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Unformatted text preview: 16.26). Although we do not prove it here, the linear Summary wave equation is satisﬁed by any wave function having the form y f (x vt ). Furthermore, we have seen that the linear wave equation is a direct consequence of
Newton’s second law applied to any segment of the string. SUMMARY
A transverse wave is one in which the particles of the medium move in a direction perpendicular to the direction of the wave velocity. An example is a wave on a
taut string. A longitudinal wave is one in which the particles of the medium move
in a direction parallel to the direction of the wave velocity. Sound waves in ﬂuids
are longitudinal. You should be able to identify examples of both types of waves.
Any onedimensional wave traveling with a speed v in the x direction can be
represented by a wave function of the form
y f (x (16.1, 16.2) vt ) where the positive sign applies to a wave traveling in the negative x direction and the
negative sign applies to a wave traveling in the positive x direction. The shape of the
wave at any instant in time (a snapshot of the wave) is obtained by holding t constant.
The superposition principle speciﬁes that when two or more waves move
through a medium, the resultant wave function equals the algebraic sum of the
individual wave functions. When two waves combine in space, they interfere to
produce a resultant wave. The interference may be constructive (when the individual displacements are in the same direction) or destructive (when the displacements are in opposite directions).
The speed of a wave traveling on a taut string of mass per unit length and
tension T is
T
(16.4)
v √ A wave is totally or partially reﬂected when it reaches the end of the medium in
which it propagates or when it reaches a boundary where its speed changes discontinuously. If a wave pulse traveling on a string meets a ﬁxed end, the pulse is reﬂected and inverted. If the pulse reaches a free end, it is reﬂected but not inverted.
The wave function for a onedimensional sinusoidal wave traveling to the
right can be expressed as
y A sin 2 (x vt ) A sin(kx t) (16.6, 16.11) where A is the amplitude, is the wavelength, k is the angular wave number,
and is the angular frequency. If T is the period and f the frequency, v, k and
can be written
v
k T f 2
2
T (16.7, 16.14)
(16.9) 2f (16.10, 16.12) You should know how to ﬁnd the equation describing the motion of particles in a
wave from a given set of physical parameters.
The power transmitted by a sinusoidal wave on a stretched string is
1
2 2A2v (16.21) 511 512 CHAPTER 16 Wave Motion QUESTIONS
1. Why is a wave pulse traveling on a string considered a
transverse wave?
2. How would you set up a longitudinal wave in a stretched
spring? Would it be possible to set up a transverse wave in
a spring?
3. By what factor would you have to increase the tension in a
taut string to double the wave speed?
4. When traveling on a taut string, does a wave pulse always
invert upon reﬂection? Explain.
5. Can two pulses traveling in opposite directions on the
same string reﬂect from each other? Explain.
6. Does the vertical speed of a segment of a horizontal, taut
string, through which a wave is traveling, depend on the
wave speed?
7. If you were to shake one end of a taut rope periodically
three times each second, what would be the period of the
sinusoidal waves set up in the rope?
8. A vibrating source generates a sinusoidal wave on a string
under constant tension. If the power delivered to the string
is doubled, by what factor does the amplitude change?
Does the wave speed change under these circumstances?
9. Consider a wave traveling on a taut rope. What is the difference, if any, between the speed of the wave and the
speed of a small segment of the rope?
10. If a long rope is hung from a ceiling and waves are sent
up the rope from its lower end, they do not ascend with
constant speed. Explain. 11. What happens to the wavelength of a wave on a string
when the frequency is doubled? Assume that the tension
in the string remains the same.
12. What happens to the speed of a wave on a taut string
when the frequency is doubled? Assume that the tension
in the string remains the same.
13. How do transverse waves differ from longitudinal waves?
14. When all the strings on a guitar are stretched to the same
tension, will the speed of a wave along the more massive
bass strings be faster or slower than the speed of a wave
on the lighter strings?
15. If you stretch a rubber hose and pluck it, you can observe
a pulse traveling up and down the hose. What happens to
the speed of the pulse if you stretch the hose more
tightly? What happens to the speed if you ﬁll the hose
with water?
16. In a longitudinal wave in a spring, the coils move back
and forth in the direction of wave motion. Does the
speed of the wave depend on the maximum speed of
each coil?
17. When two waves interfere, can the amplitude of the resultant wave be greater than either of the two original waves?
Under what conditions?
18. A so...
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 Spring '09
 MIHALISIN
 Physics

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