
562 SECOND-ORDER HYPERBOLIC EQUATIONS WITH ONE SPACE VARIABLE
◮ Domain: 0 ≤ x ≤ l. Third boundary value problem.
1
◦
. Longitudinal vibration of an elastic rod with clamped ends in the case of equal stiffness
coefficients. The following conditions are prescribed:
w = f (x) at t = 0 (initial condition),
∂
t
w = g(x) at t = 0 (initial condition ),
∂
x
w − kw = 0 at x = 0 (boundary condition),
∂
x
w + kw = 0 at x = l (boundary condition).
Solution:
w(x, t) =
∞
X
n=1
A
n
cos(λ
n
at) + B
n
sin(λ
n
at)
sin(λ
n
x + ϕ
n
),
where
A
n
=
1
kX
n
k
2
Z
l
0
sin(λ
n
x + ϕ
n
)f(x) dx, B
n
=
1
aλ
n
kX
n
k
2
Z
l
0
sin(λ
n
x + ϕ
n
)g(x) dx,
ϕ
n
= arctan
λ
n
k
, kX
n
k
2
=
Z
l
0
sin
2
(λ
n
x + ϕ
n
) dx =
l
2
+
k
k
2
+ λ
2
n
;
the λ
n
are positive roots of the ...