# D'Alembert formula

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

$$\tag{1 } \frac{\partial ^ {2} u ( t, x) }{\partial t ^ {2} } - a ^ {2} \frac{\partial ^ {2} u ( t, x) }{\partial x ^ {2} } = f( t, x),$$

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

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

 [1] V.S. Vladimirov, "Equations of mathematical physics" , MIR (1984) (Translated from Russian) [2] A.N. Tikhonov, A.A. Samarskii, "Partial differential equations of mathematical physics" , 1–2 , Holden-Day (1976) (Translated from Russian)