A tightly stretched string with fixed end points x=0 and x=l is initially in a position given by y=y_0 sin³(πx/l).It is released from rest from the initial positions.Find the displacement y(x,t)?
1 Answer
Explanation:
For 1-D wave equation (WE):
#y_("xx") = 1/c^2 y_("tt")#
Separate the variables:
#y(x,t) = X(x) tau (t)#
The WE then reads:
#X^('')/X = 1/c^2 (ddot tau)/tau = color(red)(-) k^2#
Using
#y(x,t) = (c_1 sin(k x) + c_2 cos(k x))(c_3 sin(ck t) + c_4 cos(ck t))#
The boundary conditions are
-
#y(0,t) = 0# , fixed node -
# y(l,t) = 0# , fixed node -
#y_t(x,0) = 0# , string is only released at#t = 0# -
#y(x,0) = y_0 sin^3((pi x)/l)# , initial shape
1st BC :
-
#y(0,t) = (0 + c_2 )(c_3 sin(ck t) + c_4 cos(ck t)) = 0# -
#implies c_2 = 0# .
2nd BC :
#y(l,t) = (c_1 sin k l )(c_3 sin(ck t) + c_4 cos(ck t)) = 0#
The fixed nodes are at
#k = (n pi)/l, n = 1,2,....# .
The solution thusfar:
3rd BC :
So the basis functions are
# y(x,t) = sum_1^oo b_n sin((n pi x)/l) cos((n pi c t )/l) qquad square#
4th BC :
#y(x,0) = y_0 sin^3((pi x)/l) #
By identity:
Comparing co-efficients:
So