2.7.3 Orthogonality of Eigenfunctions
Many differential EVPs from applied mechanics possess eigenfunctions that are
characterized by orthogonality:
Theorem 2.5. For a self-adjoint EVP, Ku = kLu, any two eigenfunctions
u i and u j corresponding to distinct eigenvalues k i and k j are K-orthogonal
and L-orthogonal, that is:
Z b
a
u i Ku j dx ¼
Z b
a
u i Lu j dx ¼ 0 for k i 6 ¼ k j :
ð2:54Þ
Proof. Since (k i , u i ) and (k j ,u j ) are eigenpairs, we know that Ku i = k i Lu i and
Ku j = k j Lu j . Multiply the first equation by u j and the second by u i , subtract the
two equations from each other, and integrate over x2[a;b] to obtain:
Z b
a
u j Ku i À u i Ku j
À
Á
dx ¼
Z b
a
k i u j Lu i À k j u i Lu j
À
Á
dx;
ð2:55Þ
or, since the EVP is assumed to be self-adjoint:
0 ¼ ðk i À k j Þ
Z b
a
u j Lu i dx:
ð2:56Þ
With distinct eigenvalues k i 6 ¼ k j the integral must vanish identically, that is, u i
and u j must be L-orthogonal. Multiplying Ku i = k i Lu i by u j and integrating, it is
seen that u i and u j are also K-orthogonal.
Eigenfunctions are often normalized so that
R
u j Lu j dx ¼ 1, which implies that
R
u j Ku j dx ¼ l j . The eigenfunctions are then said to be orthonormal, and to satisfy
the relations of orthonormality:
Z b
a
u i Ku j dx ¼ k j d ij ;
Z b
a
u i Lu j dx ¼ d ij ; i; j ¼ 1; 1:
ð2:57Þ
Example 2.13. For the EVP of the fixed-free vibrating rod with differential
equation c
2 u′′ = –x
2 u and boundary conditions u(0) = u′(l) = 0, the natural frequencies are given by x j = (2j – 1)pc/2l, j = 1, ∞, and the mode shapes by
u j (x) = B j sin(x j x/c), where the B j s are undetermined constants (cf. Sect. 2.5.1).
From Example 2.8 we know that the EVP is self-adjoint and thus, by Theorem 2.5,
2.7 Properties of Eigenvalues and Eigenfunctions
69