8.2 Vibration of Strings
311
Fig. 8.2 Mode shapes of a
string
8.2.1 Wave Propagation Solution
For structures having infinite or semi-infinite dimensions, the solution based on
the principles of wave propagation is particularly useful. Finite structures can also
be studied using these principles. Modelling of ground vibration in earthquake
engineering forms a typical example [1, 2].
The classical wave equation governs the case of a string undergoing small transverse vibration having constant tension T and a constant mass per unit length m and
is given by
∂
2 y
∂ x 2 =
1
c 2
∂
2 y
∂t 2
(8.21)
where c =
√
T /m.
The general solution is based on the assumption that the response y (x, t) equals
the sum of two different waves travelling in opposite directions,
y(x, t) = F 1 (x − ct) + F 2 (x + ct)
(8.22)
where F 1 and F 2 are arbitrary functions that can be differentiated twice with respect
to x and t.
311
Fig. 8.2 Mode shapes of a
string
8.2.1 Wave Propagation Solution
For structures having infinite or semi-infinite dimensions, the solution based on
the principles of wave propagation is particularly useful. Finite structures can also
be studied using these principles. Modelling of ground vibration in earthquake
engineering forms a typical example [1, 2].
The classical wave equation governs the case of a string undergoing small transverse vibration having constant tension T and a constant mass per unit length m and
is given by
∂
2 y
∂ x 2 =
1
c 2
∂
2 y
∂t 2
(8.21)
where c =
√
T /m.
The general solution is based on the assumption that the response y (x, t) equals
the sum of two different waves travelling in opposite directions,
y(x, t) = F 1 (x − ct) + F 2 (x + ct)
(8.22)
where F 1 and F 2 are arbitrary functions that can be differentiated twice with respect
to x and t.