
162 Applied calculus of variations for engineers
then the first coefficient will be simply the y
m
amplitude:
c
1
= y
m
,
since
sin(
π
L
L
2
)=sin(
π
2
)=1.
Similar, but not identical, considerations may be applied for the coefficients
of the higher normal modes.
The natural frequencies depend on the physical conditions, such as the pre-
applied tension force distribution and the material characteristics embodied
in the unit weight ρ. Specifically, the higher the tension force F in the string,
the higher the frequency becomes. A very tight string vibrates very quickly
(with high frequency), while a very loose string vibrates slowly.
11.2 The elastic membrane
We now turn our ...