Overview and Fundamental Ideas - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 9

1.3.1 Example 1-3. String with pinned ends and general initial
conditions.
Consider Equation 1.1, which does not include damping and external
excitation. In particular, we solve 2 u/ t 2 - c 2 2 u/ x 2 = 0 for the dependent
variable u(x,t) subject to the boundary conditions u(0,t) = 0 and u(L,t) = 0, plus
general initial conditions u(x,0) = (x) and u t (x,0) =

(x). We assume separable

product solutions of the form
u(x,t) = X(x)T(t) (1.17)
where X(x) and T(t) are unknown. Substitution into Equation 1.1 leads to
X"(x)/X(x) = (1/c 2 ) T"(t)/T(t). Since the left side, a function of x alone, equals a
function of t alone, each side must in turn equal the constant - . The result
X"(x)/X(x) = (1/c 2 ) T"(t)/T(t) = - (1.18)

implies two ordinary differential equations,
X"(x) + X(x) = 0 (1.19)
T"(t) + c 2 T(t) = 0 (1.20)
He re , is a positive constant (zero and negative choices lead to trivial
solutions). The assumption (or, “Ansatz”) u(x,t) = X(x)T(t) and the boundary
conditions u(0,t) = X(0)T(t) = 0 and u(L,t) = X(L)T(t) = 0 show that X(0) =
X(L) = 0. Thus, the solution X(x) = D 1 cos (x

) + D 2 sin (x ) requires our

setting X(0) = D 1 = 0 and X( L) = D 2 sin (L ) = 0.
D 2 = 0 would yield a zero solution, which violates initial conditions, so we
consider sin (L ) = 0; this requires L = n , where n = 1, 2, 3, ... may be
arbitrary. To note these possibilities, we introduce the subscripted quantity
n = ( n/L) 2

(1.21)

to which we associate the functions
X n (x) = C n sin ( nx/L) (1.22)
T n (t) = A n cos ( nct/L) + B n sin ( nct/L) (1.23)
noting that the time function is obtained by solving the equation for T(t).
Since a solution takes the form u n (x,t) = X n (x)T n (t) = {A n cos ( nct/L) +
B n sin ( nct/L)} sin ( nx/L), where we have assumed C n = 1 without loss of
generality, the more complete superposition solution satisfies
u(x,t) = u n (x,t)

= {A n cos ( nct/L) + B n sin ( nct/L)} sin ( nx/L) (1.24)

where the summation is taken from n = 1 to .
We have already used boundary conditions to select a sine expansion for
u(x,t), but the series coefficients are still undetermined. These can be fixed by