The Chemistry Maths Book, Second Edition

14.7 The vibrating string 411

and the initial value problem

(14.92)

with initial conditions (14.88), wheref(x)andg(x)are given functions.

The boundary value problem (14.91) is identical to that discussed in Section 12.6

for the particle in a box. The allowed values of the separation constant are given by

n 1 = 1 1, 2, 3, = (14.93)

and the corresponding (unnormalized) particular solutions are

(14.94)

For each value ofλ

n

, equation (14.92) is

(14.95)

where

(14.96)

and has solution

G

n

(t) 1 = 1 A

n

1 cos 1 ω

n

t 1 + 1 B

n

1 sin 1 ω

n

t (14.97)

whereA

n

andB

n

are constants determined by the initial conditions. A set of solutions

of the wave equation for the vibrating string is therefore

, n 1 = 1 1, 2, 3, = (14.98)

These solutions are called the eigenfunctions of the system. The quantitiesω

n

1 = 1 nπv 2 l

are called the eigenvalues. The set of values {ω

1

, ω

2

, ω

3

, =} is called the eigenvalue

spectrum.

Normal modes of motion

Each eigenfunction y

n

(x, 1 t)is a periodic function of time with period 2 π 2 ω

n

; the

motion is transverse harmonic motion with frequency ν

n

1 = 1 ω

n

22 π. This motion is

called the nth normal modeof the vibrating string. The first mode, withn 1 = 11 , is called

the fundamental, the second, withn 1 = 12 , is the first overtone, and so on. The motions

yxt

nx

l

AtBt

nnnnn

( ) sin,= cos +sin













π

ωω

ωλ

nn

n

l

==v

πv

dG

dt

G

n

2

2

2

+=ω 0

Fx

nx

l

n

() sin=

π

λ

n

n

l

=,

π

dG

dt

G

2

2

22

+=λv 0