What are stationary waves? Explain the formation of stationary waves and also write down the characteristics o

III. Long Answer Questions · Q9

Q. What are stationary waves? Explain the formation of stationary waves and also write down the characteristics of stationary waves.

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Concept understanding — Stationary Waves on a String

Stationary Waves on a String

Imagine holding a skipping rope at one end while your friend holds the other. If you both flick your wrists at the same time, two identical pulses travel toward each other. When they meet, they pass through each other — but for an instant, the rope's shape is the sum of both pulses. Now imagine doing this continuously, sending wave after wave from both ends.

What happens when the timing is just right? The rope doesn't seem to travel anywhere. Instead, it vibrates in place, with some points completely still and others moving wildly. That's a stationary (or standing) wave.

The Intuition: Two Waves, One Pattern

A travelling wave on a string moves energy from one end to the other. But if you send a wave down the string and it reflects back from a fixed end, the original wave and the reflected wave are identical in amplitude and frequency but travel in opposite directions. Where they overlap, they interfere.

At certain points, the two waves always cancel each other out — those points never move. They are called nodes. At other points, the waves always add up constructively, giving maximum displacement — those are antinodes. The pattern is fixed in space; the wave does not travel. Hence the name: stationary wave.

Note

A stationary wave is not a new kind of wave. It is the superposition of two identical travelling waves moving in opposite directions. No net energy flows past a node.

The Precise Statement

Let two identical waves travel along a string in opposite directions. Their displacements are:

y_1 = A \sin(kx - \omega t)

y_2 = A \sin(kx + \omega t)

By the principle of superposition, the resultant displacement is:

y = y_1 + y_2 = A \sin(kx - \omega t) + A \sin(kx + \omega t)

Using the identity $\sin P + \sin Q = 2 \sin\frac{P+Q}{2} \cos\frac{P-Q}{2}$ , we get:

y = 2A \sin(kx) \cos(\omega t)

This is the equation of a stationary wave.

y(x,t) = 2A \sin(kx) \cos(\omega t)

What This Equation Tells You

The factor $\sin(kx)$ depends only on position $x$ . The factor $\cos(\omega t)$ depends only on time $t$ . This means:

Every point on the string oscillates with the same frequency $\omega$ , but with an amplitude that varies with position: $2A |\sin(kx)|$ .
Points where $\sin(kx) = 0$ are nodes — they never move. This happens when $kx = n\pi$ , i.e. $x = n\frac{\lambda}{2}$ , for $n = 0, 1, 2, \dots$
Points where $|\sin(kx)| = 1$ are antinodes — they oscillate with maximum amplitude $2A$ . This happens when $kx = (n+\frac12)\pi$ , i.e. $x = (n+\frac12)\frac{\lambda}{2}$ .

Watch out

A common mistake is to think that the string is stationary everywhere. It is not — only the nodes are stationary. The rest of the string oscillates, but the pattern of nodes and antinodes does not move along the string.

Nodes and Antinodes: The Fixed Pattern

The distance between two consecutive nodes (or two consecutive antinodes) is $\lambda/2$ . The distance between a node and the next antinode is $\lambda/4$ .

Feature	Condition	Position
Node	$\sin(kx)=0$	$x = n\frac{\lambda}{2}$
Antinode	$	\sin(kx)

Boundary Conditions: Why Only Certain Patterns Exist …

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Step 1. A wave reflecting off a rigid boundary interferes with the still-incoming incident wave. Take two equal-amplitude, equal-frequency waves travelling in opposite directions: $y_1=A\sin(kx-\omega t)$ (rightward) and $y_2=A\sin(kx+\omega t)$ (leftward).

Step 2. Superposing them, $y=y_1+y_2$ , and simplifying using trigonometric identities gives $y(x,t)=2A\cos(\omega t)\sin(kx)$ .

Step 3. Writing this as $y=A'\cos(\omega t)$ with $A'=2A\sin(kx)$ shows every point still oscillates in SHM, but with a position-dependent amplitude $A'$ , not the same amplitude everywhere.

Step 4. This pattern does not itself travel forward or backward -- it stays fixed in place -- which is why it is called a stationary (standing) wave. Antinodes (where $A'=2A$ is maximum) form at $x=(2m+1)\lambda/4$ ; nodes (where $A'=0$ ) form at $x=n\lambda/2$ .

Step 5. Characteristics: (i) confined between rigid boundaries, does not advance;

(ii) fixed nodes and antinodes, unlike the uniform amplitude of a travelling wave; …

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Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

CBSE 2026Set ANNUAL1 mark
$Q. Fill in the blank: ____ waves does not transfer energy.$
›Reveal solutionSolution

Stationary (standing) waves do not transfer energy through the medium, unlike travelling (progressive) waves.

A stationary wave is produced by the superposition of two progressive waves of the same amplitude and frequency travelling in opposite directions (e.g., an incident wave and its reflection). The resulting pattern has fixed points called nodes, where the displacement is always zero, and antinodes, where the amplitude is maximum — but the wave pattern as a whole does not travel through the medium. Because energy is confined to oscillate back and forth between kinetic and potential …

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CBSE 2025Set ANNUAL1 markMCQ
$Q. The distance between two consecutive antinodes of a stationary wave is (A) λ (B) λ/2 (C) 2λ (D) 3λ/2$
›Reveal solutionSolution

Consecutive antinodes in a stationary wave are separated by λ/2.

A stationary wave is formed by the superposition of two identical waves travelling in opposite directions. Its displacement pattern shows fixed points of zero amplitude (nodes) and fixed points of maximum amplitude (antinodes), alternating along the medium. Nodes occur at every half wavele …

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CBSE 2025Set ANNUAL1 mark
$Q. State True or False: By stationary waves, energy can be transferred.$
›Reveal solutionSolution

The statement is False: stationary waves do not transfer energy across the medium.

A stationary wave is formed by the superposition of two identical progressive waves travelling in opposite directions. In a stationary wave, certain points (nodes) remain permanently at rest, while others (antinodes) oscillate with maximum amplitude, but the wave pattern itself does not move forward. Because there is no net transport of the disturbance from one region to another (energy simply oscillates back and forth between kinetic and potential form within each segment between …

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CBSE 2024Set ANNUAL1 markMCQ
$Q. In a stationary wave of wavelength λ, the distance between two consecutive nodes is (A) 2λ (B) λ/2 (C) λ (D) λ/4$
›Reveal solutionSolution

Consecutive nodes in a stationary wave are $\lambda/2$ apart.

A stationary wave forms from the superposition of two identical waves travelling in opposite directions. Its displacement pattern has fixed points of zero amplitude (nodes) and maximum-amplitude points (antinodes) alternating along its length. Analysing the resulting waveform shows adjacent nodes (and, separately …

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CBSE 2024Set SET-AP55001 mark
$Q. Waves that do not travel forward in a medium but remain stationary between two boundaries of the medium are called ________ waves.$
›Reveal solutionSolution

Stationary (standing) waves are formed by the superposition of two identical waves travelling in opposite directions in the same medium; the resulting pattern does not propagate — it has fixed nodes (zero displacement) and antinodes (maximum displacement).

When a wave, say y1 = A sin(ωt − kx), and its reflection travelling the opposite way, y2 = A sin(ωt + kx), superpose, the resultant is:

y = y1 + y2 = 2A cos(kx) sin(ωt)

…

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CBSE 2024Set ANNUAL1 markMCQ
$Q. In a stationary (standing) wave, the separation between two successive antinodes is:$
(a) lambda/4
(b) lambda/2
(c) lambda
(d) None of these
›Reveal solutionSolution

Successive antinodes in a stationary wave are separated by lambda/2 (half a wavelength).

In a stationary (standing) wave formed by superposition of two identical waves travelling in opposite directions, nodes (points of zero displacement) and antinodes (points of maximum displacement) occur alternately along the medium.

…

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CBSE 2022Set ANN1 mark
$Q. In the case of stationary waves, the displacement of a particle at positions of node is ________.$
›Reveal solutionSolution

A node is a point of a stationary wave that is permanently at rest, so its displacement is always zero.

A stationary (standing) wave is formed by the superposition of two identical waves travelling in opposite directions. At a node, the displacement contributions of the two component waves are always equal and opposite, so they cancel completely at every instant of time — not just momentarily, as happens for a particle passing through zero displacement in a travelling wav …

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Q.What are stationary waves? Explain the formation of stationary waves and also write down the characteristics of stationary waves.

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