Surface Effects on Bulk Plasmons

We discuss the effects of surfaces on bulk plasma oscillations in a metallic film of thickness $L$. The leading corrections to the plasma frequencies due to the surfaces are proportional to $L^{-1}$. dispersion relation of bulk modes, $\omega=\omega_p[1+\alpha(q^2/k_F^2)] $, where the allowed $q$-values are $q^{(n)}=(n\pi/\overline L)$, $\overline L\equiv L-2d_b$, and $d_b$ is a complex length characteristic of the surface (reminiscent of but different from Feibelman's $d_\perp$). An explicit expression for $d_b$ is derived. Resonant excitation by an external field, $E_0e^{-i\omega t}$, is calculated.