The Chemistry Maths Book, Second Edition

410 Chapter 14Partial differential equations

14.7 The vibrating string

We consider an elastic string, such as a guitar string, of

length land uniform linear density ρthat is stretched

and fixed at both ends under tension T. The string is

distorted transversely (pulled sideways), released and

allowed to vibrate (Figure 14.6). The transverse dis-

placement of the string is then a function of position x

and time t,

y 1 = 1 y(x, 1 t) (14.85)

and when the vibrations are small the motion of the string is described by the wave

equation

(14.86)

wherev

2

1 = 1 T 2 ρ. The boundary and initial conditions on the solutions of the equation

are

y(0, t) 1 = 1 y(l, t) 1 = 10 (14.87)

for no displacement at the ends of the string, and

y(x, 1 0) 1 = 1 f(x), (14.88)

for the initial displacement and velocity (functions).

Separation of variables

We write the displacement function (a wave function) as the product

y(x, 1 t) 1 = 1 F(x) 1 × 1 G(t) (14.89)

Substitution in the wave equation and division byy 1 = 1 FGthen gives

(14.90)

Both sides of the equation must be constant so that, with separation constant−λ

2

, the

problem in two variables reduces to the boundary value problem

F(0) 1 = 1 F(l) 1 = 10 (14.91)

dF

dx

F

2

2

2

+=,λ 0

11

2

22

2

2

F

dF

dx G

dG

dt

=

v

t

y

t

gx

=

∂

∂



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=

0

()

∂

∂

=

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∂

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22

2

2

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Figure 14.6