Mathematical Methods for Physicists: A concise introduction - Site Map

FOURIER SERIES AND INTEGRALS
Figure 4.9.
A vibrating string.
We limit the displacement to small values, so that we may set
cos ˆ 1 2 =2; sin tan ˆ dy=dx;
then
F y ˆ AP
dy
dx
x‡dx
dy
dx
x
ˆ AP d2 y
dx 2 dx:
Using Newton's second law, the equation of motion of transverse vibration of the
element becomes
Adx @2 y
@t 2 ˆ AP @2 y
@x 2 dx; or @ 2 y
@x 2 ˆ 1 @ 2 y
v 2 @t 2 ; v ˆ p
P= :
Thus the transverse displacement of the string satis®es the partial di€erential wave
equation
@ 2 y
@x 2 ˆ 1 @ 2 y
v 2 ; 0 < x < L; t > 0 …4:14†
2
@t
with the following boundary conditions: y…0; t† ˆy…L; t† ˆ0; @y=@t ˆ 0;
y…x; 0† ˆf …x†; where f …x† describes the initial shape (position) of the string,
and v is the velocity of propagation of the wave along the string.
Solution of the wave equation
To solve this boundary value problem, let us try the method of separation variables:
y…x; t† ˆX…x†T…t†:
…4:15†
Substituting this into Eq. (4.14) yields
…1=X†…d 2 X=dx 2 †ˆ…1=v 2 T†…d 2 T=dt 2 †.
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