Resume. Nous etudions une membrane vibrante avec une distribution de densite dependant d'un petit parametre e, qui converge, lorsque e \ 0, vers une densite uniforme plus une masse ponctuelle a l'origine. Nous mettons en evidence l'existence de vibrations locales, au voisinage de l'origine, et globales de la membrane. L'etude asymptotique lorsque e \ 0 est effectuee a 1'aide de la methode des developpements asymptotiques raccordes.Abstract. We study a vibrating membrane with a distribution of density depending on e, which converges, as e \ 0, to a uniform density, plus a point mass at the origin. We establish local vibrations at the vicinity of the origin and global vibrations of the membrane. The asymptotic study for e \ 0 is performed using the method of matched asymptotic expansions.1. Introduction. We consider vibrating systems containing a small region, of diameter O(e), including the origin, where the density is very much higher than elsewhere. Quite different cases arise depending on the space dimension N and the order of magnitude of the ratio e~m of densities. Many studies are devoted to this problem of concentrated masses: E. Sanchez-Palencia [1], E. Sanchez-Palencia and H. Tchatat [2], H. Tchatat [3], O. A. Oleinik [4], In this paper we study the case N = 2 (i.e., the vibrating membrane) with m >2. Using the method of matched asymptotic expansions (see for instance [5] and [6]), we derive the structure of the eigenfunctions, which is not given by other methods. It appears that there are two kinds of eigenvibration:

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