Standing Waves: λ, Mode #, Nodes & Antinodes

In summary, it is possible to set up standing waves with a wavelength of 2 meters in a string 8 meters long, but not with a wavelength of 3 meters. In a rope 6 meters long, standing waves with a wavelength of 3 meters can be established in the 2nd mode, with two nodes and a distance of 3 meters between adjacent nodes. In a string 5 cm long, the distance between adjacent nodes in antinodes in the 2nd vibratory mode will be 2.5 cm. The length of the string must be 4 cm for the lowest mode to have a wavelength of 2 cm.
  • #1
amd359
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2.2 Suppose you have a string 8 meters long. Briefly explain why it will be possible to set up standing waves
with λ= 2 meters but NOT with λ= 3 meters.
2.3 Suppose standing waves of wavelength λ= 3 meters are established in a rope 6 meters long.
i. What mode number is this?
ii. Not counting the ends, how many nodes will the standing wave have?
iii. What will be the distance, in meters, between adjacent nodes?
2.3 How long will the wavelength be in the 3rd vibratory mode of a standing wave in string 12 cm long?
2.4 What will be the distance between adjacent nodes in antinodes in the 2nd vibratory mode of a standing
wave in a string 5 cm long?
2.5 Suppose the wavelength of the lowest mode in a standing wave pattern is 2 cm. How long must the string
be?
 
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  • #2
2.1 It will be possible to set up standing waves with λ= 2 meters because the length of the string (8 meters) is an integer multiple of the wavelength (2 meters). However, it will not be possible to set up standing waves with λ= 3 meters because the length of the string (8 meters) is not an integer multiple of the wavelength (3 meters).2.3 i. This is the 2nd mode.ii. There will be two nodes.iii. The distance between adjacent nodes will be 3 meters.2.4 The distance between adjacent nodes in antinodes in the 2nd vibratory mode of a standing wave in a string 5 cm long will be 2.5 cm.2.5 The string must be 4 cm long.
 
  • #3


I can provide a response to the content regarding standing waves and their properties.

2.2 The possibility of setting up standing waves on a string depends on the relationship between the length of the string and the wavelength of the wave. In order to set up standing waves, the length of the string must be a multiple of half the wavelength. Therefore, with a string of 8 meters, it will be possible to set up standing waves with a wavelength of 2 meters (8/2=4, which is a multiple of half the wavelength), but not with a wavelength of 3 meters (8/3=2.67, which is not a multiple of half the wavelength).

2.3 i. The mode number for standing waves is determined by the number of nodes and antinodes present in the wave. In this case, with a wavelength of 3 meters and a string length of 6 meters, the mode number would be 2 (since there would be 2 nodes and 2 antinodes).

ii. Not counting the ends, there would be 4 nodes in the standing wave (2 on each side).

iii. The distance between adjacent nodes would be half the wavelength, so it would be 1.5 meters (3/2=1.5).

2.3 The wavelength in the 3rd vibratory mode of a standing wave on a string 12 cm long would be 4 cm. This is because the length of the string is a multiple of half the wavelength (12/4=3, which is a multiple of half the wavelength).

2.4 In the 2nd vibratory mode of a standing wave on a string 5 cm long, the distance between adjacent nodes and antinodes would be 1.25 cm. This is because the length of the string is a multiple of half the wavelength (5/4=1.25, which is a multiple of half the wavelength).

2.5 The length of the string must be twice the wavelength in order to set up the lowest mode of a standing wave. Therefore, with a wavelength of 2 cm, the string must be at least 4 cm long.
 

1. What are standing waves?

Standing waves are a type of wave that occurs when two waves with the same frequency and amplitude travel in opposite directions and interfere with each other. This results in a stationary pattern of nodes (points of no displacement) and antinodes (points of maximum displacement).

2. What is the significance of the wavelength (λ) in standing waves?

The wavelength of a standing wave is the distance between two consecutive nodes or antinodes. It is directly related to the frequency and speed of the wave, and can be used to calculate these values. In standing waves, the wavelength remains constant while the amplitude varies.

3. What is the mode number in standing waves?

The mode number in standing waves represents the number of nodes or antinodes present in the wave. For example, in the fundamental mode (also known as the first harmonic), there is one node and one antinode. In the second harmonic, there are two nodes and two antinodes, and so on.

4. How are nodes and antinodes related in standing waves?

Nodes and antinodes are always present in pairs in standing waves. Nodes occur at points of no displacement, while antinodes occur at points of maximum displacement. The distance between a node and its adjacent antinode is equal to half the wavelength of the wave.

5. What are some real-world applications of standing waves?

Standing waves have many practical uses, including in musical instruments, such as stringed instruments and wind instruments. They are also used in medical imaging techniques, such as ultrasound, and in industrial applications, such as measuring distances in laser interferometers. Additionally, standing waves are studied in research to better understand the behavior of waves and their properties.

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