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:
   1. The ends of the string must be nodes because they are fixed and cannot move.
   2. The pattern of nodes and antinodes depends on the length of the string and the wavelength of the wave.

Harmonics and Frequencies

Standing waves on a string can form different patterns, called harmonics, depending on the frequency of the wave.

1. First Harmonic:
   1. The simplest pattern, with two nodes and one antinode.
   2. The wavelength is $$\lambda_1 = 2L$$.
2. Second Harmonic:
   1. A more complex pattern, with three nodes and two antinodes.
   2. The wavelength is $$\lambda_2 = L$$.
3. Third Harmonic:
   1. An even more complex pattern, with four nodes and three antinodes.
   2. The wavelength is $$\lambda_3 = \frac{2L}{3}$$.

Standing Waves in Pipes

1. Standing waves can also form in pipes, which are used in wind instruments like flutes and clarinets.
2. The behavior of standing waves in pipes depends on whether the ends of the pipe are open or closed.

Pipes with Both Ends Open

1. Boundary Conditions:
   1. Both ends of the pipe are antinodes because the air molecules at the open ends can oscillate freely.
2. First Harmonic:
   1. The simplest pattern has two antinodes at the ends and one node in the middle.
   2. The wavelength is $\lambda_1 = 2L$.

For the $n$-th harmonics with both ends open, the wavelength and length of the pipe are related as follows:

$$L=\frac{n\lambda}{2}$$

Pipes with One End Closed

1. Boundary Conditions:
   1. The closed end is a node (since the air molecules cannot move), and the open end is an antinode.
2. First Harmonic:
   1. The simplest pattern has one node at the closed end and one antinode at the open end.
   2. The wavelength is four times the length of the pipe: $$
      \lambda_1 = 4L
      $$

For the $n$-th harmonics with one end open and another closed, the wavelength and length of the pipe are related as follows:

$$L = \frac{n\lambda}{4} \, \text{where} \, n \, \text{must be odd.}$$

Why Standing Waves Matter

Standing waves are essential in many real-world applications, particularly in music and engineering.