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)
[a1]	R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , 2 , Interscience (1965) (Translated from German)