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In general, when solving a 2nd order PDE (such as the wave equation below) for $$u(x,t), \quad x \in(-\infty,\infty), \: t\in (0,\infty)$$ it should be sufficient to provide initial conditions $u(x,0)$ and $\partial_t u(x,0)$. However when I try solve this numerically in Mathematica 

NDSolve[{D[u[x, t], {t, 2}] == D[u[x, t], {x, 2}], 
u[x, 0] == E^(-x^2), Derivative[0, 1][u][x, 0] == 1}, u, {x, -7, 7}, {t, 0, 4}]

I get the error

NDSolve::bcart: Warning: an insufficient number of boundary conditions 
have been specified for the direction of independent variable x. 
Artificial boundary effects may be present in the solution.

The only thing I can think of is that when solving numerically I have introduced fictitious new boundaries to the problem by simply specifying a finite range {x,-7,7}. If this is the problem, how do I get around it? (NB the real PDE I'm trying to solve is more complicated and does not describe localized waves).

Also, oddly, if I take out the second I.C.

Derivative[0, 1][u][x, 0] == 1

It doesn't complain but it seems to invent its own boundary conditions and gives the wrong solution.

Edit: the actual PDE I'm solving is of the form $$ -\partial_t^2\phi + \sqrt{\frac{2M}{r}} \left( \frac{3}{2r}\partial_t\phi + 2\partial_t \partial_r \phi \right) + \frac{1}{r^2} \partial_r \left[ r (r-2M)\partial_t \phi\right] -m^2\phi =0,$$

with the initial conditions $\phi(r>>1, t) = 10 - t^{1/3} $ and \partial_r\phi(r>>1,t) = 0$. Any tips regarding the best way to get around the issues above in this case would be amazing.

Rudyard
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    "It doesn't complain but it seems to invent its own boundary conditions and gives the wrong solution. " Yes, for more information, check this: https://mathematica.stackexchange.com/q/73961/1871 Then how to circumvent? It really depends. For your specific problem, absorbing boundary condition (ABC) can be used: https://mathematica.stackexchange.com/a/128524/1871 But do notice there's no universal technique for dealing with this type of problem, so, please be more specific if ABC isn't suitable for you. You can also have a look at those posts tagged with [boundary-condition-at-infinity] – xzczd Jun 06 '18 at 15:47
  • Thank you for the links, I'm looking at them now. My specific problem is solving the PDE: $-\partial_t^2\phi + \sqrt{\frac{2M}{r}} \left( \frac{3}{2r}\partial_t\phi + 2\partial_t \partial_r \phi \right) + \frac{1}{r^2} \partial_r \left[ r (r-2M)\partial_t \phi\right] -m^2 =0,$ with the initial conditions $\phi(r>>1, t) = 10 - t^{1/3} $ and $\partial_r\phi(r>>1,t) = 0$ (and $M$ and $m$ are specified numbers). – Rudyard Jun 07 '18 at 16:50
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    Well, these are not initial conditions (i.c.), they're boundary conditions (b.c.) at infinity in $r$ direction. What's the condition in $t$ direction then? Is this an equation already studied in literature? Can you show us some background information? – xzczd Jun 08 '18 at 06:20

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