Question

Consider the following problem of vibration of a string. A tightly stretched string with fixed end points $x=0$ and $x=L$ is initially in a position given by $y(x,0) = f(x)$. It is released from this rest position and allowed to vibrate. The mathematical representation of this problem which depicts the displacement of the string $y(x,t)$ at different times for different values of $x, 0 \le x \le L$ is

• A. \( \frac{\partial y}{\partial x} = C^2 \frac{\partial^2 y}{\partial t^2}, y(x,0)=f(x), y(0,t)=0, y(L,t)=0 \)
• B. \( \frac{\partial y}{\partial t} = C^2 \frac{\partial^2 y}{\partial x^2}, y(0,t)=f(x), y(0,L)=0, y(L,0)=0, (\frac{\partial y}{\partial t})_{t=0}=0 \)
• C. \( \frac{\partial^2 y}{\partial x^2} = C^2 \frac{\partial^2 y}{\partial t^2}, y(x,0)=f(x), (\frac{\partial y}{\partial x})_{x=0}=0, y(0,L)=0 \)
• D. \( \frac{\partial^2 y}{\partial t^2} = C^2 \frac{\partial^2 y}{\partial x^2}, y(x,0)=f(x), y(0,t)=0, y(L,t)=0, (\frac{\partial y}{\partial t})_{t=0}=0 \)

Hint:

• 1D Wave Equation: \(y_{tt} = c^2 y_{xx}\).
• Fixed ends at \(x=0, x=L \implies y(0,t)=0, y(L,t)=0\).
• Initial displacement: \(y(x,0) = f(x)\).
• Released from rest \(\implies\) initial velocity \(y_t(x,0) = 0\).

Answer 1

The Correct Option is D

The problem describes the vibration of a string, which is governed by the one-dimensional wave equation. The standard form of the 1D wave equation for displacement \(y(x,t)\) is: \[ \frac{\partial^2 y}{\partial t^2} = c^2 \frac{\partial^2 y}{\partial x^2} \] where \(c\) is the wave speed (here denoted as C). Boundary Conditions (fixed end points):
Since the string is fixed at \(x=0\) and \(x=L\), the displacement at these points must be zero for all time \(t\): \(y(0,t) = 0\) for \(t \ge 0\) \(y(L,t) = 0\) for \(t \ge 0\)
Initial Conditions: 
1. Initial position (displacement) is given by \(y(x,0) = f(x)\). 
2. The string is "released from this rest position". This means its initial velocity is zero. The initial velocity is \(\frac{\partial y}{\partial t}\) at \(t=0\). 
So, \( \left(\frac{\partial y}{\partial t}\right)_{t=0} = 0 \) for \(0 \le x \le L\). 
Let's check the options: 
(a) Equation is wrong (\(\frac{\partial y}{\partial x}\) vs \(\frac{\partial^2 y}{\partial t^2}\)). 
(b) Equation is of heat equation type (\(\frac{\partial y}{\partial t}\)). Boundary conditions are incorrectly stated. 
(c) Equation form is incorrect (\(\frac{\partial^2 y}{\partial x^2} = C^2 \frac{\partial^2 y}{\partial t^2}\) should be \( \frac{\partial^2 y}{\partial t^2} = C^2 \frac{\partial^2 y}{\partial x^2} \)). 
Boundary conditions are mixed/incorrect. (d) Equation: \( \frac{\partial^2 y}{\partial t^2} = C^2 \frac{\partial^2 y}{\partial x^2} \) (Correct wave equation). Initial position: \(y(x,0)=f(x)\) (Correct). 
Boundary conditions: \(y(0,t)=0, y(L,t)=0\) (Correct for fixed ends). 
Initial velocity: \((\frac{\partial y}{\partial t})_{t=0}=0\) (Correct for release from rest). 
Option (d) correctly represents the mathematical model for this problem.