Q.Show that a stretched string vibrates with all harmonics.
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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.
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:
y1=Asin(kx−ωt)
y2=Asin(kx+ωt)
By the principle of superposition, the resultant displacement is:
y=y1+y2=Asin(kx−ωt)+Asin(kx+ωt)
Using the identity sinP+sinQ=2sin2P+Qcos2P−Q, we get:
y=2Asin(kx)cos(ωt)
This is the equation of a stationary wave.
y(x,t)=2Asin(kx)cos(ωt)
What This Equation Tells You
The factor sin(kx) depends only on position x. The factor cos(ωt) depends only on time t. This means:
- Every point on the string oscillates with the same frequency ω, 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=nπ, i.e. x=n2λ, for n=0,1,2,…
- Points where ∣sin(kx)∣=1 are antinodes — they oscillate with maximum amplitude 2A. This happens when kx=(n+21)π, i.e. x=(n+21)2λ.
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 λ/2. The distance between a node and the next antinode is λ/4.
| Feature | Condition | Position |
|---|---|---|
| Node | sin(kx)=0 | x=n2λ |
| Antinode | $ | \sin(kx) |
Boundary Conditions: Why Only Certain Patterns Exist …
Since f_n = n f1 for every positive integer n (1st, 2nd, 3rd, ... harmonics all allowed, both even and odd), a stretched string vibrates with all harmonics of the fundamental — unlike an air column closed at one end, which supports only odd harmonics. …
[!TLDR]
A string of length L fixed at both ends supports standing waves with nodes at both ends; the allowed wavelengths are lambda_n = 2L/n (n=1,2,3,...), giving frequencies f_n = n v / (2L) = n * f1, where f1 = v/2L is the fundamental frequency.
Method …
- CBSE 2026Set ANNUAL1 markQ.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 …
- CBSE 2025Set ANNUAL1 markMCQQ.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 …
- CBSE 2025Set ANNUAL1 markQ.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 …
- CBSE 2024Set ANNUAL1 markMCQQ.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 λ/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 …
- CBSE 2024Set SET-AP55001 markQ.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)
…
- CBSE 2024Set ANNUAL1 markMCQQ.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.
…
- CBSE 2022Set ANN1 markQ.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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