Question

Consider the initial value problem for the one-dimensional wave equation Write as clear as Ou Ou ot (4) possible , some work has been hard to follow Thanks! a(z,0) = e-r2 (a) Determine the solution u(r,t) of (4) (b) Sketch the solution in the xu-plane at t = 0, t = 1 , and (c) Which direction does the wave travel? 2

Answer 1

The Auxiliary equation for the given partial differential equation is

${dt\over 1}={dx\over -3}={du\over 0}$ --------------------(1)

From the first two fractions we have

$-3dt=dx \\ \Rightarrow -3t+c_1=x ~~~~~~~~~~\text{on integrating}\\ \Rightarrow c_1=x+3t$

ALso from second and third fractions of (1) we get

$du=0 \\\Rightarrow u=c_2$

hence the general solution of (4) is

$c_2=\phi(c_1) \\\Rightarrow u(x,t)=\phi(x+3t)$

Since

$u(x,0)=e^{-x^2} \\\Rightarrow \phi(x)=e^{-x^2}$

hence $u(x,t)=e^{-(x+3t)^2}$

b) The different sketches are

c) The waves are traveling in negative (left) direction