Complex Variables and Partial Differential Equations (3130005)

Question-5c

BE | Semester-3 Summer-2020 | 27-10-2020

Q5) (c)

→ We know that the solution of wave equation

\frac{\partial^{2} u}{\partial t^{2}} = c^{2} \frac{\partial^{2} u}{\partial x^{2}}

is given by

u (x, t) = \sum_{n = 1}^{\infty} A_{n} \sin \frac{nπx}{L} \cos \frac{nπct}{L} - - - - - [1]

→ Using ,

u (x, 0) = k (\sin x - \sin 2 x)

, we get

u (x, 0) = \sum_{n = 1}^{\infty} A_{n} \sin nx

\Rightarrow k (\sin x - \sin 2 x) = A_{1} \sin x + A_{2} \sin 2 x + A_{3} \sin 3 x + . . .

\Rightarrow A_{1} = k, A_{2} = - k, A_{3} = 0

So, the solution is,

u (x, t) = A_{1} \sin x \cos ct + A_{2} \sin 2 x \cos 2 ct (∵ 1)

u (x, t) = k \sin x \cos t - k \sin 2 x \cos 2 t (∵ c = 1)