C.4.1 Formation and properties of standing waves Notes

Formation of Standing Waves through Superposition

To understand this, let's break down the process:

1. Two identical waves:
   1. Imagine two waves with the same speed, wavelength, and amplitude.
   2. One wave travels to the right, while the other travels to the left.
2. Superposition:
   1. When these waves meet, they overlap and combine according to the principle of superposition.
   2. This means the displacement of the resulting wave is the sum of the displacements of the two individual waves.
3. Interference:
   1. The waves interfere with each other, creating regions of constructive and destructive interference.
   2. In constructive interference, the displacements add up, resulting in a larger amplitude.
   3. In destructive interference, the displacements cancel each other out, resulting in zero displacement.

Nodes and Antinodes

1. In a standing wave, certain points remain fixed while others oscillate with maximum amplitude.
2. These points are called nodes and antinodes.

Nodes: Points of Zero Displacement

This occurs because of destructive interference between the two waves.

Antinodes: Points of Maximum Displacement

This occurs due to constructive interference, where the crests or troughs of the two waves align.

Relationship Between Nodes and Antinodes

1. Distance: The distance between two consecutive nodes (or two consecutive antinodes) is half a wavelength ($\frac{\lambda}{2}$).
2. Position: Antinodes are always located halfway between two nodes.

Characteristics of Standing Waves

Standing waves have several unique features that distinguish them from traveling waves:

1. No Energy Transfer:
   1. Standing waves do not transfer energy along the medium.
   2. The energy is confined between the nodes and antinodes.
2. Fixed Pattern:
   1. The pattern of nodes and antinodes remains stationary, even though the particles of the medium oscillate.
3. Varying Amplitude:
   1. The amplitude of oscillation varies along the wave.
   2. It is zero at the nodes and maximum at the antinodes.

Standing Waves on Strings

Standing waves can be observed on strings with fixed ends, such as those in musical instruments.

1. Fixed Ends:
   1. When a wave reaches a fixed end, it reflects and travels back in the opposite direction.
   2. This creates two identical waves moving in opposite directions on the string.
2. Boundary Conditions: