Question

If two antinodes and three nodes are formed in a distance of \(1.21 \AA\), then the wavelength of the stationary wave is (A) \(2.42 \AA\) (B) \(6.05 \AA\) (C) \(3.63 \AA\) (D) \(1.21 \AA\)

Short answer

The wavelength of the stationary wave is 1.21 Å, which corresponds to option (D). So, the correct answer is (D) \(1.21 \AA\).

Step-by-step solution

Determine the segments between nodes and antinodes

The given information states that there are two antinodes and three nodes within a distance of 1.21 Å. In a stationary wave, an antinode is formed between every pair of adjacent nodes. Therefore, there are two segments between nodes and antinodes. Each of these segments is half a wavelength long.

Calculate the total wavelength

Since we have two segments that are each half a wavelength long, their total length will be equal to one complete wavelength. The total length of these segments is given as 1.21 Å. Therefore, the wavelength of the stationary wave is 1.21 Å.

Identify the correct answer

The wavelength of the stationary wave is 1.21 Å, which corresponds to option (D). So, the correct answer is (D) \(1.21 \AA\).

Key concepts

Antinodes

In a stationary wave, antinodes are points where the wave reaches its maximum amplitude. These points occur due to the constructive interference of the wave's components. When two waves overlap and are perfectly in phase at a certain location, they amplify each other's displacement leading to an antinode. These points appear midway between adjacent nodes where the wave's motion is most vigorous. Antinodes are essential in analyzing and understanding wave patterns since they help in identifying regions of maximum energy transfer.

• Antinodes are located between two nearby nodes.
• They represent the points of maximum vibration in the standing wave.

Cleverly using antinodes, we can visualize and determine properties like wavelengths and the frequency of waves in various applications, ranging from musical instruments to acoustical engineering.

Nodes

Nodes are pivotal locations in a stationary wave where the wave's amplitude is zero. These are points of destructive interference where the wave components cancel each other out perfectly. In the context of stationary waves, nodes remain fixed in position, meaning there is no movement at these points. They form the backbone structure of the wave because the entire wave pattern revolves around nodes and antinodes.

• Nodes occur at the points where two waves overlap in such a way that they cancel each other out.
• There is no movement at nodes, making them points of zero amplitude.

Visualizing nodes helps in determining the segments of a wave cycle and is instrumental in calculating the wave's certain properties such as speed and wavelength. They also play a critical role in understanding wave phenomena in various mediums.

Wavelength Calculation

Wavelength is a fundamental concept in wave physics, which describes the distance over which the wave's shape repeats. It is measured as the distance between two consecutive points in phase, such as from one crest to the next or from one trough to another in the wave. In stationary waves, the challenge often lies in deducing this distance based on known quantities like nodes and antinodes.
In the given problem, with two antinodes and three nodes forming within a distance of 1.21 Å, understanding the wavelength involves linking these points:

• For stationary waves, each segment between a node and an antinode represents half of a wavelength.
• The presence of two antinodes and three nodes signifies that there are two such segments.
• Adding these segments together gives the full wavelength.

Thus, since the total distance of these segments is 1.21 Å, the problem concludes that the complete wavelength of the stationary wave is 1.21 Å. This understanding showcases the importance of geometrically interpreting nodes and antinodes to make precise calculations in wave mechanics.