Introduction

Wave equation:

u_tt = a² u_xx

Characteristics:

General solution:

u(x,t) = f₁(x-at) + f₂(x+at)

Here f₁(x-at) is a function that moves to the right with speed a; a 'right-going wave'. And f₂(x-at) is a function that moves to the left with speed a; a 'left-going wave'.

D'Alembert solution:

Assume no boundaries, . Then we can solve for f₁ and f₂ in terms of the given initial string displacement f(x)=u(x,0) and initial velocity g(x)=u_t(x,0) to give: