Question of Consistency as between the Eulerian Viewpoint and the Lagrangian Viewpoint

Abstract

Considered in this paper is the problem, in one dimension, of forced oscillations and standing waves of sound in a cube-shaped resonator, the vibrating wall of the resonator being represented as a delta-function source that is then expanded in the normal modes of the resonator. Calculations are carried out first from the Eulerian point of view and then, secondly, from the Lagrangian point of view. The uncovering of a discrepancy in this connection, i.e., as between the Eulerian and Lagrangian points of view, necessitates modification of the usual Lagrangian equation of continuity so that it be consistent with the Eulerian equation of continuity in the presence of sources, upon which modification the two viewpoints, Eulerian and Lagrangian, are found to be consistent. Obtained also, in connection with this question of consistency, is the wave equation for the Lagrangian particle displacement in the presence of sources, once sources have been specified from the Eulerian point of view. [Work supported by the U. S. Air Force Systems Command.]