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A formula expressing the solution of the Cauchy problem for the wave equation with one spatial variable. Let the given functions  $  \phi $, 
$  \psi $
belong, respectively, to the spaces  $  C  ^ {2} ( - \infty , + \infty ) $
and  $  C  ^ {1} ( - \infty , + \infty ) $, 
and let  $  f( t, x) $
be continuous together with the first derivative with respect to  $  x $
in the half-plane  $  \{ t \geq  0,  - \infty < x < + \infty \} $. 
Then the classical solution  $  u( t, x) $
in  $  \{ t > 0,  - \infty < x < \infty \} $
of the [[Cauchy problem|Cauchy problem]]

$$ \tag{1 }

\frac{\partial   ^ {2} u ( t, x) }{\partial  t  ^ {2} }
 - a  ^ {2}

\frac{\partial   ^ {2} u ( t, x) }{\partial  x  ^ {2} }
  =  f( t, x),
$$

$$ \tag{2 }
\left . u( t, x) \right | _ {t=} 0  =  \phi ( x), \left . 
\frac{\partial 
u ( t, x) }{\partial  t }
 \right | _ {t=} 0  =  \psi ( x) ,
$$

is expressed by d'Alembert's formula:

$$ 
u( t, x)  =  
\frac{1}{2a}
 \int\limits _ { 0 } ^ { t }   \int\limits _ {x- a( t- \tau ) } ^ { {x+ }  a( t- \tau ) } f ( \tau , \xi )  d \xi  d \tau +
$$

$$ 
+ 

\frac{1}{2a}
 \int\limits _ { x- } at ^ { x+ }  at \psi ( \xi ) \ 
d \xi + 
\frac{1}{2}
 [ \phi ( x+ at) + \phi ( x- at) ] .
$$

If the functions  $  \phi $
and  $  \psi $
are given and satisfy the above smoothness conditions on the interval  $  \{ | x - x _ {0} | < aT \} $, 
and if  $  f( t, x) $
satisfies it in the triangle

$$ 
Q _ {x _ {0}  }  ^ {T}  =  \{ | x - x _ {0} | < a( T- t) ,\ 
t\geq  0 \} ,
$$

then d'Alembert's formula gives the unique solution of the problem (1), (2) in  $  Q _ {x _ {0}  }  ^ {T} $. 
The requirements on the given functions may be weakened if one is interested in solutions in a certain generalized sense. For instance, it follows from d'Alembert's formula that if  $  f $
is integrable with respect to any triangle  $  Q _ {x _ {0}  }  ^ {T} $, 
if  $  \psi $
is locally integrable and if  $  \phi $
is continuous, the weak solution of Cauchy's problem (1), (2) may be defined as a uniform limit (in any  $  Q _ {x _ {0}  }  ^ {T} $) 
of classical solutions with smooth data and is also expressed by d'Alembert's formula.

The formula was named after J. d'Alembert (1747).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.S. Vladimirov,   "Equations of mathematical physics" , MIR  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.N. Tikhonov,   A.A. Samarskii,   "Partial differential equations of mathematical physics" , '''1–2''' , Holden-Day  (1976)  (Translated from Russian)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Courant,   D. Hilbert,   "Methods of mathematical physics. Partial differential equations" , '''2''' , Interscience  (1965)  (Translated from German)</TD></TR></table>