Physical Chemistry Third Edition

14.3 Classical Waves 635

1.5

1.0

Wave displacement

0.5

0.0

2 0.5

2 1.0

2 1.5

0.0 0.2 0.4 0.6

x/L

(a)

0.8 1.0

1.5

1.0

Wave displacement

0.5

0.0

2 0.5

2 1.0

2 1.5

0.0 0.2 0.4 0.6

x/L

(b)

0.8 1.0

1.5

1.0

Wave displacement

0.5

0.0

2 0.5

2 1.0

2 1.5

0.0 0.2 0.4 0.6

x/L

(d)

0.8 1.0

1.5

1.0

Wave displacement

0.5

0.0

2 0.5

2 1.0

2 1.5

0.0 0.2 0.4 0.6

x/L

(c)

0.8 1.0

Figure 14.7 Standing Waves in a Flexible String.(a) The wave function forn1.

(b) The wave function forn2. (c) The wave function forn3. (d) The wave function

forn4.

A string does not usually move as described by a single harmonic. Alinear combi-

nationof harmonics can satisfy the wave equation:

z(x,t)

∑∞

n 1

an(t) sin

(nπx

L

)

(14.3-24)

The fact that a linear combination of solutions can be a solution to the wave equa-
tion is called theprinciple of superposition. When a string in a musical instrument is
struck or bowed, it moves according to some linear combination of harmonics. A violin
sounds different from a piano because of the difference in the strengths of various
harmonics.
The Fourier series is named for
Jean Baptiste Joseph Fourier,
1768–1830, a famous French
mathematician and physicist.

The linear combination shown in Eq. (14.3-24) is called aFourier sine series.

Fourier cosine series also exist, which are linear combinations of cosine functions.

TheFourier coefficientsa 1 ,a 2 ,...must depend ontto satisfy the wave equation.

With the initial condition that the string is passing through its equilibrium position at