D’Alembert’s Solution for Wave Equation

D’Alembert’s Solution for Wave Equation
In this lecture we discuss the one dimensional wave equation. We review some of the physical situations in which the wave equations describe the dynamics of the physical system.
Suppose we have the wave equation u_tt-c^2 u_xx=0 and we wish to solve it given the conditions u(x,0)=F(x) and u_t (x,0)=G(x).
We change variables to r=x+ct and s=x-ct. The general solution of this PDE is u(x,t)=F(x+ct )+G(x-ct)
Example 1: Solve the initial value problem u_tt-4u_xx=0
with u(x,0)=0 and u_t (x,0)=tan?x
Solution : u(x,t)=F(x+2t )+G(x-2t)
u(x,0)=0 ? 0=F(x+0 )+G(x-0)
F(x )+G(x)=0 ?(1)
u_t (x,t)=2 F^ (x+2t )-2G^ (x-2t)
u_t (x,0)=tan?x ? 2 F^ (x)-2G^ (x)=tan?x
2 F(x)-2G(x)=-ln?cos?(x) ? (2)
2×equ(1)+equ(2) ? 4 F(x)=-ln?cos?(x)
F(x)=-( 1 )/( 4 ) ln?cos?(x) and G(x)=( 1 )/( 4 ) ln?cos?(x)
So F(x+2t )=-( 1 )/( 4 ) ln?cos?(x+2t ) and G(x-2t )=( 1 )/( 4 ) ln?cos?(x-2t )
Then u(x,t)=( 1 )/( 4 ) ln?cos?(x-2t ) -( 1 )/( 4 ) ln?cos?(x+2t )
Or u(x,t)=( 1 )/( 4 ) ln??cos?(x-2t )/cos?(x+2t ) ?





Example 2: Solve the initial value problem u_tt-a^2 u_xx=0
with u(x,0)=0 and u_t (x,0)=( 1 )/(1+x^2 )
Solution :
u(x,t)=F(x+at )+G(x-at)
u(x,0)=0 ? 0=F(x+0 )+G(x-0)
F(x )+G(x)=0 ?(1)
u_t (x,t)=a F^ (x+at )-aG^ (x-at)
u_t (x,0)=( 1 )/(1+x^2 ) ? aF^ (x)-aG^ (x)=( 1 )/(1+x^2 )
aF(x)-aG(x)=tan^(-1)?x ? (2)
a×equ(1)+equ(2) ? 2a F(x)=tan^(-1)?x
F(x)=( 1 )/( 2a ) tan^(-1)?x and G(x)=-( 1 )/( 2a ) tan^(-1)?x
So F(x+at )=( 1 )/( 2a ) tan^(-1)?(x+at ) and G(x-at )=( 1 )/( 2a ) tan^(-1)?(x-at )
Then u(x,t)=( 1 )/( 2a ) tan^(-1)?(x+at )-( 1 )/( 2a ) tan^(-1)?(x-at )


H.W: Solve the initial value problems
1. u_tt-a^2 u_xx=0 with u(x,0)=0 and u_t (x,0)=sin?x
2. u_tt-9u_xx=0 with u(x,0)=0 and u_t (x,0)=e^2x