Chapter 18

Superposition and Standing Waves

Quick Quiz 18.1

Two pulses are traveling toward each other at 10 cm/s on a long string, as shown in the figure below. Sketch the shape of the string at t = 0.6 s.

Answer: The shape of the string at t = 0.6 s is shown below.

Quick Quiz 18.1

Quick Quiz 18.2

Two pulses move in opposite directions on a string and are identical in shape except that one has positive displacements of the elements of the string and the other has negative displacements. At the moment that the two pulses completely overlap on the string,

(a) the energy associated with the pulses has disappeared

(b) the string is not moving

(c) the string forms a straight line

(d) the pulses have vanished and will not reappear

Answer: (c). The pulses completely cancel each other in terms of displacement of elements of the string from equilibrium, but the string is still moving. A short time later, the string will be displaced again and the pulses will have passed each other.

Quick Quiz 18.2

Quick Quiz 18.3

Consider a standing wave on a string as shown in the figure below. Define the velocity of elements of the string as positive if they are moving upward in the figure. At the moment the string has the shape shown at the bottom of part (a), the instantaneous velocity of elements along the string

(a) is zero for all elements (b) is positive for all elements

(c) is negative for all elements (d) varies with the position of the element

Answer: (a). The pattern shown at the bottom of Figure 18.9a corresponds to the extreme position of the string. All elements of the string have momentarily come to rest.

Quick Quiz 18.3

Quick Quiz 18.4

Continuing with the scenario in question 3, at the moment the string has the shape shown at the bottom of part b of the figure above, the instantaneous velocity of elements along the string

(a) is zero for all elements

(b) is positive for all elements

(c) is negative for all elements

(d) varies with the position of the element

Answer: (d). Near a nodal point, elements on one side of the point are moving upward at this instant and elements on the other side are moving downward.

Quick Quiz 18.4

Quick Quiz 18.5

When a standing wave is set up on a string fixed at both ends,

(a) the number of nodes is equal to the number of antinodes

(b) the wavelength is equal to the length of the string divided by an integer

(c) the frequency is equal to the number of nodes times the fundamental frequency

(d) the shape of the string at any time is symmetric about the midpoint of the string.

Answer: (d). Choice (a) is incorrect because the number of nodes is one greater than the number of antinodes. Choice (b) is only true for half of the modes; it is not true for any odd-numbered mode. Choice (c) would be correct if we replace the word nodes with antinodes.

Quick Quiz 18.5

Quick Quiz 18.6

A wine glass can be shattered through resonance by maintaining a certain frequency of a high-intensity sound wave. The figure below shows a side view of a wine glass vibrating in response to such a sound wave. If an integral number of waves "fit" around the circumference of the vibrating rim, how many wavelengths fit around the rim in part (a)?

(a) 1

(b) 2

(c) 3

(d) 4

(e) 5

Courtesy of P

rofessor Thom

as D. R

ossing, Northern Illinois

University

Answer: For each natural frequency of the glass, the standing wave must “fit” exactly around the rim. In Figure 18.17a we see three antinodes on the near side of the glass, and thus there must be another three on the far side. This corresponds to three complete waves. In a top view, the wave pattern looks like this (although we have greatly exaggerated the amplitude):

Quick Quiz 18.6

Quick Quiz 18.7

A pipe open at both ends resonates at a fundamental frequency fopen. When one end is covered and the pipe is

again made to resonate, the fundamental frequency is fclosed.

Which of the following expressions describes how these two resonant frequencies compare?

(a) fclosed = fopen

(b) fclosed = 1/2 fopen

(c) fclosed = 2 fopen

(d) fclosed = 3/2 fopen

Answer: (b). With both ends open, the pipe has a fundamental frequency given by Equation 18.11: fopen = v / 2L. With one end closed, the pipe has a fundamental frequency given by Equation 18.12:

Quick Quiz 18.7

openclosed fL

v

L

vf 2

122

14

Quick Quiz 18.8

Balboa Park in San Diego has an outdoor organ. When the air temperature increases, the fundamental frequency of one of the organ pipes

(a) stays the same

(b) goes down

(c) goes up

(d) is impossible to determine

Answer: (c). The increase in temperature causes the speed of sound to go up. According to Equation 18.11, this will result in an increase in the fundamental frequency of a given organ pipe.

Quick Quiz 18.8

Quick Quiz 18.9

You are tuning a guitar by comparing the sound of the string with that of a standard tuning fork. You notice a beat frequency of 5 Hz when both sounds are present. You tighten the guitar string and the beat frequency rises to 8 Hz. In order to tune the string exactly to the tuning fork, you should

(a) continue to tighten the string

(b) loosen the string

(c) impossible to determine

Answer: (b). Tightening the string has caused the frequencies to be farther apart, based on the increase in the beat frequency.

Quick Quiz 18.9