Stationary waves

Stationary waves (also called standing waves) are formed when two waves of the same frequency, amplitude, and speed travel in opposite directions and superpose. Unlike progressive (travelling) waves, stationary waves do not transfer energy from one place to another.

Principle of Superposition

When two or more waves meet at a point, the resultant displacement at that point is the algebraic sum of the displacements due to each wave.

Formation of Stationary Waves

• Stationary waves are produced by the superposition of two identical waves travelling in opposite directions.
• This can occur, for example, when a wave is reflected back along its path (e.g., on a stretched string fixed at both ends, or in an air column).

Graphical Representation

• At certain points, the two waves always cancel each other out (destructive interference), producing nodes (points of zero amplitude).
• At other points, the two waves always reinforce each other (constructive interference), producing antinodes (points of maximum amplitude).

Nodes and Antinodes

• Node: Point where the displacement is always zero.
• Antinode: Point where the displacement is maximum.

The distance between two adjacent nodes (or two adjacent antinodes) is half a wavelength ($\frac{\lambda}{2}$).

Experiments Demonstrating Stationary Waves

(a) Stretched String

• A string fixed at both ends can be made to vibrate using a mechanical oscillator.
• Stationary waves are formed, with nodes at the fixed ends.
• The pattern depends on the frequency and length of the string.

(b) Air Columns

• Air columns in tubes (e.g., organ pipes) can form stationary waves.
• For a tube closed at one end, a node forms at the closed end and an antinode at the open end.
• For a tube open at both ends, antinodes form at both ends.

(c) Microwaves

• Microwaves reflected from a metal plate can interfere with the incident waves to form stationary waves.
• A detector moved along the path detects points of minimum and maximum intensity (nodes and antinodes).

Determining Wavelength from Nodes and Antinodes

• Measure the distance between several consecutive nodes (or antinodes).
• The distance between two adjacent nodes (or antinodes) is $\frac{\lambda}{2}$.
• The wavelength $\lambda$ can be calculated as:

Summary Table: Stationary vs Progressive Waves

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Stationary waves are a key example of the superposition principle in action, and understanding their formation and properties is essential for further studies in wave physics.