To find which solution to the Helmholz equation with nonzero right-hand side

${\displaystyle \Delta v(x)+k^{2}v(x)=-F(x),\quad x\in \mathbb {R} ^{3},}$ 

with some fixed ${\displaystyle k>0}$ , corresponds to the outgoing waves, one considers the wave equation with the source term,

${\displaystyle (\Delta -\partial _{t}^{2})u(x,t)=-F(x)e^{-ikt},\quad t\geq 0,\quad x\in \mathbb {R} ^{3},}$ 

with zero initial data ${\displaystyle u(x,0)=0,\,\partial _{t}u(x,0)=0}$ . A particular solution to the Helmholtz equation corresponding to outgoing waves is obtained as the limit

${\displaystyle v(x)=\lim _{t\to +\infty }u(x,t)e^{ikt}}$ 

for large times.^[1][3]

• Limiting absorption principle
• Sommerfeld radiation condition