Derivadas

18
Superposition and Standing Waves
CHAPTER OUTLINE
18.1
18.2
18.3
18.4
18.5
18.6
18.7
18.8

Superposition and Interference
Standing Waves
Standing Waves in a String Fixed
at Both Ends
Resonance
Standing Waves in Air Columns
Standing Waves in Rod and
Membranes
Beats: Interference in Time
Nonsinusoidal Wave Patterns

ANSWERS TO QUESTIONS
Q18.1

No. Waves with all waveforms interfere. Waves withother wave shapes are also trains of disturbance that
add together when waves from different sources move
through the same medium at the same time.

*Q18.2 (i)

(ii)

*Q18.3

Q18.4

If the end is fixed, there is inversion of the pulse
upon reflection. Thus, when they meet, they cancel
and the amplitude is zero. Answer (d).
If the end is free, there is no inversion on
reflection. When they meet, theamplitude is
2 A = 2 ( 0.1 m ) = 0.2 m. Answer (b).

In the starting situation, the waves interfere constructively. When the sliding section is moved
out by 0.1 m, the wave going through it has an extra path length of 0.2 m = λ 4, to show partial
interference. When the slide has come out 0.2 m from the starting configuration, the extra path
length is 0.4 m = λ 2, for destructive interference. Another0.1 m and we are at r2 − r1 = 3λ 4
for partial interference as before. At last, another equal step of sliding and one wave travels one
wavelength farther to interfere constructively. The ranking is then d > a = c > b.
No. The total energy of the pair of waves remains the same. Energy missing from zones of
destructive interference appears in zones of constructive interference.

*Q18.5 Answer (c).The two waves must have slightly different amplitudes at P because of their different
distances, so they cannot cancel each other exactly.
Q18.6

Damping, and non–linear effects in the vibration turn the energy of vibration into internal energy.

*Q18.7 The strings have different linear densities and are stretched to different tensions, so they carry
string waves with different speeds and vibratewith different fundamental frequencies. They are all
equally long, so the string waves have equal wavelengths. They all radiate sound into air, where
the sound moves with the same speed for different sound wavelengths. The answer is (b) and (e).
12

⎛T ⎞
v
, where v = ⎜ ⎟
⎝ μ⎠
2L
1
L−1 will be reduced by a factor . Answer (f ).
2
1
μ −1 2 will be reduced by a factor
. Answer (e).
2

*Q18.8 Thefundamental frequency is described by f1 =
(i)

If L is doubled, then f1

(ii)

If μ is doubled, then f1

(iii) If T is doubled, then f1

T will increase by a factor of 2 . Answer (c).

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*Q18.9

Answer (d). The energy has not disappeared, but is still carried by the wave pulses. Each particle of the string still has kineticenergy. This is similar to the motion of a simple pendulum.
The pendulum does not stop at its equilibrium position during oscillation—likewise the particles of
the string do not stop at the equilibrium position of the string when these two waves superimpose.

*Q18.10 The resultant amplitude is greater than either individual amplitude, wherever the two waves are
nearly enough in phase that 2Acos(φ 2)is greater than A. This condition is satisfied whenever the
absolute value of the phase difference φ between the two waves is less than 120°. Answer (d).
Q18.11

What is needed is a tuning fork—or other pure-tone generator—of the desired frequency. Strike
the tuning fork and pluck the corresponding string on the piano at the same time. If they are precisely in tune, you will hear a single pitch withno amplitude modulation. If the two pitches are a
bit off, you will hear beats. As they vibrate, retune the piano string until the beat frequency goes
to zero.

*Q18.12 The bow string is pulled away from equilibrium and released, similar to the way that a guitar
string is pulled and released when it is plucked. Thus, standing waves will be excited in the bow
string. If the arrow leaves from the...