Physical Chemistry Third Edition

14.3 Classical Waves 629

14.6Assume that a particle of massmmoves only in the

zdirection and is subject to a constant force given by

Fz−mg. Solve its equation of motion, findingvzandz

as functions of time for arbitrary values ofz(0) andvz(0),

the values at time zero.

14.7The frequency of oscillation of several diatomic

molecules is as follows:

a.HF 1. 2406 × 1014 s−^1

b.HCl 8. 9665 × 1013 s−^1

c.HBr 7. 9414 × 1013 s−^1

d.Br2 9. 7528 × 1012 s−^1

Find the force constant for the bond in each molecule.

Comment on your results, since all of these molecules

have single bonds.

14.8 a. Show that a vertical spring with a massmsuspended

from it is lengthened by an amount equal tomg/k

wheregis the acceleration due to gravity, andkis the

spring constant.

b.Ifzis the displacement of the mass from its equili-

brium position, show that the potential energy is

given by

VV 0 +

kz′^2

2

+constant

whereV 0 is the potential energy of the spring at its

equilibrium length without the mass attached to it, and

wherez′is the displacement of the mass from its new

equilibrium position.

c.Show that the frequency of oscillation is the same as in

the horizontal position. An object of mass 0.250 kg is

suspended from a spring withk 5 .55 Nm−^1. Find the

distance by which the spring is lengthened, the period,

and the frequency.

14.3 Classical Waves

A wave is an oscillating displacement that depends on position and time. Waves that

are described by classical mechanics include sound waves, waves on the surface of a

body of water, and vibrations of strings in musical instruments. In a water wave the

displacement is the distance to a point on the surface from its equilibrium position,

and in a sound wave the displacement is the difference between the pressure and its

equilibrium value. A region of positive displacement is called acrest, and a region

of negative displacement is called atrough. A location where the displacement of a

wave equals zero is called anode. The distance from one crest to the next is called the

wavelengthλ. Theperiodτof a wave is the time for the first return of the oscillating

object to an initial state. Thefrequencyνis the reciprocal of the period, or the number

of oscillations per second.

There are two principal types of waves. Atraveling wavepropagates (moves along)

like waves on the surface of a body of water. Astanding wavesuch as the vibra-

tion of a string in a musical instrument does not propagate, but has stationary nodes.

Figure 14.5 represents some features of traveling and standing waves. The traveling

wave in Figure 14.5a moves to the right without changing shape, whereas the standing

wave in Figure 14.5b oscillates between stationary nodes.

One important property of waves isinterference. When two waves of the same

type come to the same location, their displacements add. If two crests or two troughs

coincide, a displacement of larger magnitude results. This addition is calledconstruc-

tive interference. If a crest of one wave and a trough of another wave coincide, they

will partially or completely cancel each other. This cancellation is calleddestructive

interference.

A property that arises from interference isdiffraction. If a water wave encoun-

ters a post, there will be a reflected wave with circular crests that moves out in all

directions from the post. The reflected waves from a row of equally spaced posts can