(a) A string is stretched between two fixed supports and a note of sound is heard when it is plucked at its middle point.
(i) How standing waves are produced ?
(ii) Derive an expression for the frequency of sound produced by the first two modes of vibration.
(b) Where will a man

Question

(a) A string is stretched between two fixed supports and a note of sound is heard when it is plucked at its middle point.
(i) How standing waves are produced ?
(ii) Derive an expression for the frequency of sound produced by the first two modes of vibration.
(b) Where will a man hear a louder sound in the case of stationary wave (node or anti node) ?

Accepted Answer

>

> [!TLDR]
> A plucked string reflects its wave at both fixed ends; the incident and reflected waves superpose to give a standing wave with fixed nodes at the ends. Only wavelengths λₙ = 2L/n fit this condition, giving discrete frequencies νₙ = nv/2L. The fundamental (n=1) and first overtone (n=2) are ν₁ = v/2L and ν₂ = v/L. Sound is loudest at an antinode.
>
> **(a)(i) How standing waves are produced**
>
> When the string is plucked, a transverse wave travels along it in both directions. At each fixed support the wave is reflected (with a phase reversal, since the support cannot move). The forward-travelling wave and its reflection are two waves of the same amplitude and frequency moving in *opposite* directions on the same string. By the principle of superposition they combine to form a **standing (stationary) wave** — a wave pattern that does not travel, with fixed points of zero displacement (**nodes**) and fixed points of maximum displacement (**antinodes**).
>
> Because the string is clamped at both ends, the displacement at x = 0 and x = L must always be zero — the ends are forced to be nodes. Only those wave patterns that place a node exactly at both ends can exist as a persistent (resonant) standing wave on the string; all other wavelengths destructively interfere with themselves over time and die out.
>
> **(a)(ii) Frequency of the first two modes**
>
> Let the string have length L. A standing wave with p complete loops (each loop = half a wavelength, since a loop runs node-to-node) fits the string when
>
> L = n·(λₙ/2),  n = 1, 2, 3, …
>
> ⟹ λₙ = 2L/n
>
> The wave speed on the string is v = √(T/μ), where T is the tension and μ the mass per unit length. Since v = νλ,
>
> νₙ = v/λₙ = nv/2L
>
> - **First mode (fundamental, n = 1):** one loop (a single antinode at the centre).
>  λ₁ = 2L ⟹ ν₁ = v/2L
>
> - **Second mode (first overtone, n = 2):** two loops, with a node at the midpoint.
>  λ₂ = L ⟹ ν₂ = v/L = 2ν₁
> …

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

Q.(a) A string is stretched between two fixed supports and a note of sound is heard when it is plucked at its middle point.

Concept understanding — Stationary Waves on a String

Stationary Waves on a String

The Intuition: Two Waves, One Pattern

The Precise Statement

What This Equation Tells You

Nodes and Antinodes: The Fixed Pattern

Boundary Conditions: Why Only Certain Patterns Exist …

Q.(a) A string is stretched between two fixed supports and a note of sound is heard when it is plucked at its middle point.

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