In Eq. 5.32b, Vp is the phase velocity
where 
and y is amplitude of vibration at time t at distance x from fixed end.
Using method of separation of variable to solve the PDE (Eq. 5.32b) we write
y = f(x)g(t) |
(5.33) |
|
(5.34) |
And,
|
(5.35) |
substituting in Eq. (5.32b), we get
|
(5.36) |
or,
|
(5.37) |
where 
is the separation constant.
Since, each side of the Eq. (5.37) is a function of a single variable,
|
(5.38) |
or,
|
(5.39) |
The characteristic equation is
|
(5.40) |
or,
m1 = ± i αVp |
(5.41) |
So, the general solution is
|
(5.42) |
