本研究建立了層依賴力常數、局部法線方向波動條件、TRC 篩選、邊界反射與有限薄膜特徵模態之間的分析關係。其中，q_z^(ζ) 僅描述指定粗粒化區段之局部法線方向響應，不能單獨判定聲子模態之侷限深度、偏振方向或表面權重。完整模態判讀仍須結合特徵向量、層向權重、位移分量與表面投影體相聲子頻帶。;This study investigates phonon responses in ultrathin face-centered cubic (FCC) (001) aluminum and silver films. An analytical framework is developed by combining the finite-film slab method， layer-dependent force constants， local coarse-grained wave equations， a normal-direction transmission-line model， and the transverse resonance condition (TRC). Under the harmonic approximation and central-force spring model， the finite-film dynamical matrix includes first- and second-neighbor atomic interactions， retains the in-plane Bloch phase， and uses reflection coefficients to describe normal-direction reflections caused by free surfaces or external-material fixed endpoints. The interlayer force constants are adjusted through a surface modification function to compare the effects of surface weakening， surface strengthening， and different modification depths. This function is a controlled parametric model and does not represent a layer-resolved fit to the surface force constants of a specific material.
To establish the local normal response of each interlayer section， the region between adjacent atomic layers is represented by a local coarse-grained displacement field， from which a three-component local wave equation is derived using the elastic energy density. The zz element is subsequently extracted from the plane-wave matrix to establish the section-dependent local normal wavenumber q_z^(ζ)， local cutoff angular frequency， and scalar characteristic impedance. For each in-plane wave vector and trial angular frequency along a high-symmetry path in the surface Brillouin zone (SBZ)， q_z^(ζ) is first obtained for each section from the local dispersion relation. The input impedances are then recursively evaluated from both sides of the film to select candidate solutions satisfying the TRC. The selected solutions are used to calculate the reflection coefficients and multiple-reflection corrections， after which the finite-film dynamical matrix is constructed to obtain the characteristic angular frequencies and eigenvectors. The TRC impedance residuals remain at extremely small levels in all cases， while the imaginary parts of the characteristic angular frequencies obtained from the dynamical matrix remain much smaller than their real parts. These two quantities separately examine the numerical quality of the TRC selection and eigenvalue calculation.
The silver-film results show that increasing the number of atomic layers increases the number of characteristic angular-frequency branches according to 3N_z and generally reduces the spacing between adjacent branches. Under the surface-weakening parameters examined in this study， changing the film thickness does not cause the section nearest the surface to change from a propagation-like response to another type of local response. In contrast， the direction and depth dependence of the interlayer force-constant modification directly change the local cutoff angular frequency of each section. Surface weakening lowers the local cutoff angular frequency in near-surface sections， so the real part remains the primary nonzero component of the local normal wavenumber. Surface strengthening raises the local cutoff angular frequency and produces a non-negligible imaginary part of the wavenumber for some TRC solutions. When the strengthened force-constant profile varies more gradually toward the film interior， the evanescent-like characteristic can extend into more internal sections.
A comparison between free surfaces and external-material fixed endpoints shows that， when the force constants inside the film are unchanged， the mapping of q_z^(ζ) over the complete in-plane-wave-vector and angular-frequency plane of a given section is not directly altered by the boundary condition. However， the fixed endpoints change the input impedances， reflection coefficients， TRC-selected angular frequencies， and boundary diagonal terms of the dynamical matrix. Consequently， the local-normal-wavenumber curves sampled along the TRC solutions remain different from those of the free-surface case. Increasing the equivalent force constants of the external regions raises their local cutoff angular frequencies， causing some solutions to exhibit evanescent-like characteristics outside the film. The fixed endpoints also break the invariance under uniform translation of the entire film， shifting the original zero-frequency acoustic-phonon branch at the SBZ center to a finite frequency. Compared with the free-surface case， the low-frequency Rayleigh-like branch under the fixed-endpoint condition is no longer clearly separated from the bulk band. A boundary-stiffened branch separated from the main bulk band also appears in the high-frequency region.
The layer weights and displacement components of the eigenvectors further show that the frequency range of a dispersion curve alone is insufficient to determine the complete mode type. Representative low-frequency branches of the free-surface silver film have pronounced weights at the upper and lower surface layers， but their dominant displacement directions vary with the in-plane wave vector. For example， the dominant polarization changes from a shear-horizontal characteristic near the X̄ point to a shear-vertical characteristic near the M̄ point. When higher-frequency branches approach or enter a dense surface-projected bulk phonon band， their weights can extend from the surfaces into the film interior， exhibiting surface-resonance or bulk-like characteristics. The external-material fixed endpoints redistribute the weights of some branches from the outermost layers toward the subsurface layers and film interior， but the changes are not identical among different branches. The aluminum-film results further show that the same analytical procedure can be applied to other FCC metallic films. Under the harmonic approximation， the characteristic angular-frequency scale of a material is jointly determined by its atomic mass and force constants.
This study establishes an analytical relationship among layer-dependent force constants， local normal wave conditions， TRC selection， boundary reflection， and the characteristic modes of finite films. The local normal wavenumber q_z^(ζ) describes only the local normal response of a specified coarse-grained section. It cannot independently determine the localization depth， polarization direction， or surface weight of a phonon mode. Complete mode identification still requires the eigenvectors， layer-resolved weights， displacement components， and surface-projected bulk phonon bands.
