Physical Chemistry Third Edition

634 14 Classical Mechanics and the Old Quantum Theory

The velocity att0is

νz(0)z 0

(nπc

L

)

sin

(nπx

L

)

(14.3-19)

Our second initial condition has specified the initial velocity. Instead of specifying the

maximum amplitude we could have specified that the initial velocity is given by this

function.

Exercise 14.6

a.Show by substitution that Eq. (14.3-16) satisfies Eq. (14.3-3).

b.What is the effect on the wave function of replacingnby its negative?

c.What is the relationship between the value ofnand the number of nodes?

The wavelengthλfor our standing wave is the distance such that the argument of

the sine function in Eq. (14.3-15) changes by 2π. This gives

λ

2 L

n

(14.3-20)

The wavelength isquantized. That is, it can take on a certain value for each value of

n, but no values between these allowed values. There is a relation betweenκandλ:

λ

2 L

n



2 π

κ

(14.3-21)

The relation betweenλandκis independent of the value ofn.

The periodτof our standing wave is the time required for the argument of the time

factor sin(nπct/L) to change by 2π, so that

2 π

nπcτ

L

or τ

2 L

nc

(14.3-22)

The frequencyνis the number of oscillations per unit time and equals the reciprocal

of the period:

ν

nc

2 L



n

2 L

√

T

ρ

(14.3-23)

The frequency is quantized. It is directly proportional to the integern, inversely propor-

tional to the length of the string, directly proportional to the square root of the tension

force, and inversely proportional to the square root of the mass per unit length.

The strings of real musical instruments obey our formulas to a good approximation.

In musical acoustics, the standing wave withn1 is called thefundamentalor the

first harmonic, the standing wave withn2 is called thefirst overtoneor thesecond

harmonic, and so on. Figure 14.7 represents the coordinate factorφfor several values

ofn. Each graph in the figure represents the position of the string at a time when the

time factorηequals unity. For other times, the string oscillates between the position

given by this graph and its negative.