Explain how waves interfere in a closed pipe.

In a closed pipe, waves interfere by creating standing waves through the superposition of incident and reflected waves.

In more detail, when a wave is produced in a closed pipe, it travels down the pipe and upon reaching the closed end, it is reflected back. This reflected wave then travels back along the pipe and meets the next incident wave. This meeting of the incident and reflected waves is what we call interference. The type of interference that occurs depends on the phase difference between the two waves. If the waves meet in phase, constructive interference occurs, leading to an increase in amplitude. If they meet out of phase, destructive interference occurs, resulting in a decrease in amplitude or even cancellation of the wave.

The interference of waves in a closed pipe leads to the formation of standing waves. Standing waves are characterised by nodes and antinodes. Nodes are points of complete destructive interference where the displacement of the medium is always zero. Antinodes, on the other hand, are points of maximum displacement resulting from constructive interference. In a closed pipe, there is always a node at the closed end and an antinode at the open end.

The fundamental frequency, or first harmonic, of a closed pipe is when there is only one node-antinode pair along the length of the pipe. This occurs when the length of the pipe is equal to a quarter of the wavelength of the wave. Higher harmonics, or overtones, occur when there are multiple node-antinode pairs along the pipe. These correspond to frequencies that are odd multiples of the fundamental frequency.

The interference of waves in a closed pipe is a fascinating phenomenon that is not only important in understanding the physics of waves, but also has practical applications in musical instruments. For example, the notes produced by wind instruments like the clarinet or the flute are determined by the interference of sound waves within the instrument's pipe.