Physical Chemistry Third Edition

638 14 Classical Mechanics and the Old Quantum Theory

so thatνtct/λ,or

ν

c

λ

(14.3-34)

Equation (14.3-34) is the general relation between wavelength and frequency for a

traveling wave. This important equation holds for all kinds of waves, including sound

waves and electromagnetic waves. If we replaceλby 2L/nandcby

√

T/ρwe obtain

ν

n

2 L

√

T

ρ

(14.3-35)

which is identical to Eq. (14.3-23), the formula for the frequency of a standing wave.

Using Eqs. (14.3-30) and (14.3-34), we can now write

z(x,t)Asin

(

2 πx

λ

− 2 πνt

)

(14.3-36)

Exercise 14.11

In addition to waves in strings, other oscillations such as sound waves obey wave equations.

The speed of sound in air at sea level and 20◦Cis343ms−^1. Find the wavelength of a sound

wave with a frequency of 440 s−^1 , or 440 Hertz. (This frequency is the frequency of “A” above

middle “C” in the musical scale.)

Two traveling waves moving in opposite directions can interfere to produce a standing

wave. The two traveling waves

zR(x,t)Asin(κx−κct) (14.3-37a)

and

zL(x,t)Asin(κx+κct) (14.3-37b)

interfere to give

z(x,t)zR(x,t)+zL(x,t)A[sin(κx+κct)+sin(κx−κct)] (14.3-38)

which by the use of trigonometric identities is the same as the standing wave

z(x,t) 2 Asin(κx) cos(κct) (14.3-39)

Exercise 14.12

Use trigonometric identities to obtain Eq. (14.3-39) from Eq. (14.3-38).

James Clerk Maxwell, 1831–1879, was The Wave Theory of Electromagnetic Radiation
a British physicist who made
fundamental contributions to
electrodynamics, gas kinetic theory, and
thermodynamics.

In 1865, Maxwell developed a mathematical theory of electromagnetism. According

to this theory the electric fieldEEE and the magnetic fieldHcan oscillate like waves,

constituting electromagnetic radiation. Examples of electromagnetic radiation are