The fundamental mode of vibration of a stretched string is shown below.
a) Draw the second and third harmonics. (1)
b) Prove that frequencies produced in the string are in the ratio of 1 : 2 : 3. (2)
c) Let the fundamental frequency is 45 Hz and the length of the wire is 87.5 cm.

Question

The fundamental mode of vibration of a stretched string is shown below.
a) Draw the second and third harmonics. (1)
b) Prove that frequencies produced in the string are in the ratio of 1 : 2 : 3. (2)
c) Let the fundamental frequency is 45 Hz and the length of the wire is 87.5 cm. If linear density of the wire is 4.0 x 10^-2 kg/m. Find the tension in the string. (2)

Accepted Answer

![Figure — Part (a) asks to draw the second and third harmonics of a stretched string fixed at both ends; NCERT Fig 14.13](/figures/waves/first-six-harmonics-vibrations-stretched-string-fixed-fig-14-13.svg)

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> [!TLDR]
> A string fixed at both ends can only support standing waves with nodes at the two fixed ends, which forces the wavelengths (and hence frequencies) into the ratio $1:2:3:\dots$ — this is why plucking a string produces a fundamental plus a full harmonic series. Using $T=\mu(2Lf_1)^2$ with the given numbers, the tension works out to about **248.1 N**.
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> **Setup — the fundamental mode shown in the figure**
>
> $$	ext{Figure given: A stretched string fixed at both ends (shown as hatched supports) vibrating in its fundamental mode: a single symmetric lens/eye-shaped loop bulging between the two fixed ends (one antinode at the centre, nodes at the two ends).}$$
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> This is the **first harmonic (fundamental)**: one loop, node–antinode–node, wavelength $\lambda_1 = 2L$.
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> **a) The second and third harmonics**
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> A drawn diagram cannot be rendered in this text answer, but the exact shape each one takes is fully specified below (this is the standard way these are drawn in the NCERT figure, and is sufficient to reproduce them by hand):
>
> - **Second harmonic ($n=2$):** the string vibrates in **two equal loops** of opposite phase, side by side. There are nodes at both fixed ends **and** one additional node exactly at the midpoint of the string; antinodes sit at the centre of each loop (at $L/4$ and $3L/4$ from one end). Wavelength $\lambda_2 = L$ (one full wavelength fits the string).
> - **Third harmonic ($n=3$):** the string vibrates in **three equal loops**. There are nodes at the two fixed ends plus **two** additional internal nodes (at $L/3$ and $2L/3$ from one end), with antinodes at the centre of each of the three loops. Wavelength $\lambda_3 = 2L/3$.
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> In general, the $n$th harmonic has $n$ loops, $(n+1)$ nodes total (including the two ends),

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

Q.The fundamental mode of vibration of a stretched string is shown below.

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.The fundamental mode of vibration of a stretched string is shown below.

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