UPSC 2020 Maths Optional Paper 2 Q8a — Step-by-Step Solution

20 marks · Section B

Wave equation · PDEs · asked 7× in 13 yrs · Read the full method →

Question

One end of a tightly stretched flexible thin string of length $l$ is fixed at the origin and the other at $x=l$ . It is plucked at $x=l/3$ so that it assumes initially the shape of a triangle of height $h$ in the $x$ - $y$ plane. Find the displacement $y$ at any distance $x$ and at any time $t$ after the string is released from rest. Take $\frac{\text{horizontal tension}}{\text{mass per unit length}}=c^2$ .

Technique

d’Alembert/Fourier (separation of variables); cosine-in-time modes for the rest condition; half-range sine coefficients of the triangular initial profile.

Solution

Step 1 — Governing equation and boundary/initial conditions

The transverse displacement satisfies the one-dimensional wave equation

\frac{\partial^2 y}{\partial t^2}=c^2\frac{\partial^2 y}{\partial x^2},\qquad 0<x<l,

with

y(0,t)=0,\quad y(l,t)=0\quad(\text{fixed ends}),\qquad \frac{\partial y}{\partial t}(x,0)=0\quad(\text{released from rest}),

and initial shape (triangle, peak $h$ at $x=l/3$ ):

y(x,0)=f(x)= \begin{cases}\dfrac{3h}{l}\,x, & 0\le x\le \dfrac{l}{3},\\[2mm] \dfrac{3h}{2l}\,(l-x), & \dfrac{l}{3}\le x\le l.\end{cases}

(At $x=l/3$ both branches give $h$ .)

Step 2 — Separation of variables

The solution satisfying the fixed-end conditions and zero initial velocity is

y(x,t)=\sum_{n=1}^{\infty} b_n\sin\frac{n\pi x}{l}\cos\frac{n\pi c t}{l},

since each mode $\sin\frac{n\pi x}{l}\cos\frac{n\pi ct}{l}$ has zero velocity at $t=0$ and vanishes at $x=0,l$ .

Step 3 — Fourier coefficients from the initial shape

b_n=\frac{2}{l}\int_0^l f(x)\sin\frac{n\pi x}{l}\,dx =\frac{2}{l}\!\left[\int_0^{l/3}\frac{3h}{l}x\sin\frac{n\pi x}{l}dx+\int_{l/3}^{l}\frac{3h}{2l}(l-x)\sin\frac{n\pi x}{l}dx\right].

Carrying out the integration (integration by parts),

b_n=\frac{9h}{\pi^2 n^2}\sin\frac{n\pi}{3}.

(Equivalently, the standard plucked-string result $b_n=\dfrac{2hl^2}{\pi^2 n^2\,a(l-a)}\sin\dfrac{n\pi a}{l}$ with $a=l/3$ .) Note $\sin\frac{n\pi}{3}=0$ for $n=3,6,9,\dots$ , so every third harmonic is absent.

Step 4 — Displacement

Answer

\boxed{\;y(x,t)=\frac{9h}{\pi^2}\sum_{n=1}^{\infty}\frac{1}{n^2}\sin\frac{n\pi}{3}\,\sin\frac{n\pi x}{l}\,\cos\frac{n\pi c t}{l}.\;}