D'Alembert's Formula

In mathematics, and specifically partial differential equations (PDEs), d´Alembert's formula is the general solution to the one-dimensional wave equation:
:$u_-c^2u_=0,\, u(x,0)=g(x),\, u_t(x,0)=h(x),$
for $-\infty < x<\infty,\,\, t>0$

It is named after the mathematician Jean le Rond d'Alembert, who derived it in 1747 as a solution to the problem of a vibrating string.

Details

The characteristics of the PDE are $x \pm ct = \mathrm$ (where $\pm$ sign states the two solutions to quadratic equation), so we can use the change of variables $\mu = x + ct$ (for the positive solution) and $\eta = x-ct$ (for the negative solution) to transform the PDE to $u_ = 0$. The general solution of this PDE is $u(\mu,\eta) = F(\mu) + G(\eta)$ where $F$ and $G$ are $C^1$ functions. Back in $x, t$ coordinates,

:$u(x,t) = F(x+ct) + G(x-ct)$
:$u$ is $C^2$ if $F$ and $G$ are $C^2$.

This solution $u$ can be interpreted as two waves with constant velocity $c$ moving in opposite directions along the x-axis.

Now consider this solution with the Cauchy data $u(x,0)=g(x), u_t(x,0)=h(x)$.

Using $u(x,0) = g(x)$ we get $F(x) + G(x) = g(x)$.

Using $u_t(x,0) = h(x)$ we get $cF'(x)-cG'(x) = h(x)$.

We can integrate the last equation to get
$cF(x)-cG(x)=\int_^x h(\xi) \, d\xi + c_1.$

Now we can solve this system of equations to get
$F(x) = \frac\left(-cg(x)-\left(\int_^x h(\xi) \, d\xi +c_1 \right)\right)$
$G(x) = \frac\left(-cg(x)+\left(\int_^x h(\xi) d\xi +c_1 \right)\right).$

Now, using $u(x,t) = F(x+ct)+G(x-ct)$

d'Alembert's formula becomes:
u(x,t) = \frac\left (x-ct) + g(x+ct)\right+ \frac \int_^ h(\xi) \, d\xi.

Generalization for inhomogeneous canonical hyperbolic differential equations

The general form of an inhomogeneous canonical hyperbolic type differential equation takes the form of:
$u_ - c^2 u_ = f(x,t),\, u(x,0)=g(x),\, u_t(x,0)=h(x),$
for $-\infty < x < \infty, \,\, t > 0, f \in C^2(\R^2,\R)$.

All second order differential equations with constant coefficients can be transformed into their respective canonic forms. This equation is one of these three cases: Elliptic partial differential equation, Parabolic partial differential equation and Hyperbolic partial differential equation.

The only difference between a homogeneous and an inhomogeneous (partial) differential equation is that in the homogeneous form we only allow 0 to stand on the right side ($f(x,t) = 0$), while the inhomogeneous one is much more general, as in $f(x,t)$ could be any function as long as it's continuous and can be continuously differentiated twice.

The solution of the above equation is given by the formula:
$u(x,t) = \frac\bigl( g(x+ct) + g(x-ct)\bigr) + \frac \int_^ h(s)\, ds + \frac \int_0^t \int_^ f(s,\tau) \, ds \, d\tau .$

If $g(x) = 0$, the first part disappears, if $h(x) = 0$, the second part disappears, and if $f(x) = 0$, the third part disappears from the solution, since integrating the 0-function between any two bounds always results in 0.

See also

*D'Alembert operator
*Mechanical wave
*Wave equation

Notes

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Partial differential equations