Question

using D' Alembert method, find the deflection of a vibrating string of unit length having fixed ends, with initial velocity zero and initial deflection f(x)=asin²nx

Answer 1

Solution:

1-D wave equation can be written as:

$\dfrac{\partial^{2} u}{\partial t^{2}}=c^{2} \dfrac{\partial^{2} u}{\partial x^{2}}$

Here $f(x)=a\sin^2nx$

Using $D^{\prime}$ Alembert's method, the solution can be given as:

$y(x,t)=\dfrac12[f(x+ct)+f(x-ct)] \\=\dfrac12[a\sin^2n(x+ct)+a\sin^2n(x-ct)] \\=\dfrac a2[\sin^2(nx+nct)+\sin^2(nx-nct)]$

$=\dfrac a2[\dfrac{1-\cos 2(nx+nct)}{2}+\dfrac{1-\cos 2(nx-nct)}{2}]$ [Using $\sin^2t=\dfrac{1-\cos 2t}2$ ]

$=\dfrac a4[{2-\cos 2(nx+nct)}-\cos 2(nx-nct)] \\=\dfrac a4[{2-2\cos (\dfrac{2(nx+nct)-2(nx-nct)}{2}}) \cos (\dfrac{2(nx+nct)+2(nx-nct)}{2})]$

$=\dfrac a4[2-2\cos (nct) \cos (nx)] \\=\dfrac a2[1-\cos (nct) \cos (nx)]$