Physical Chemistry Third Edition

636 14 Classical Mechanics and the Old Quantum Theory

1

0

n 53

n 52

n 51

Sum of

3 harmonics

0.0 0.2 0.4

x/L

(a)

0.6 0.8 1.0

Wave displacement

21

1

0

n 53

n 52

n 51

Sum of

3 harmonics

0.0 0.2 0.4

x/L

(b)

0.6 0.8 1.0

Wave displacement

21

Figure 14.8 The Superposition of Three Harmonics of a Flexible String.(a) At time

tL/ 4 c. (b) At timet 3 L/ 4 c.

t0, the following linear combination is a solution of the wave equation

z(x,t)

∑∞

n 1

Ansin

(nπx

L

)

sin

(nπct

L

)

(14.3-25)

whereA 1 ,A 2 ,...are constants.

Exercise 14.7

Show by substitution that the series in Eq. (14.3-25) satisfies Eq. (14.3-3).

The different harmonics exhibit constructive interference and destructive interference

that continually change, because the different harmonics have different frequencies.

Figure 14.8 shows a linear combination of three harmonics withA 1 1,A 2  0 .2,

andA 3  0 .1. Figure 14.8a shows the sum at timetL/(4c), and Figure 14.8b shows

the sum att 3 L/(4c).

Traveling Waves

In a string of finite length with stationary ends, only standing waves can occur. Traveling

waves can occur in a very long string. A wave function that satisfies Eq. (14.3-3) and

corresponds to a traveling wave is

z(x,t)Asin(κx−κct) (14.3-26)

Exercise 14.8

Show by substitution that the function in Eq. (14.3-26) satisfies Eq. (14.3-3).