In wave physics, a standing (stationary) wave is formed under which condition?

Introduction / Context:
Standing waves, also called stationary waves, are an important concept in wave physics. They appear on stretched strings, in air columns of musical instruments, and in microwave cavities. This question checks whether you know the actual physical condition required for a standing wave to form, rather than just any overlap of waves.

Given Data / Assumptions:

• We are dealing with mechanical or electromagnetic waves traveling in a medium or along a string.
• A wave can be reflected from a boundary such as a fixed end or a closed pipe.
• The question asks under what condition a standing wave pattern appears.
• We assume linear superposition is valid, so overlapping waves add algebraically.

Concept / Approach:
A standing wave is produced when two waves of the same frequency, same amplitude, and same type travel in opposite directions and interfere continuously. In practice this usually happens when a traveling wave reflects back on itself from a boundary, and the incident and reflected waves superpose. The nodes (points of zero displacement) and antinodes (points of maximum displacement) appear at fixed positions, giving the impression that the wave is not traveling even though energy is still present in the system.

Step-by-Step Solution:
Step 1: Consider a string fixed at one end. A wave pulse travels along the string and is reflected from the fixed end. Step 2: If a continuous sinusoidal wave is sent, a reflected wave of the same frequency and almost the same amplitude returns in the opposite direction. Step 3: When the incident and reflected waves have the same frequency and amplitude, their superposition creates regions where the displacements always cancel (nodes) and regions where they reinforce (antinodes). Step 4: The pattern of nodes and antinodes remains fixed in space. This is called a standing or stationary wave. Step 5: Therefore, the correct condition is that a traveling wave and its reflection of the same frequency and amplitude superpose.

Verification / Alternative check:
Mathematically, adding two sinusoidal waves of equal amplitude A and angular frequency omega traveling in opposite directions gives a displacement y = 2*A*sin(kx)*cos(omega*t). This expression has a spatial part and a time part separated, with fixed nodes where sin(kx) = 0. This confirms that equal frequency and equal amplitude opposite traveling waves create a standing wave. Simply having a large amplitude or any overlap of different waves does not guarantee such a fixed pattern.

Why Other Options Are Wrong:
The amplitude of a wave becomes greater than its wavelength: Amplitude and wavelength have different dimensions; a large amplitude does not create a standing wave by itself.
Two waves of different frequencies simply cross each other once: Waves of different frequencies produce a changing interference pattern, not a stable standing wave.
The speed of the wave becomes zero or nearly zero: Wave speed is determined by the medium and tension; standing waves do not require the wave speed to vanish.

Common Pitfalls:
Students sometimes think that any interference of waves is a standing wave. In reality, the pattern is only stationary when the overlapping waves have the same frequency and amplitude and travel in opposite directions. Another confusion is to think that standing waves are motionless; in fact, the medium oscillates in place, and energy is still present even though the pattern does not travel.

Final Answer:
A standing wave occurs when a traveling wave and its reflected wave of the same frequency and amplitude superpose.