I am building a program to numerically solve the euler fluid equations based on finite volume discretization and am having trouble on a step where the eigenvalues of a cell need to be calculated according to the picture below. The eigenvalues for each cell are calculated by dotting the velocity vector with the normal of the face and adding C, the speed of sound.
Dotting the velocity vector with the normal of the face of the cell make sense. Therefore, to my understanding, you must have 4 eigenvalues, one for each face, for each cell. I am confused by the next part, which says to calculate the average of the neighboring cells at each face. The notation used does not differentiate between the left or right face, making it seem like there is only one eigenvalue for each computational direction (for each cell). I am having trouble understanding the exact representation of this term and its meaning.