Standing waves – problems and solutions
1. The 3-m string tied at one end and the other end is connected to the vibrator. When the vibrator is vibrated, the string formed a stationary wave, as shown in the figure below.
Determine the position of the 5th antinode from the fixed end.
Solution :
Distance between two nodes = 3 meters / 5 = 3/5 meters.
The distance between the first node and the fixed end = 3/5 meters
The distance between the second node and the fixed end = 2 (3/5 meters) = 6/5 meters
The distance between the third node and the fixed end = 3 (3/5 meters) = 9/5 meters
The distance between the fourth node and the fixed end = 4 (3/5 meters) = 12/5 meters
Distance between node and antinode = 1/2 (3/5 meters) = 3/10 meters.
Distance between the fifth antinode and the fixed end = distance between the fourth node and the fixed end + distance between node and antinode = 12/5 + 3/10 = 24/10 + 3/10 = 27/10 = 2.7 meters.
2. As shown in the figure below, one end connected to the vibrator and another end is fixed. If string’s length is 1.5 meters, find the distance between the fourth node and vibrator.
Solution :
Distance between two nodes = 1.5 meters / 11 = 1.5 / 11 meters.
Distance between the first node and vibrator = 1.5 / 11 meters
Distance between the second node and vibrator = 2 (1.5 / 11 meters) = 3/11 meters
Distance between the third node and vibrator = 3 (1.5 / 11 meters) = 4.5 / 11 meters
Distance between the fourth node and vibrator = 4 (1.5 / 11 meters) = 6/11 meters = 0.54 meters
3. A string has both ends kept fixed, produces a fundamental tone with a frequency of 420 Hz. Determine the third overtone.
A. 840 Hz
B. 1260 Hz
C. 1680 Hz
D. 2940 Hz
Known :
The fundamental frequency (f1) = 420 Hz
Both ends are kept fixed.
Wanted : the third overtone
Solution :
First overtone (f2) = 2 f1 = 2 (420 Hz) = 840 Hz
Second overtone (f3) = 3 f1 = 3 (420 Hz) = 1260 Hz
Third overtone (f4) = 4 f1 = 4 (420 Hz) = 1680 Hz
The correct answer is C.
4. The wavelength of the first overtone of a string is 40 cm. If the speed of sound wave in air is 340 m/s, determine the third overtone.
A. 850 Hz
B. 1600 Hz
C. 1700 Hz
D. 3200 Hz
Known :
Wavelength of the first overtone (λ) = 40 cm = 0.4 meters
The speed of the sound wave in air (v) = 340 meters/second
Wanted: frequency of the third overtone
Solution :
Below is a figure of a standing wave on a string with both ends is kept fixed. First, calculate the length of the string using the wavelength of the first overtone. After that, before calculating the frequency of the third overtone, first, calculate the wavelength of the third overtone.
Wavelength of the first overtone :
Length of string (L) = 2. ½ λ
Length of string (L) = λ
Length of string (L) = 0.4 meters
Wavelength of the third overtone :
L = 2 λ
0.4 = 2 λ
λ = 0.4 / 2
λ = 0.2 meters
Frequency of the third overtone :
f = v / λ
f = 340 : 0.2
f = 1700 Hertz
The correct answer is C.
5. A tube open at both ends with length of 40 cm produces a fundamental tone with frequency of 420 Hz. Determine the second overtone.
A. 380 Hz
B. 460 Hz
C. 840 Hz
D. 1260 Hz
Known :
Length of pipe (L) = 40 cm = 0.4 meters
Frequency of the fundamental tone (f1) = 420 Hertz
Wanted: Frequency of the second overtone (f3)
Solution :
If fundamental tone (f1) = 420 Hertz then the second overtone (f3) = 3 f1 = 3 (420 Hertz) = 1260 Hertz
The correct answer is D.
6. Sound wave at closed tube has a wave pattern similar to…
A. Wave propagation on a string
B. Wave propagation on-the air column
C. Standing wave on a string fixed at one end
D. Standing wave on a string fixed at both ends
Solution :
A closed tube is a tube that is open at one end but closed at other, as shown in the figure above.
The correct answer is C.
20 conceptual questions and answers related to standing waves:
1. Question: What is a standing wave?
Answer: A standing wave is a wave pattern that appears to remain stationary, with nodes and antinodes, resulting from the interference of two waves traveling in opposite directions.
2. Question: How are nodes and antinodes different?
Answer: Nodes are points of zero amplitude, where the wave remains at rest, while antinodes are points of maximum amplitude.
3. Question: Can standing waves form in any medium?
Answer: Standing waves can form in any medium that allows wave propagation, such as strings, air columns, and water.
4. Question: How is a standing wave’s frequency related to its harmonic number?
Answer: The fundamental frequency (first harmonic) is the lowest frequency of a standing wave. Higher harmonics have frequencies that are integer multiples of the fundamental frequency.
5. Question: What determines the locations of nodes in a standing wave?
Answer: Nodes are formed where waves traveling in opposite directions destructively interfere, canceling each other out.
6. Question: How are standing waves created on a string fixed at both ends?
Answer: When a wave traveling down a string reflects at the fixed end, it interferes with incoming waves, creating a standing wave pattern if the frequencies match certain conditions.
7. Question: What’s the relationship between wavelength and the length of the medium in a standing wave?
Answer: For a string fixed at both ends, the length of the medium is an integer multiple of half the wavelength of the standing wave.
8. Question: Why don’t standing waves transfer energy across the medium?
Answer: While individual particles in the medium oscillate, the overall wave pattern remains stationary, so there’s no net energy transport in any particular direction.
9. Question: Can you observe standing waves in open tubes?
Answer: Yes, standing waves can be formed in open tubes, but the boundary conditions differ from closed tubes, affecting node and antinode positions.
10. Question: What happens to the standing wave when the tension in a string is increased?
Answer: Increasing the tension increases the wave speed, which can change the frequency and the pattern of the standing wave.
11. Question: How is resonance related to standing waves?
Answer: Resonance occurs when an external force or vibration matches a system’s natural frequency, creating a pronounced standing wave.
12. Question: What is the fundamental frequency?
Answer: The fundamental frequency, or first harmonic, is the lowest frequency at which a system can support a standing wave.
13. Question: How are overtones related to harmonics?
Answer: Overtones are frequencies above the fundamental frequency. The first overtone corresponds to the second harmonic, the second overtone to the third harmonic, and so on.
14. Question: Why is there no displacement at nodes?
Answer: At nodes, the two interfering waves are out of phase by 180°, resulting in destructive interference and zero displacement.
15. Question: Can standing waves be polarized?
Answer: Standing waves on a string are transverse and thus have a direction of oscillation (polarization). However, in a medium like air, standing sound waves are longitudinal and don’t exhibit polarization.
16. Question: Why do musical instruments use the principle of standing waves?
Answer: Musical instruments often produce sound by creating standing waves, with different harmonics producing different musical notes.
17. Question: How is wave speed related to the frequency and wavelength of a standing wave?
Answer: Wave speed (v) is the product of frequency (f) and wavelength (λ): v = f x λ.
18. Question: Can two different frequencies produce standing waves in the same medium?
Answer: Yes, as long as the frequencies correspond to the conditions for standing waves in that medium. Each frequency represents a different harmonic.
19. Question: How do standing waves in closed tubes differ from those in open tubes?
Answer: In closed tubes, one end has a node while the other end has an antinode. In open tubes, both ends have antinodes.
20. Question: Can standing waves be formed with any frequency in a given medium?
Answer: No, only specific frequencies that meet the boundary conditions of the medium will produce standing waves.
Understanding standing waves is crucial in various fields, from music to telecommunications, as they represent fundamental wave behaviors under specific conditions.
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