14.3 Classical Waves
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14.6 Assume that a particle of mass m moves only in the
z direction and is subject to a constant force given by
F z −mg. Solve its equation of motion, finding v z and z
as functions of time for arbitrary values of z(0) and v z (0),
the values at time zero.
14.7 The frequency of oscillation of several diatomic
molecules is as follows:
a. HF 1.2406 × 10 14 s −1
b. HCl 8.9665 × 10 13 s −1
c. HBr 7.9414 × 10 13 s −1
d. Br2 9.7528 × 10 12 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 mass m suspended
from it is lengthened by an amount equal to mg/k
where g is the acceleration due to gravity, and k is the
spring constant.
b. If z is the displacement of the mass from its equilibrium position, show that the potential energy is
given by
V V 0 +
kz 2
2
+ constant
where V 0 is the potential energy of the spring at its
equilibrium length without the mass attached to it, and
where z 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 with k 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 a crest, and a region
of negative displacement is called a trough. A location where the displacement of a
wave equals zero is called a node. The distance from one crest to the next is called the
wavelength λ. The period τ of a wave is the time for the first return of the oscillating
object to an initial state. The frequency ν is the reciprocal of the period, or the number
of oscillations per second.
There are two principal types of waves. A traveling wave propagates (moves along)
like waves on the surface of a body of water. A standing wave such as the vibration 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 is interference. 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 called constructive 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 called destructive
interference.
A property that arises from interference is diffraction. If a water wave encounters 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
629
14.6 Assume that a particle of mass m moves only in the
z direction and is subject to a constant force given by
F z −mg. Solve its equation of motion, finding v z and z
as functions of time for arbitrary values of z(0) and v z (0),
the values at time zero.
14.7 The frequency of oscillation of several diatomic
molecules is as follows:
a. HF 1.2406 × 10 14 s −1
b. HCl 8.9665 × 10 13 s −1
c. HBr 7.9414 × 10 13 s −1
d. Br2 9.7528 × 10 12 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 mass m suspended
from it is lengthened by an amount equal to mg/k
where g is the acceleration due to gravity, and k is the
spring constant.
b. If z is the displacement of the mass from its equilibrium position, show that the potential energy is
given by
V V 0 +
kz 2
2
+ constant
where V 0 is the potential energy of the spring at its
equilibrium length without the mass attached to it, and
where z 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 with k 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 a crest, and a region
of negative displacement is called a trough. A location where the displacement of a
wave equals zero is called a node. The distance from one crest to the next is called the
wavelength λ. The period τ of a wave is the time for the first return of the oscillating
object to an initial state. The frequency ν is the reciprocal of the period, or the number
of oscillations per second.
There are two principal types of waves. A traveling wave propagates (moves along)
like waves on the surface of a body of water. A standing wave such as the vibration 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 is interference. 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 called constructive 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 called destructive
interference.
A property that arises from interference is diffraction. If a water wave encounters 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