Mathematical equation: Solution of Wave equation

"Equation Formula"

\ (\large \frac{\partial^2u}{\partial t^2}=a^2\frac{\partial^2u}{\partial x^2}\quad\normalsize (0<x<l, t>0) \)

where \ (a\) is a positive real number.

"Typical boundary conditions"

{Both ends fixed} First class homogeneous boundary condition + First class homogeneous boundary condition

\ (\large \left. u\right|_{x=0}=0\)

\ (\large \left. u\right|_{x=l}=0\)

{One end fixed one end open} First class homogeneous boundary condition + second class homogeneous boundary condition

\ (\large \left. u\right|_{x=0}=0\)

\ (\large \left \frac{\partial u}{\partial x}\right|_{x=l}=0\)

Solution

1. Separating variables

Since the equation and condition are linear, the function to be solved \ (U (x,t) \) can separate the variable

\ (\large u (x,t) =x (x) t (t) \)

You can then adjust both sides of the equation to a formula that contains only one argument, which is generally written as

\ (\large \frac{x "(x)}{x (x)}=\frac{1}{a^2}\frac{t" (t)}{t (t)}=-\lambda\)

2.

Mathematical equation: Solution of Wave equation