Briefly explain the difference between travelling waves and standing waves.

III. Long Answer Questions · Q2

Q. Briefly explain the difference between travelling waves and standing waves.

Tamil Nadu DgeTextbookSubjectiveImportance★★★★★est

10% · 7/73 Questions

✓ Free question

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

A string fixed at both ends (like a guitar string) must have nodes at both ends. This forces the string to vibrate only at specific wavelengths:

L = n\frac{\lambda_n}{2} \quad \Rightarrow \quad \lambda_n = \frac{2L}{n}, \quad n = 1, 2, 3, \dots

The corresponding frequencies are:

f_n = \frac{v}{\lambda_n} = \frac{nv}{2L}

These are the natural frequencies or harmonics of the string. The $n=1$ pattern is the fundamental (first harmonic), $n=2$ is the second harmonic, and so on.

Important

A stationary wave on a string fixed at both ends can only exist at these discrete frequencies. Pluck the string at any other frequency, and you get a messy travelling wave that quickly dies out.

The Key Difference from Travelling Waves

In a travelling wave, the phase $\phi = kx - \omega t$ changes with both position and time — the whole waveform moves. In a stationary wave, the phase factor $\cos(\omega t)$ is the same for all points; every point reaches its maximum displacement at the same instant. The wave does not travel — it stands.

Stationary waves on a string are a core NCERT Class 11 Physics topic under Waves, and 'stationary wave equation class 11 physics' or 'standing waves on a string important questions' are frequently searched by board and JEE Main aspirants. The node-antinode framework built here is directly reused in the sonometer and organ-pipe topics later in the same chapter.

Step 1. A travelling (progressive) wave continuously advances forward or backward through the medium at its wave velocity, carrying energy along with it; every particle of the medium vibrates with the same amplitude, only differing in phase.

Step 2. A stationary (standing) wave, formed when two travelling waves of equal amplitude and frequency but moving in opposite directions superpose, does NOT itself advance through the medium -- it remains fixed in place.

Step 3. In a stationary wave, certain fixed points (nodes) never vibrate at all, while other fixed points (antinodes) vibrate with maximum amplitude $2A$ ; all other points vibrate with amplitudes in between, unlike the uniform amplitude of a travelling wave.

Step 4. A travelling wave transports energy continuously forward through the medium, while a stationary wave transports no net energy at all -- energy merely sloshes locally back and forth between neighbouring nodes and antinodes.

✓Final answer

Travelling waves advance through the medium with uniform particle amplitude and carry energy forward; stationary waves are a fixed interference pattern of nodes (zero amplitude) and antinodes (maximum amplitude), formed by two oppositely-directed equal waves, and transport no net energy.

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 form within each segment bounded by nodes (and no net displacement pattern propagates), there is no net transport of energy from one point of the medium to another, unlike in a travelling wave.

✓Final answer
$Stationary (standing) waves do not transfer energy.$
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 wavelength, and antinodes lie exactly midway between consecutive nodes — so consecutive antinodes are also separated by exactly $\lambda/2$ .

✓Final answer
$(B) λ/2.$
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 nodes), a stationary wave does not transfer energy from one point to another, unlike a progressive (travelling) wave, which does carry energy along its direction of propagation.

✓Final answer
$False — energy is not transferred by stationary waves.$
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, adjacent antinodes) are always spaced exactly $\lambda/2$ apart, where $\lambda$ is the wavelength of the component travelling waves.

✓Final answer
$\lambda/2$
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)

Notice the spatial part, cos(kx), is now separated from the time part, sin(ωt) — the wave shape does not travel; instead, every point just oscillates up and down with an amplitude 2A cos(kx) that depends on its fixed position x. Points where cos(kx) = 0 never move (nodes); points where cos(kx) = ±1 oscillate with maximum amplitude (antinodes). Since the pattern stays fixed in space between the boundaries (e.g., a stretched string fixed at both ends), these are called stationary or standing waves.

✓Final answer
$Stationary (standing) waves.$
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.

Adjacent nodes are separated by lambda/2, adjacent antinodes are also separated by lambda/2, and the separation between a node and the next antinode is lambda/4.

✓Final answer
$The correct option is (b) lambda/2 — the distance between two successive antinodes is half the wavelength.$
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 wave. This makes the node's displacement zero at all times, unlike an antinode, which oscillates with maximum amplitude.

✓Final answer
$The displacement of a particle at a node is zero, at all instants of time.$

←Previous question · 11.13Q1 — Discuss how ripples are formed in still water.→Next question · 11.13Q3 — Show that the velocity of a travelling wave produced in a string is v=…

Q.Briefly explain the difference between travelling waves and standing waves.

🎓Unlock everything free for 14 days

Unlock everything free for 14 days