Nodes and Antinodes: System Closed at Both Ends

Nodes and antinodes are important in physics. While studying standing sound waves, it is essential to calculate the nodes and antinodes formula. This starts with a detailed understanding of standing sound waves associated with nodes and antinodes.

Let us go through the basic behaviour of standing sound waves, nodes, and antinodes. First, we’ll understand the quick nodes and antinodes formula, followed by the nodes and antinodes studied for systems closed at both ends. Further, we will go through the list of questions that help students understand the associated terms and have a detailed explanation of the possible doubts. 

What are standing sound waves?

Standing waves are created when two waves of the same frequency interfere in the same medium while travelling opposite each other. These sound waves have certain points called nodes and antinodes where there is no displacement of the particles. A common example of standing sound waves is those created by stringed musical instruments.

What are nodes and antinodes?

The nodes in the standing waves are the points where the amplitude of the vibration is zero, whereas the antinodes are the points where the amplitude of the vibration is at maximum. A vibrating guitar chord being the wave, its ends can be called nodes, whereas the antinodes lie between two nodes. 

This results in the superposition of two or more progressive waves propagating opposite each other.

What is the antinode formula?

For any standing wave, it is easy to find the nodes and antinodes formula. The formula for the amplitude of a standing wave is 2aSinkx, where x represents the position of nodes or antinodes.

For antinode:

2aSinkx = maximum, which is possible when Sinkx = 1

Sinkx = Sin ((n + ½) )

kx = n + ½

x = n + ½π

Nodes and antinodes for the system closed at both ends

After knowing what nodes and antinodes are, it is easy to study them in closed systems, which is like a closed pipe where the air molecules are not free to move. The displaced standing wave has a node at the closed ends because there is no parallel back and forth motion to the closed tube. The open end of any of the tubes has an antinode, as the molecules can vibrate horizontally parallel to the tube’s length. 

Let us consider a simple system closed at both ends. Let L be the length of the closed tube, and λ is the wavelength. The standing wave, thus, has one antinode and one node. The corresponding standing wave in such a system can be represented by:

L = λ / 4

Thus, λ = 4L

The fundamental frequency of the standing wave is given by-

f1 = v / λ

or

f1 = v / 4L

Hence, only odd-numbered harmonics are observed in any closed system at both ends. The closed system has a node at one end and an antinode at the other. Further, the standing waves in the closed system can be termed as the air pressure along the tube’s length. The pressure at the open end constantly has a node, while the pressure at the closed end varies and has an antinode.

Conclusion

A system closed at both ends has a node on one side and an antinode on the other side. The nodes and antinodes formula helps calculate the standing sound wave’s details in a closed system and concludes that only odd-harmonics occur. No boundary displacement takes place when a rigid body is present. This states that the amplitude is always zero at the boundary, where nodes are formed.