^{-}}^{0^{+}}),

(2)

where

{[ϕ]}_{0^{-}}^{0^{+}} ≔ ϕ (0^{+}, t) - ϕ (0^{-}, t)

denotes the discontinuity across x = 0. The reduced jump model is

\begin{align} \frac{\partial^{2} ϕ}{\partial x^{2}} - \frac{1}{c^{2}} \frac{\partial^{2} ϕ}{\partial t^{2}} & = 0, & x \neq 0, \end{align}

(3a)

\begin{align} {[ϕ]}_{0^{-}}^{0^{+}} = 2 B_{μ} (t) h ϕ_{x} |_{0}, {[ϕ_{x}]}_{0^{-}}^{0^{+}} & = 0, \end{align}

(3b)

where ϕ_x|₀ denotes the common value ϕ_x(0^±, t) implied by the continuity condition

{[ϕ_{x}]}_{0^{-}}^{0^{+}} = 0

. Here, B_μ(t) is an effective blockage coefficient associated with the instantaneous opening. Introducing the blockage ratio

μ (t) = \frac{h_{p} (t)}{h} \in (0,1),

(4)

we write

B_{μ} (t) = B (μ (t)),

where

B

is a closure relating the opening to the effective compact-element strength. For an infinitesimally thin plate, the standard thin-obstruction model is

B (μ) = - \frac{2}{π} \ln [\sin (\frac{π}{2} (1 - μ))] .

(5)

In the periodically driven setting considered later, the opening (and hence B_μ(t)) is taken to be T_p-periodic, where the pumping period is T_p = 2π/ω_p and ω_p is the pumping angular frequency. Rather than prescribing a specific waveform, we represent the periodic blockage coefficient in Fourier form

B_{μ} (t) = \sum_{q \in Z} {\hat{B}}_{q} e^{- i q ω_{p} t}, {\hat{B}}_{q} = \frac{1}{T_{p}} \int_{0}^{T_{p}} B_{μ} (t) e^{i q ω_{p} t} d t,

(6)

with

{\hat{B}}_{- q} = \bar{{\hat{B}}_{q}}

when B_μ(t) is real-valued. If the plate motion is prescribed through μ(t), then B_μ(t) follows from (5), and the coefficients

{\hat{B}}_{q}

are obtained from (6). Alternatively, one may prescribe a periodic B_μ(t) directly (typically with B_μ(t) ≥ 0 so that inversion of (5) remains well defined) and interpret (5) as the corresponding instantaneous opening relation. Further discussion of thin-obstruction models and the blockage coefficient can be found (Koukouraki et al., 2025; Mei et al., 2005).

3. Static diaphragm

As a preliminary validation step, and to assess the range of applicability of the plane-wave jump description from Section 2.1, we first consider scattering by a stationary infinitesimally thin rigid diaphragm of constant penetration height h_p, mounted at x = 0. In this static configuration, the opening is fixed, so the blockage ratio μ = h_p/h is constant and the effective blockage coefficient $B_{μ} = B (μ)$ in (3b) is also constant. In the Fourier representation introduced later, this corresponds to the special case B_μ(t) ≡ B_μ, with only the zero Fourier coefficient nonzero, namely, ${\hat{B}}_{0} = B_{μ}$ and ${\hat{B}}_{q} = 0$ for q ≠ 0.

3.1. Mode-matching formulation for arbitrary frequency regime

We assume time-harmonic dependence at angular frequency ω > 0 with the convention e^−iωt. Accordingly, $Φ (x, y, t) = R {φ (x, y) e^{- i ω t}},$ where φ(x, y) is the complex acoustic potential amplitude and k = ω/c is the acoustic wavenumber. For a fixed diaphragm penetration height h_p ∈ (0, h), the acoustic domain is time-independent, $Ω = D \ Γ_{p}$ , and the field satisfies the Helmholtz problem

\begin{align} Δ φ + k^{2} φ & = 0, & (x, y) \in Ω, \end{align}

(7a)

\begin{align} \partial_{y} φ & = 0, & y = - h, 0, x \in R, \end{align}

(7b)

where ∂_y denotes differentiation with respect to y. On the diaphragm segment, that is, the blocked part of the cross-section, the no-penetration condition gives

\partial_{x} φ (0, y) = 0, y \in (- h, - (h - h_{p})),

(8)

where ∂_x denotes differentiation with respect to x. On the complementary open part of the interface,

S_{o} = {(0, y) : y \in (- (h - h_{p}), 0)},

(9)

both acoustic pressure and axial particle velocity are continuous across x = 0, so that

φ (0^{+}, y) = φ (0^{-}, y), \partial_{x} φ (0^{+}, y) = \partial_{x} φ (0^{-}, y), y \in (- (h - h_{p}), 0) .

(10)

To impose the radiation condition and the interface constraints, we expand the solution in the rigid-wall transverse eigenfunctions ${ψ_{m} (y)}_{m \geq 0}$ on y ∈ [−h, 0], defined by the Neumann Sturm–Liouville problem

ψ_{m}^{″} (y) + α_{m}^{2} ψ_{m} (y) = 0, ψ_{m}^{'} (- h) = ψ_{m}^{'} (0) = 0, α_{m} = \frac{m π}{h},

where primes denote differentiation with respect to y. This yields the orthonormal basis

ψ_{m} (y) = \{\begin{cases} \frac{1}{\sqrt{h}}, & m = 0, \\ \sqrt{\frac{2}{h}} \cos (\frac{m π (y + h)}{h}), & m \geq 1, \end{cases} β_{m} = \sqrt{k^{2} - α_{m}^{2}} = \sqrt{k^{2} - {(\frac{m π}{h})}^{2}} .

(11)

The branch is chosen such that $I (β_{m}) \geq 0$ , ensuring exponential decay of evanescent contributions as |x| → ∞. Imposing modal radiation then gives the standard upstream and downstream modal expansions

\begin{align} φ (x < 0, y) & = (e^{i β_{0} x} + R e^{- i β_{0} x}) ψ_{0} (y) + \sum_{m = 1}^{\infty} A_{m} e^{- i β_{m} x} ψ_{m} (y), \end{align}

(12a)

\begin{align} φ (x > 0, y) & = T e^{i β_{0} x} ψ_{0} (y) + \sum_{m = 1}^{\infty} B_{m} e^{i β_{m} x} ψ_{m} (y), \end{align}

(12b)

where R and T are the reflection and transmission coefficients of the incident plane mode, and ${A_{m}}_{m \geq 1}$ and ${B_{m}}_{m \geq 1}$ are higher-mode amplitudes.

Because the upstream and downstream ducts are identical, it is convenient to decompose the solution into symmetric and antisymmetric parts with respect to x = 0,

φ = φ_{s} + φ_{a}, φ_{s} (- x, y) = φ_{s} (x, y), φ_{a} (- x, y) = - φ_{a} (x, y) .

(13)

This decomposition converts the original two-sided scattering problem into two one-sided problems. On the opening S_o, symmetry implies

\partial_{x} φ_{s} (0, y) = 0, φ_{a} (0, y) = 0, y \in (- (h - h_{p}), 0),

(14)

while on the rigid diaphragm segment the no-penetration condition applies to both parts,

\partial_{x} φ_{s} (0, y) = 0, \partial_{x} φ_{a} (0, y) = 0, y \in (- h, - (h - h_{p})) .

(15)

Let R_s and R_a denote the plane-mode reflection factors associated with the symmetric and antisymmetric subproblems, respectively. Then, the physical reflection and transmission coefficients are recovered from

R = \frac{1}{2} (R_{s} + R_{a}), T = \frac{1}{2} (R_{s} - R_{a}) .

(16)

In the present lossless setting, |R_s| = |R_a| = 1. Moreover, (14) and (15) imply ∂_xφ_s(0, y) = 0 for all y ∈ (−h, 0), so the symmetric subproblem corresponds to reflection from a rigid termination and hence R_s = 1. Therefore, all non-trivial frequency dependence is contained in the antisymmetric factor R_a.

For the antisymmetric subproblem in the upstream half-duct x ≤ 0, we expand

φ_{a} (x \leq 0, y) = \sum_{m = 0}^{\infty} a_{m} (x) ψ_{m} (y),

(17)

where a_m(x) is the axial modal amplitude of the mth rigid-wall transverse mode. We decompose each modal amplitude into the incident contribution and a reflected/radiating part,

a_{m} (x) = a_{m, 0} (x) + a_{m, r} (x), a_{m, 0} (x) = δ_{m 0} e^{i β_{0} x}, a_{m, r} (x) = C_{m} e^{- i β_{m} x},

(18)

where δ_m0 is the Kronecker delta and C_m is the complex reflection amplitude of mode m. The antisymmetric plane-mode reflection factor is R_a = C₀.

The opening condition φ_a(0, y) = 0 on S_o is imposed weakly by projection onto an orthonormal basis ${{\hat{ψ}}_{p} (y)}_{p \geq 0}$ defined on the open interval y ∈ [−(h − h_p), 0]:

{\hat{ψ}}_{p} (y) = \{\begin{cases} \frac{1}{\sqrt{h - h_{p}}}, & p = 0, \\ \sqrt{\frac{2}{h - h_{p}}} \cos (\frac{p π (y + (h - h_{p}))}{h - h_{p}}), & p \geq 1, \end{cases} y \in [- (h - h_{p}), 0] .

(19)

Projecting φ_a(0, y) = 0 onto

{{\hat{ψ}}_{p}}

yields

F^{†} a (0) = 0,

(20)

where

a (0) = {(a_{0} (0), a_{1} (0), \dots)}^{⊤}

is the vector of modal amplitudes at x = 0, ^† denotes the conjugate transpose, and the overlap matrix F has entries

{(F)}_{m p} = \int_{- (h - h_{p})}^{0} ψ_{m} (y) {\hat{ψ}}_{p} (y) d y .

(21)

Next, we represent the axial velocity trace ∂_xφ_a(0, y) in a form that automatically accounts for the rigid/open partition of the interface:

{\partial_{x} φ_{a}|}_{x = 0} = \{\begin{cases} 0, & y \in (- h, - (h - h_{p})), \\ \sum_{p = 0}^{\infty} c_{p} {\hat{ψ}}_{p} (y), & y \in (- (h - h_{p}), 0), \end{cases}

(22)

where c_p are unknown coefficients and

c = {(c_{0}, c_{1}, \dots)}^{⊤}

is their coefficient vector. Differentiating (17) with respect to x and projecting onto {ψ_m} gives

a^{'} (0) = F c,

(23)

where

a^{'} (0) = {(a_{0}^{'} (0), a_{1}^{'} (0), \dots)}^{⊤}

and primes on a_m denote differentiation with respect to x. On the other hand, differentiating (18) and evaluating at x = 0 yields a diagonal Dirichlet-to-Neumann relation in modal space,

a^{'} (0) = Y (a_{0} (0) - a_{r} (0)),

(24)

where a₀(0) = (1,0,0,… )^⊤ is the incident modal vector at x = 0,

a_{r} (0) = {(C_{0}, C_{1}, \dots)}^{⊤}

is the reflected modal vector at x = 0, and the diagonal matrix Y has entries

{(Y)}_{m m^{'}} = i β_{m} δ_{m m^{'}} .

Since a(0) = a₀(0) + a_r(0), the constraint (20) implies

F^{†} a_{r} (0) = - F^{†} a_{0} (0) .

(25)

Combining (23)–(25) gives an explicit expression for the reflected modal vector at the interface,

a_{r} (0) = a_{0} (0) - Y^{- 1} F c,

(26)

where the opening-velocity coefficients are determined by

c = 2 {(F^{†} Y^{- 1} F)}^{- 1} F^{†} a_{0} (0) .

(27)

After truncating the modal system to a finite size, a_r(0) can be computed numerically. The antisymmetric reflection factor is then R_a = C₀, and the physical coefficients R and T follow from (16) with R_s = 1.

3.2. Low-frequency approximation: The plane-wave regime

For sufficiently low frequencies, that is, below the first transverse cut-off, the acoustic field is nearly uniform across y ∈ [−h, 0] and can be represented accurately by the one-dimensional jump model (3a) and (3b). We now derive the plane-wave reflection and transmission factors for the static jump model and relate them to the general periodic representation adopted later.

Assuming time-harmonic dependence at angular frequency ω > 0 with the convention e^−iωt, we write

ϕ (x, t) = R {f (x) e^{- i ω t}}, k = \frac{ω}{c} .

In the static case, the blockage coefficient is constant, B_μ(t) ≡ B_μ. In the Fourier representation (6), this corresponds to

{\hat{B}}_{0} = B_{μ}

and

{\hat{B}}_{q} = 0

for all q ≠ 0, so the sideband coupling reduces to a single uncoupled harmonic.

We seek the plane-wave solution on the two half-lines in the form

f^{-} (x) = e^{ikx} + R_{PWA} e^{- i k x}, x < 0, f^{+} (x) = T_{PWA} e^{ikx}, x > 0,

where R_PWA and T_PWA denote the plane-wave approximation (PWA) reflection and transmission coefficients. From continuity of axial volume velocity, expressed as the derivative jump condition

{[f^{'}]}_{0^{-}}^{0^{+}} = 0

, we obtain

i k (1 - R_{PWA}) = i k T_{PWA} \Rightarrow T_{PWA} = 1 - R_{PWA} .

The potential jump condition in (3b) gives

T_{PWA} - (1 + R_{PWA}) = 2 B_{μ} h i k (1 - R_{PWA}),

where h is the channel height and B_μ is the constant blockage coefficient associated with the fixed blockage ratio μ = h_p/h. Substituting T_PWA = 1 − R_PWA yields

R_{PWA} = - \frac{i k h B_{μ}}{1 - i k h B_{μ}}, T_{PWA} = \frac{1}{1 - i k h B_{μ}} .

(28)

In the lossless setting, the plane-wave power balance is

| R_{PWA} |^{2} + | T_{PWA} |^{2} = 1 .

Figure 2 illustrates the reconstructed two-dimensional acoustic field for the representative blockage ratio μ = 0.5 at two nondimensional frequencies Ω = ωh/c = kh. At Ω = 1, the scattered field exhibits noticeable transverse structure near the diaphragm, indicating the role of higher-order, primarily evanescent, duct modes in satisfying the mixed conditions at x = 0. By contrast, at Ω = 0.2 the solution is nearly uniform across the channel height, with transverse variations confined to a small neighbourhood of the diaphragm. This is consistent with the plane-wave regime, where the fundamental mode dominates and the diaphragm is well represented by the compact jump relation.

Figure 2.

Two-dimensional profiles of the time-harmonic acoustic velocity potential reconstructed by the mode-matching solution for a static diaphragm of blockage ratio μ = h_p/h: (a) μ = 0.5 with nondimensional frequency Ω = ωh/c = 1 and (b) μ = 0.5 with nondimensional frequency Ω = ωh/c = 0.2.

Figure 3 shows the magnitudes of the plane-mode reflection and transmission coefficients, |R| and |T|, as functions of the nondimensional frequency Ω = ωh/c for μ = 0.75 and μ = 0.85. As Ω → 0, the diaphragm behaves as a compact obstruction, so that |T| → 1 and |R| → 0. In this long-wavelength limit, the PWA agrees closely with the multimodal results, confirming the validity of the effective jump description when transverse variations are weak.

Figure 3.

Magnitudes of the plane-mode reflection and transmission coefficients (vertical axis, ‘magnitudes |R|, |T|’), computed from the mode-matching formulation for blockage ratios μ = 0.75 and μ = 0.85 as functions of the nondimensional frequency Ω = ωh/c (horizontal axis). The plane-wave approximation (PWA) is given by equation (28) and is plotted only over its formal validity range Ω ≤ 1.

As Ω increases, the multimodal curves depart from the PWA because higher-order duct modes become increasingly important in satisfying the mixed boundary and interface conditions. The blockage ratio controls the strength of the effective discontinuity: larger μ produces stronger reflection and lower transmission over a wider frequency range. A marked change occurs as Ω approaches the first transverse cut-off, Ω ≃ π, where the first higher duct mode changes from evanescent to propagating. Near this threshold, the scattered field becomes highly sensitive to the diaphragm geometry, and the PWA is no longer sufficient; the full multimodal formulation is then required.

4. Periodically modulated diaphragm

We now consider scattering of plane acoustic waves by an infinitesimally thin diaphragm whose effective blockage varies periodically in time. In the long-wavelength regime discussed in Section 2.1, the diaphragm is represented by the reduced plane-wave jump model (3a) and (3b). The actuation enters this model through the time-periodic blockage coefficient B_μ(t), which is itself determined by the prescribed motion of the penetration height h_p(t) via the thin-obstruction closure (5). Throughout this work we adopt the single-harmonic prescription

B_{μ} (t) = B_{μ, 0} + B_{μ, 1} \cos (ω_{p} t), {\hat{B}}_{0} = B_{μ, 0}, {\hat{B}}_{\pm 1} = B_{μ, 1} / 2,

so that the actuation is fully characterised by three independent physical parameters with clear physical roles. The mean blockage B_μ,0 fixes the equilibrium opening

\bar{μ} = ⟨ μ (t) ⟩

about which the baffle oscillates and controls the static component of the scattering response; in the limit B_μ,1 → 0 the diaphragm reduces to the static configuration of Section 3 with |R|, |T| given by (28). The modulation amplitude B_μ,1 (equivalently, the cycle-averaged opening excursion Δμ≔μ_max − μ_min) sets the depth of the actuation and, through the off-diagonal entries

{\hat{B}}_{\pm 1}

of the Toeplitz coupling matrix (39), directly governs the energy transfer from the carrier (n = 0) into the first sidebands r_±1, t_±1; all sidebands r_n, t_n for n ≠ 0 vanish as B_μ,1 → 0. The pumping (modulation) angular frequency ω_p, together with the incident frequency ω, fixes the spectral spacing ω_n − ω_n−1 = ω_p of the generated Floquet harmonics in (32) and hence the location of the sidebands relative to the carrier; the dimensionless ratio ω_p/ω also governs the transition between the slow, quasistatic regime (ω_p ≪ ω) of Section 4.2 and the strongly non-adiabatic regime in which the sideband amplitudes depend explicitly on ω_p. The interplay of these three parameters is illustrated quantitatively in Figures 4 –10.

Figure 4.

Effect of modulation depth and pumping ratio on the time-domain blockage and the reflected Floquet spectrum. Panels (a), (c), (e) show the blockage ratio μ(t) = h_p(t)/h over one, two, and three pumping periods, respectively, for three modulation depths B_μ,1/B_max ∈ {0.10, 0.25, 0.50} (legend in panel (e)). Panels (b), (d), (f) show the corresponding magnitudes of the reflected Floquet coefficients |r_n| versus the nondimensional sideband frequencies Ω_n = Ω + nΩ_p for the pumping ratios Ω_p/Ω ∈ {1/8, 1/4, 1/2} (legend in panel (f)); the vertical dotted line marks the carrier frequency Ω. The frequency window is widened from (b) to (d) to (f) to display additional positive sidebands.

Figure 5.

Reflected Floquet amplitudes |r_n| versus the pumping-to-incident frequency ratio ω_p/ω for the plane-wave acoustic jump model. The incident frequency is fixed at Ω = ωh/c = 0.1, and the modulation parameters are chosen so that the inferred blockage ratio spans μ(t) ∈ [0, μ_max] with μ_max = 0.95. Panel (a) shows −1 ≤ n ≤ 3, while panel (b) isolates the negative indices −5 ≤ n ≤ −2. In both panels, the vertical axis (“reflection amplitude |r_n| (log scale)”) uses a logarithmic scale. Vertical dotted lines mark the commensurate pumping ratios ω_p/ω = 1/|n| at which ω_n = 0 for the negative indices.

Figure 6.

Classification of Floquet sidebands for the modulated diaphragm. For each index n, the frequency ω_n = ω + nω_p determines whether the harmonic is propagating (ω_n > 0, retained), static (ω_n = 0, excluded from power sums), or omitted (ω_n < 0). Dashed lines indicate the thresholds ω_n = 0 at ω_p/ω = 1/|n| for n < 0.

Figure 7.

Spatio-temporal magnitude of the acoustic pressure field |p(x, t)| reconstructed from the truncated Floquet solution in the plane-wave regime. Panels correspond to different pumping ratios: (a) Ω_p/Ω = 1/8 (slow modulation), (b) Ω_p/Ω = 1/4 (baseline), and (c) Ω_p/Ω = 1/2 (faster modulation). The dashed vertical line indicates the diaphragm location x = 0. Increasing Ω_p/Ω enhances temporal mixing and produces stronger modulation of the transmitted wave field downstream, consistent with redistribution of energy among Floquet sidebands.

Figure 8.

Spatio-temporal magnitude of the acoustic pressure field |p(x, t)| reconstructed from the Floquet solution for different mean blockage levels. The modulation depth is fixed at B₁/B_max = 0.15, while the mean component is varied: (a) B₀/B_max = 0.15 (deep modulation about a low mean), (b) B₀/B_max = 0.35, and (c) B₀/B_max = 0.55. The dashed vertical line indicates the diaphragm location x = 0. Increasing the mean blockage enhances the effective compact strength of the diaphragm, leading to stronger reflection and a progressively more uniform transmitted field downstream.

Figure 9.

Comparison between the full Floquet solution and the quasistatic (QS) approximation for the time-modulated acoustic diaphragm in the plane-wave regime, with fixed incident nondimensional frequency Ω = ωh/c = 0.1. Panels (a)–(c) show the reflection amplitudes |r_n| (grey bars) obtained from the truncated Floquet system and the QS predictions $| {\tilde{r}}_{n} |$ (blue circles) for three pumping ratios: (a) ω_p/ω = 10⁻³, (b) ω_p/ω = 1/6, and (c) ω_p/ω = 1/4. Panel (d) reports the relative discrepancy δ|r_n| as a function of ω_p/ω for the harmonics n = 0, ±1, ±2.

Figure 10.

(a) Magnitude of the fundamental reflected Floquet coefficient |r₀| versus the nondimensional incident frequency Ω = ωh/c for a static diaphragm at μ_max = 0.95 and at the mean blockage μ_mean ≈ 0.698, together with the periodically modulated diaphragm for two pumping frequencies Ω_p = ω_ph/c. The QS prediction for the fundamental is indicated by open circles. (b) Dependence of |r₀| on the normalised pumping rate Ω_p/Ω = ω_p/ω for three fixed incident frequencies Ω.

4.1. Floquet formulation

We assume that the blockage coefficient B_μ(t) is T_p-periodic, where T_p = 2π/ω_p and ω_p > 0 is the pumping angular frequency, and adopt the Fourier representation (6). For the commonly used single-harmonic choice

B_{μ} (t) = B_{μ, 0} + B_{μ, 1} \cos (ω_{p} t),

the Fourier coefficients are

{\hat{B}}_{0} = B_{μ, 0}

{\hat{B}}_{\pm 1} = B_{μ, 1} / 2

, and

{\hat{B}}_{q} = 0

for |q|≥ 2. The corresponding time-dependent blockage ratio μ(t) = h_p(t)/h ∈ (0, 1) can be written as

μ (t) = 1 - \frac{2}{π} \arcsin [\exp (- \frac{π}{2} B_{μ} (t))],

(29)

which follows directly from the thin-obstruction closure (5).

We consider a right-going incident plane wave of angular frequency ω > 0 impinging on the modulated diaphragm from x = −∞. Guided by Floquet theory, we seek the plane-wave potential in the form

ϕ (x, t) = R \{e^{- i ω t} f (x, t)\}, f (x, t) = f (x, t + T_{p}),

(30)

where f(x, t) is T_p-periodic in time. We expand f as a Fourier series,

f (x, t) = \sum_{n \in Z} f_{n} (x) e^{- i n ω_{p} t},

(31)

where f_n(x) are the Floquet sideband amplitudes. The associated sideband angular frequencies are

ω_{n} = ω + n ω_{p},

(32)

with corresponding wavenumbers

k_{n} = \frac{ω_{n}}{c} .

(33)

Substituting (30) and (31) into (3a) for x ≠ 0 and projecting onto $e^{- i n ω_{p} t}$ yields, for each retained index n,

\frac{d^{2} f_{n}}{d x^{2}} + k_{n}^{2} f_{n} = 0, x \neq 0 .

(34)

The jump model (3b) implies continuity of flux,

{[f_{n}^{'}]}_{0^{-}}^{0^{+}} = 0,

(35)

and a jump in the potential. Using (6) together with orthogonality of the time harmonics, the projected jump condition becomes

{[f_{n}]}_{0^{-}}^{0^{+}} = 2 h \sum_{q \in Z} {\hat{B}}_{q} f_{n - q}^{'} (0) .

(36)

Here

{[\cdot]}_{0^{-}}^{0^{+}}

denotes the jump across x = 0, and

f_{m}^{'} (0)

denotes the common value of

f_{m}^{'} (0^{\pm})

implied by (35). In computations, the sum in (36) is restricted to the retained index set, with contributions from discarded harmonics omitted. For single-harmonic modulation, (36) reduces to the nearest-neighbour coupling

{[f_{n}]}_{0^{-}}^{0^{+}} = 2 B_{μ, 0} h f_{n}^{'} (0) + B_{μ, 1} h (f_{n - 1}^{'} (0) + f_{n + 1}^{'} (0)) .

For each retained sideband we impose outgoing behaviour as x → ±∞ and write

\begin{align} f_{n}^{-} (x) & = δ_{n 0} e^{i k_{n} x} + r_{n} e^{- i k_{n} x}, x < 0, \end{align}

(37a)

\begin{align} f_{n}^{+} (x) & = t_{n} e^{i k_{n} x}, x > 0, \end{align}

(37b)

where δ_n0 is the Kronecker delta, and r_n and t_n are the reflection and transmission coefficients of the nth Floquet sideband. Enforcing (35) gives

t_{n} = δ_{n 0} - r_{n} .

(38)

Substituting (37a), (37b) and (38) into (36) yields a linear coupled system for the reflection coefficients {r_n}, with the coupling range determined by the harmonic content of B_μ(t). We truncate to the finite retained index set

I_{N} = {n \in Z : | n | \leq N, ω_{n} > 0},

and introduce the vectors

r = {(r_{n})}_{n \in I_{N}},

t = {(t_{n})}_{n \in I_{N}},

and

b = {(δ_{n 0})}_{n \in I_{N}} .

Define the Toeplitz-type matrix

T_{\hat{B}} \in C^{| I_{N} | \times | I_{N} |}

{(T_{\hat{B}})}_{n, n^{'}} = {\hat{B}}_{n - n^{'}}, n, n^{'} \in I_{N},

(39)

where

{\hat{B}}_{q}

is taken as zero whenever q lies outside the retained modulation range, and truncation is enforced by dropping terms associated with discarded indices. Let

K = d i a g {(k_{n})}_{n \in I_{N}}

denote the diagonal wavenumber matrix. Then, the truncated system can be written compactly as

(2 I - 2 i h T_{\hat{B}} K) r = - 2 i h k_{0} \hat{B}, \hat{B} = {({\hat{B}}_{n})}_{n \in I_{N}},

(40)

where I is the identity matrix and k₀ = ω/c is the incident wavenumber. Hence,

r = - {(2 I - 2 i h T_{\hat{B}} K)}^{- 1} 2 i h k_{0} \hat{B}, t = b - r .

(41)

For single-harmonic modulation, $T_{\hat{B}}$ has only three nonzero diagonals, and (40) reduces to a tridiagonal system.

For each sideband, (34) admits the conserved flux $I {\bar{f_{n}} f_{n}^{'}}$ . In the lossless setting, the sum of the fluxes over propagating retained harmonics is conserved across the diaphragm. For the retained propagating set $I_{N}$ , we define the transmitted and reflected power coefficients by

T + R = 1, T = \sum_{n \in I_{N}} | t_{n} |^{2} \frac{k_{n}}{k_{0}}, R = \sum_{n \in I_{N}} | r_{n} |^{2} \frac{k_{n}}{k_{0}},

(42)

where k₀ = ω/c corresponds to the incident wave. This identity provides an a posteriori check on truncation accuracy.

The retained-harmonic set $I_{N} = {n \in Z : | n | \leq N, ω_{n} > 0}$ is controlled by a single integer truncation index N, which we use as a numerical convergence parameter. The truncation error of any retained sideband is monitored through

e_{n} (N) ≔ | | r_{n}^{(N)} | - | r_{n}^{(N - 1)} | |, e (N) ≔ \max_{n \in I_{N - 1}} e_{n} (N),

together with the power-balance residual

Δ_{PB} (N) ≔ | 1 - (T + R) |

from (42). For the single-harmonic actuation laws considered in this work, both diagnostics decay rapidly in N in the plane-wave regime. At the representative parameters Ω = 0.1 and μ_max = 0.95 used in Sections 4.1.1–4.2, e(N) falls below 10⁻⁶ already at N = 8 and below 10⁻¹⁰ at N = 12, while Δ_PB(N) remains at machine precision

\sim 1 0^{- 14}

for every truncation level since (40) and (41) enforce flux balance exactly. All Floquet computations reported below therefore use N = 12 as the default; doubling N to 24 leaves the displayed amplitudes unchanged to graphical accuracy. This fast decay reflects the nearest-neighbour coupling structure of

T_{\hat{B}}

for single-harmonic modulation, which makes the off-diagonal influence on r_n decay geometrically with |n|.

The Floquet system (40) and (41) is built on the reduced plane-wave jump model (3a) and (3b), and therefore inherits the long-wavelength restriction Ω < π identified in Section 2.1. A natural extension is to combine the multimodal mode-matching analysis of Section 3.1 with the time-periodic interface conditions, so that each Floquet sideband carries its own multimodal trace at the diaphragm. This couples the two infinite indices, the modal index m and the Floquet index n, through a doubly indexed overlap matrix and removes the cut-off restriction, albeit at substantially higher algebraic cost. The diagnostics already indicated that the plane-wave reduction is accurate within the parameter window of interest, and the structure of the corresponding fully multimodal Floquet system follows transparently from the present formulation; a detailed treatment is therefore deferred to future work.

4.1.1. Generation of harmonics

Using the Floquet solution (41) together with the power-balance diagnostic (42), we examine how periodic actuation redistributes reflected energy among Floquet sidebands. To obtain a pronounced sideband response in the long-wavelength regime, we consider a strong modulation of the blockage ratio μ(t) = h_p(t)/h such that μ(t) ∈ [0, μ_max] with μ_max = 0.95. We adopt a single-harmonic waveform

B_{μ} (t) = B_{μ, 0} + B_{μ, 1} \cos (ω_{p} t),

for which the only nonzero Fourier coefficients are

{\hat{B}}_{0} = B_{μ, 0}, {\hat{B}}_{\pm 1} = \frac{B_{μ, 1}}{2}, {\hat{B}}_{q} = 0 (| q | \geq 2) .

For later reference, we denote by $B_{\max} ≔ \max_{t \in [0, T_{p}]} B_{μ} (t)$ the maximum actuation level; for the single-harmonic choice above, B_max = B_μ,0 + B_μ,1. The corresponding blockage ratio μ(t) varies periodically within each pumping cycle, so that the opening fraction 1 − μ(t) oscillates between nearly open and near-closure. We introduce the nondimensional incident and pumping frequencies Ω≔ωh/c, and Ω_p≔ω_ph/c, so that Ω_p/Ω = ω_p/ω. The periodic modulation transfers energy from the carrier component (n = 0) into neighbouring Floquet sidebands located at Ω_n = Ω + nΩ_p. Typically, the first sidebands n = ±1 are dominant, while higher-order contributions decay rapidly with |n|, supporting the use of modest Floquet truncation in the long-wavelength regime.

Figure 4 summarises the influence of the actuation parameters on the instantaneous blockage and, through Floquet coupling, on the reflected sideband spectrum. The left column shows μ(t) over one, two, and three pumping periods. Increasing the modulation depth B_μ,1/B_max increases the excursion of μ(t) within each cycle; for the single-harmonic waveform this corresponds to increasing $| {\hat{B}}_{\pm 1} |$ while keeping the mean coefficient ${\hat{B}}_{0}$ fixed. The repetition over multiple periods confirms the strict periodicity assumed in (31).

The right column reports the reflected sideband magnitudes |r_n| at the discrete frequencies Ω_n = Ω + nΩ_p. For all pumping ratios shown, the reflected content remains concentrated near the carrier, and the amplitudes decay as |n| increases. Varying Ω_p/Ω primarily changes the spacing of the sidebands on the frequency axis: smaller Ω_p/Ω gives more closely spaced sidebands, whereas larger Ω_p/Ω produces fewer, more widely separated ones. Even in the enlarged frequency windows of panels (b), (d), and (f), the higher positive sidebands remain weak, again supporting harmonic truncation in the long-wavelength regime.

Figure 5 fixes the incident frequency at Ω = ωh/c = 0.1 and varies the pumping-to-incident frequency ratio ω_p/ω ∈ [0, 2] (equivalently, Ω_p/Ω ∈ [0, 2]). Panel (a) shows the components −1 ≤ n ≤ 3, while panel (b) isolates the higher negative indices −5 ≤ n ≤ −2 on a logarithmic scale. Over the full range, the reflected spectrum is dominated by the carrier and its nearest neighbours (n = 0, ±1), whereas higher positive orders remain weaker. For n ≥−1, the dependence on ω_p/ω is smooth; in the adiabatic limit ω_p/ω → 0, the first sidebands become nearly symmetric, |r₋₁|≃|r₁|, consistent with the quasistatic approximation in Section 4.2.

For negative indices, the sideband frequency ω_n = ω + nω_p decreases as ω_p increases, so panel (b) exhibits sharp variations near the thresholds ω_n ≈ 0, namely, ω_p/ω ≈ 1/|n|. This behaviour reflects the retained-harmonic convention: indices with ω_n < 0 are omitted, whereas the borderline case ω_n = 0 corresponds to a static potential component, which carries no time-harmonic pressure and no power flux and is therefore excluded from power sums. Figure 6 summarises this classification in the (n, ω_p/ω) plane and clarifies why k_n = ω_n/c → 0 as ω_p/ω → 1/|n| for n < 0.

In particular, if the incident and pumping frequencies are commensurate, ω = mω_p with $m \in N^{*}$ , then ω_n = (m + n)ω_p and

r_{n} = 0 for n < - m, when ω = m ω_{p},

(43)

since ω_n < 0 for n < − m. At such commensurate ratios the truncated linear system (40) simplifies because the coefficients associated with n < − m are removed by (43).

For the single-harmonic pumping law, that is, when only ${\hat{B}}_{0}$ and ${\hat{B}}_{\pm 1}$ are nonzero, the projected jump condition (36) reduces to nearest-neighbour coupling and explicit relations among the lowest-order sidebands can be derived. For example, when m = 1 (i.e. ω = ω_p), one has ω₋₁ = 0 and hence r₋₂ = r₋₃ = ⋯ = 0. Using (36)–(38), this gives

r_{- 1} = i h k_{0} {\hat{B}}_{- 1} (r_{0} - 1), k_{0} = \frac{ω}{c},

(44)

and the n = 0 equation decouples from r₋₁ because k₋₁ = 0 at ω = ω_p, yielding

r_{1} = \frac{i h k_{0} {\hat{B}}_{0} + r_{0} (1 - i h k_{0} {\hat{B}}_{0})}{2 i h k_{0} {\hat{B}}_{1}} .

(45)

More generally, after imposing (43), the remaining coefficients r_n (for n ≥−m) are coupled through (40). As an illustration, solving a low-order truncated commensurate system retaining harmonics up to n = 2 gives

r_{2} = - \frac{\{2 i h k_{0} {\hat{B}}_{0} + (4 {\hat{B}}_{0}^{2} - 4 {\hat{B}}_{1} {\hat{B}}_{- 1}) h^{2} k_{0}^{2} + [2 - 6 i h k_{0} {\hat{B}}_{0} - (4 {\hat{B}}_{0}^{2} - 4 {\hat{B}}_{1} {\hat{B}}_{- 1}) h^{2} k_{0}^{2}] r_{0}\}}{12 {\hat{B}}_{1} {\hat{B}}_{- 1} h^{2} k_{0}^{2}} .

(46)

Equation (46) depends on the chosen truncation; the full Floquet solution is still given by (41).

Having established the sideband structure and the retained/omitted classification, we now reconstruct the spatio–temporal acoustic field to provide a direct time-domain interpretation of the Floquet spectra. Using the truncated solution (41), we evaluate the pressure magnitude |p(x, t)| over one pumping period, with the diaphragm location marked by the dashed line at x = 0.

Figure 7 shows |p(x, t)| for three pumping ratios Ω_p/Ω ∈ {1/8, 1/4, 1/2}. For slow modulation, the downstream field (x > 0) remains close to a monochromatic travelling wave in magnitude, while weak temporal beating is visible upstream because of the small reflected sidebands. At the baseline ratio, the modulation generates a clear space–time beating pattern on both sides of the interface, indicating stronger transfer of energy from the carrier (n = 0) into the nearest harmonics (n = ±1). For faster modulation, the pattern exhibits more rapid temporal variation and stronger interference, consistent with enhanced harmonic coupling and broader redistribution of reflected and transmitted energy among the retained Floquet components.

These reconstructions complement the spectral results: increasing the pumping ratio mainly changes the temporal beating rate through the sideband spacing Ω_n = Ω + nΩ_p and amplifies the observable modulation of the transmitted field, while the diaphragm at x = 0 remains the sole source of harmonic generation in the reduced jump description (3a) and (3b).

We next isolate the influence of the mean blockage at fixed modulation depth in order to separate the baseline scattering strength from the time-dependent mixing.

Figure 8 presents |p(x, t)| for fixed B₁/B_max = 0.15 while varying the mean component B₀/B_max ∈ {0.15, 0.35, 0.55}. For low mean blockage, the diaphragm oscillates about a relatively weak obstruction and the downstream field exhibits pronounced temporal modulation and interference. As the mean blockage increases, the transmitted field becomes more coherent in space–time magnitude and its temporal variability decreases, indicating increasing dominance of the carrier component. For sufficiently high mean blockage, the diaphragm behaves as a strong compact scatterer throughout the cycle: the upstream field shows pronounced standing-wave structure associated with enhanced reflection, while the downstream magnitude becomes comparatively uniform in time. This behaviour is consistent with the plane-wave jump model, in which the mean value of the compact parameter primarily controls the overall scattering strength, whereas the oscillatory component drives redistribution among Floquet harmonics.

The space–time diagnostics in Figures 7 and 8 also motivate the quasistatic (adiabatic) limit considered next: as Ω_p/Ω → 0, the beating becomes slow and the response is well approximated by an instantaneous static scattering problem evaluated at the current blockage level.

4.2. Quasistatic approximation

Floquet theory quantifies sideband generation and its dependence on the pumping frequency ω_p. Numerically, for sufficiently slow pumping, the dominant coefficients (n = −1, 0, 1) depend only weakly on ω_p, suggesting the onset of an adiabatic, or quasistatic (QS), regime as ω_p → 0.

Assuming ω_p ≪ ω, we approximate the scattering as instantaneously static: at each time t, the static plane-wave reflection and transmission factors are evaluated at the instantaneous blockage B_μ(t), neglecting $O (ω_{p} / ω)$ corrections. Specifically, let

R_{PWA} (t) ≔ - \frac{i k_{0} h \sum_{q \in Z} {\hat{B}}_{q} e^{- i q ω_{p} t}}{1 - i k_{0} h \sum_{q \in Z} {\hat{B}}_{q} e^{- i q ω_{p} t}}, T_{PWA} (t) ≔ \frac{1}{1 - i k_{0} h \sum_{q \in Z} {\hat{B}}_{q} e^{- i q ω_{p} t}},

(47)

where k₀ = ω/c is the incident wavenumber. In the QS approximation, the T_p-periodic functions R_PWA(t) and T_PWA(t) admit the Fourier expansions

R_{PWA} (t) = \sum_{n \in Z} {\tilde{r}}_{n} e^{- i n ω_{p} t}, T_{PWA} (t) = \sum_{n \in Z} {\tilde{t}}_{n} e^{- i n ω_{p} t},

(48)

with coefficients

{\tilde{r}}_{n} = \frac{1}{T_{p}} \int_{0}^{T_{p}} R_{PWA} (t) e^{i n ω_{p} t} d t, {\tilde{t}}_{n} = \frac{1}{T_{p}} \int_{0}^{T_{p}} T_{PWA} (t) e^{i n ω_{p} t} d t .

(49)

For a general periodic waveform, ${\tilde{r}}_{n}$ and ${\tilde{t}}_{n}$ can be evaluated numerically from (49), for example, by applying an FFT to uniform samples of R_PWA(t) and T_PWA(t) over one pumping period.

Closed-form expressions are available when the modulation contains only the 0 and ± 1 Fourier components, that is,

B_{μ} (t) = {\hat{B}}_{0} + {\hat{B}}_{1} e^{- i ω_{p} t} + {\hat{B}}_{- 1} e^{i ω_{p} t}, {\hat{B}}_{- 1} = \bar{{\hat{B}}_{1}} .

(50)

Substituting (50) into (47) shows that T_PWA(t) has the rational form

\begin{array}{l} T_{PWA} (t) = \frac{1}{1 - i k_{0} h ({\hat{B}}_{0} + {\hat{B}}_{1} e^{- i ω_{p} t} + {\hat{B}}_{- 1} e^{i ω_{p} t})} . \end{array}

Introducing the phase variable φ = ω_pt and using T_p = 2π/ω_p, the definition (49) becomes

{\tilde{t}}_{n} = \frac{1}{2 π} \int_{0}^{2 π} \frac{e^{i n φ}}{1 - i k_{0} h ({\hat{B}}_{0} + {\hat{B}}_{1} e^{- i φ} + {\hat{B}}_{- 1} e^{i φ})} d φ .

(51)

Define

β ≔ 1 - i k_{0} h {\hat{B}}_{0}, γ ≔ 2 i k_{0} h \sqrt{{\hat{B}}_{1} {\hat{B}}_{- 1}}, θ ≔ \arg ({\hat{B}}_{1}),

(52)

so that

{\hat{B}}_{1} = \sqrt{{\hat{B}}_{1} {\hat{B}}_{- 1}} e^{- i θ}

and

{\hat{B}}_{- 1} = \sqrt{{\hat{B}}_{1} {\hat{B}}_{- 1}} e^{i θ}

. Then,

\begin{array}{l} {\hat{B}}_{1} e^{- i φ} + {\hat{B}}_{- 1} e^{i φ} = 2 \sqrt{{\hat{B}}_{1} {\hat{B}}_{- 1}} \cos (φ + θ), \end{array}

and the denominator in (51) reduces to β − γ cos(φ + θ). Shifting ψ = φ + θ and using periodicity gives

{\tilde{t}}_{n} = e^{- i n θ} \frac{1}{2 π} \int_{0}^{2 π} \frac{e^{i n ψ}}{β - γ \cos ψ} d ψ .

(53)

To evaluate (53), we apply the unit-circle substitution z = e^iψ, for which dψ = dz/(iz) and cos ψ = (z + z⁻¹)/2. This converts (53) into the contour integral over |z| = 1

\begin{array}{l} {\tilde{t}}_{n} = e^{- i n θ} \frac{1}{2 π i} \oint_{| z | = 1} \frac{2 z^{n}}{2 β z - γ (z^{2} + 1)} d z = e^{- i n θ} \frac{1}{2 π i} \oint_{| z | = 1} \frac{2 z^{n}}{- γ (z - z_{+}) (z - z_{-})} d z, \end{array}

where

\begin{array}{l} z_{\pm} = \frac{β}{γ} \pm \sqrt{{(\frac{β}{γ})}^{2} - 1}, z_{+} z_{-} = 1 . \end{array}

With the branch choice

R (\sqrt{β^{2} - γ^{2}}) > 0

, exactly one of these roots lies inside the unit circle. Evaluating the residue at that pole yields

{\tilde{t}}_{n} = \frac{e^{- i n θ}}{\sqrt{β^{2} - γ^{2}}} a^{- | n |}, a ≔ \frac{β}{γ} \pm \sqrt{{(\frac{β}{γ})}^{2} - 1},

(54)

where the sign in the definition of a is chosen so that |a| > 1 (equivalently, the corresponding pole z_in = 1/a lies inside |z| = 1). Therefore,

{\tilde{r}}_{n} = δ_{0 n} - {\tilde{t}}_{n} .

(55)

The square-root branch is chosen so that

R (\sqrt{β^{2} - γ^{2}}) > 0

. For numerical robustness, one may equivalently select

a = β / γ \pm \sqrt{{(β / γ)}^{2} - 1}

so that |a| > 1, ensuring that a^−|n| decays as |n| increases. If

{\hat{B}}_{\pm 1} = 0

, the diaphragm is static and the QS series reduces to

{\tilde{r}}_{0} = R_{PWA}

and

{\tilde{t}}_{0} = T_{PWA}

, with

{\tilde{r}}_{n} = {\tilde{t}}_{n} = 0

for n ≠ 0. Moreover, (54) and (55) are independent of ω_p, as expected in the adiabatic regime; when θ = 0 (even-in-time modulation), the QS spectrum is symmetric in n.

A few practical considerations are in order regarding the numerical evaluation of (54) and (55). The quantities $β = 1 - i k_{0} h {\hat{B}}_{0}$ and $γ = 2 i k_{0} h \sqrt{{\hat{B}}_{1} {\hat{B}}_{- 1}}$ are computed directly from (52), with ${\hat{B}}_{- 1} = \bar{{\hat{B}}_{1}}$ for real B_μ(t). The complex square root $\sqrt{β^{2} - γ^{2}}$ is evaluated on the principal branch after first forming β² − γ² as a complex number; should the resulting real part be negative, the sign is reversed so as to enforce $R (\sqrt{β^{2} - γ^{2}}) > 0$ , the branch used in deriving (54). Of the two reciprocal candidates $a_{\pm} = β / γ \pm \sqrt{{(β / γ)}^{2} - 1}$ , the one with |a| > 1 is retained, which is well defined except in the degenerate limit |β| = |γ|. The factor a^−|n| is then evaluated as exp(−|n| log a) rather than by repeated multiplication, so that floating-point error does not amplify with |n|, and the closed-form coefficients follow as ${\tilde{t}}_{n} = e^{- i n θ} a^{- | n |} / \sqrt{β^{2} - γ^{2}}$ and ${\tilde{r}}_{n} = δ_{0 n} - {\tilde{t}}_{n}$ . The static case ${\hat{B}}_{1} = 0$ , detected by |γ| < ɛ with a small tolerance (e.g. ɛ = 10⁻¹⁴), reverts to ${\tilde{t}}_{0} = T_{PWA}$ , ${\tilde{r}}_{0} = R_{PWA}$ and ${\tilde{t}}_{n} = {\tilde{r}}_{n} = 0$ (n ≠ 0); the borderline case |β| = |γ|, for which |a| = 1, corresponds to a pole on the unit circle and is excluded from the long-wavelength regime considered here. We have cross-checked (54) against direct FFT evaluation of (49) from T_PWA(t) sampled on a uniform grid of 2¹² points over one pumping period, and against the truncated Floquet system (41) in the adiabatic limit ω_p/ω → 0; agreement to 10⁻¹² in $| {\tilde{t}}_{n} |$ and $| {\tilde{r}}_{n} |$ is obtained for all |n| ≤ 12 across the parameter ranges considered in Figure 9.

To quantify the range of validity of the QS approximation relative to the full Floquet solution (41), we fix the incident frequency and vary the pumping rate. Throughout Figure 9, we take Ω = ωh/c = 0.1 and compare |r_n| with $| {\tilde{r}}_{n} |$ . We monitor the relative discrepancy

\begin{array}{l} δ | r_{n} | = \frac{‖ r_{n} | - | {\tilde{r}}_{n} | |}{| {\tilde{r}}_{n} |} . \end{array}

Figure 9 shows that the QS approximation is highly accurate for very slow modulation. In panel (a), the Floquet spectrum is essentially symmetric in n, and the QS prediction is almost indistinguishable from the full solution over the displayed harmonics. As the pumping rate increases, as seen in panels (b) and (c), small but systematic deviations first appear in higher-order sidebands, whereas the dominant components near n = 0 remain well captured. The error curves in panel (d) confirm that δ|r_n| is smallest in the adiabatic limit ω_p/ω → 0 and increases as ω_p/ω grows, with higher-order harmonics showing the greatest sensitivity to faster modulation. Now for the fundamental component n = 0, we compare a static diaphragm with a time-modulated diaphragm. For the actuation used in Figure 10, the maximum blockage is μ_max = 0.95, while the corresponding mean blockage implied by the periodic law is μ_mean ≈ 0.698 (time average over one pumping period). Figure 10(a) shows that, over the long-wavelength range considered, the modulated fundamental reflection satisfies

| R_{PWA} (μ_{mean}) | < | r_{0} | < | R_{PWA} (μ_{\max}) |,

that is, the time-dependent diaphragm primarily redistributes energy among Floquet sidebands, while the fundamental component remains bounded between the static responses associated with the mean and extreme blockages. Moreover, the modulated curves for Ω_p = ω_ph/c = 0.2 and Ω_p = 0.1 remain close, indicating that |r₀| depends only weakly on the pumping frequency in this plane-wave regime.

Figure 10(b) further confirms that the sensitivity of the fundamental component to pumping increases with incident frequency. For smaller Ω, the curve is nearly flat in Ω_p/Ω, whereas for larger Ω noticeable departures from the QS prediction appear as Ω_p/Ω increases. This is consistent with the adiabatic interpretation: the QS reduction is most accurate when ω_p ≪ ω and the incident frequency remains well within the plane-wave validity range.

5. Conclusions

We have presented a systematic analysis of linear acoustic scattering in a rigid duct containing an infinitesimally thin vertical rigid baffle whose penetration height varies in time. The study combines multimodal mode-matching analysis, long-wavelength reduction, and Floquet theory in order to capture both the static scattering behaviour and the harmonic generation produced by periodic actuation of the baffle.

For the reference configuration with a static baffle, a two-dimensional boundary-value problem was formulated and solved using a truncated modal expansion. This provided quantitative reflection and transmission characteristics across the full frequency range and established a benchmark for assessing reduced descriptions. In the low-frequency regime, the baffle was shown to admit an effective compact representation through axial jump conditions, yielding a reduced model that accurately reproduces the plane-wave scattering response within its validity range. This confirms that the dominant physical effect of a shallow rigid insert at long wavelengths is governed primarily by local continuity and flux-balance constraints rather than detailed transverse structure.

When the baffle penetration is periodically modulated in time, the scattering process becomes intrinsically multi-frequency. A Floquet formulation was therefore developed to compute the coupled harmonic reflection and transmission coefficients. The results demonstrate clear redistribution of acoustic energy among temporal sidebands, with the strength of harmonic generation controlled by the modulation amplitude, frequency, and the underlying static scattering properties. In the limit of slow modulation, a quasistatic (adiabatic) approximation was derived and shown to provide an accurate and computationally efficient alternative to the full Floquet system, thereby clarifying the transition between dynamic and effectively steady scattering regimes.

Moreover, the analysis highlights the central role of accurate interface modelling and multimodal coupling in predicting scattering from time-varying internal elements in duct acoustics. The framework developed here is general and can be extended to more complex configurations, including compliant or elastic inserts, dissipative linings, branched or multi-duct geometries, and coupled fluid–structure interaction systems. It also provides a foundation for studying active or space–time modulated acoustic devices aimed at frequency conversion, non-reciprocal transport, or adaptive noise control. Future work may therefore focus on nonlinear effects, broadband optimisation, and experimental validation of time-modulated duct elements.

From an engineering standpoint, the time-modulated diaphragm analysed here serves as a deliberately simple, yet practically realisable, building block for a number of vibration- and acoustics-control technologies. In active noise-control devices for HVAC and exhaust ducts, a motorised or piezoelectrically driven blade whose immersion depth is modulated at ω_p behaves, in the long-wavelength regime, as a tunable compact element: the mean blockage ${\hat{B}}_{0}$ controls the broadband insertion loss while the modulation amplitude ${\hat{B}}_{\pm 1}$ governs the redistribution of reflected and transmitted energy among the sidebands ω ± ω_p, providing a low-complexity actuation mechanism for shifting tonal energy out of an annoying band into spectral regions where additional passive treatment, or psychoacoustic masking, is more effective. The same hardware can also serve as the active element of a non-reciprocal acoustic component, since the periodic time modulation breaks time-reversal symmetry at the interface and pumping at the appropriate phase and amplitude steers different sidebands into different spatial directions, an effect that is otherwise unavailable in time-invariant linear ducts. The Floquet sideband structure quantified here is the duct-acoustic analogue of the conversion processes exploited in time-modulated electromagnetic media (Engheta, 2023; Gazalet et al., 2013) and in low-frequency mechanical metamaterials (Cai et al., 2020; Chen et al., 2026; Jamil et al., 2022; Mei et al., 2012), and the closed-form quasistatic expression (54) provides explicit conversion efficiencies in the slow-pumping regime that can be used directly as a design formula for frequency converters and parametric amplifiers. Finally, because the present model exposes the control variables $({\hat{B}}_{0}, {\hat{B}}_{\pm 1}, ω_{p})$ explicitly, it lends itself to model-based control design and to model-based fault detection of moving in-duct hardware. These directions are likely to become increasingly attractive as electromechanical actuation in ducts becomes cheaper and as low-frequency metamaterial concepts move from the laboratory into engineering practice.

Footnotes

ORCID iD

Aqsa Yaseen

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2026R764), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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