Question

Use d'Alembert's formula to solve the wave equation c²∂x²/∂2u = ∂t²/∂2u −[infinity]0 with the following and initial conditions u(x,0)=xsin(x), uₑ(x,0)=cos(2x)

Answer 1

The question pertains to solving a wave equation using d'Alembert's formula, where the initial displacement and velocity are given as functions of position.

Step-by-step explanation:

The question is about using d'Alembert's formula to solve the wave equation with given initial conditions. D'Alembert's formula, which applies to one-dimensional wave equations, allows for the solution to be expressed in terms of two arbitrary functions that represent the initial displacement and velocity of the wave.

Given the initial conditions u(x,0) = x sin(x) and u_t(x,0) = cos(2x), the formula enables us to write the solution as u(x,t) = 0.5[ f(x+ct) + f(x-ct) ] + (1/2c) ∫ g(s)ds, where f(x) represents the initial displacement and g(x) the initial velocity distribution. The functions f and g need to be determined from the initial conditions provided.