d'Alembert's formula

Short description: Mathematical solution

In mathematics, and specifically partial differential equations (PDEs), d´Alembert's formula is the general solution to the one-dimensional wave equation:

[math]\displaystyle{ u_{tt}-c^2u_{xx}=0,\, u(x,0)=g(x),\, u_t(x,0)=h(x), }[/math]

for [math]\displaystyle{ -\infty \lt x\lt \infty,\,\, t\gt 0 }[/math]

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

[math]\displaystyle{ u(x,t) = F(x+ct) + G(x-ct) }[/math]

[math]\displaystyle{ u }[/math] is [math]\displaystyle{ C^2 }[/math] if [math]\displaystyle{ F }[/math] and [math]\displaystyle{ G }[/math] are [math]\displaystyle{ C^2 }[/math].

This solution [math]\displaystyle{ u }[/math] can be interpreted as two waves with constant velocity [math]\displaystyle{ c }[/math] moving in opposite directions along the x-axis.

Now consider this solution with the Cauchy data [math]\displaystyle{ u(x,0)=g(x), u_t(x,0)=h(x) }[/math].

Using [math]\displaystyle{ u(x,0) = g(x) }[/math] we get [math]\displaystyle{ F(x) + G(x) = g(x) }[/math].

Using [math]\displaystyle{ u_t(x,0) = h(x) }[/math] we get [math]\displaystyle{ cF'(x)-cG'(x) = h(x) }[/math].

We can integrate the last equation to get [math]\displaystyle{ cF(x)-cG(x)=\int_{-\infty}^x h(\xi) \, d\xi + c_1. }[/math]

Now we can solve this system of equations to get [math]\displaystyle{ F(x) = \frac{-1}{2c}\left(-cg(x)-\left(\int_{-\infty}^x h(\xi) \, d\xi +c_1 \right)\right) }[/math] [math]\displaystyle{ G(x) = \frac{-1}{2c}\left(-cg(x)+\left(\int_{-\infty}^x h(\xi) d\xi +c_1 \right)\right). }[/math]

Now, using [math]\displaystyle{ u(x,t) = F(x+ct)+G(x-ct) }[/math]

d'Alembert's formula becomes: [math]\displaystyle{ u(x,t) = \frac{1}{2}\left[g(x-ct) + g(x+ct)\right] + \frac{1}{2c} \int_{x-ct}^{x+ct} h(\xi) \, d\xi. }[/math]

Generalization for inhomogeneous canonical hyperbolic differential equations

The general form of an inhomogeneous canonical hyperbolic type differential equation takes the form of: [math]\displaystyle{ u_{tt} - c^2 u_{xx} = f(x,t),\, u(x,0)=g(x),\, u_t(x,0)=h(x), }[/math] for [math]\displaystyle{ -\infty \lt x \lt \infty, \,\, t \gt 0, f \in C^2(\R^2,\R) }[/math].

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 ([math]\displaystyle{ f(x,t) = 0 }[/math]), while the inhomogeneous one is much more general, as in [math]\displaystyle{ f(x,t) }[/math] 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: [math]\displaystyle{ u(x,t) = \frac{1}{2}\bigl( g(x+ct) + g(x-ct)\bigr) + \frac{1}{2c} \int_{x-ct}^{x+ct} h(s)\, ds + \frac{1}{2c} \int_0^t \int_{x-c(t-\tau)}^{x+c(t-\tau)} f(s,\tau) \, ds \, d\tau . }[/math]

If [math]\displaystyle{ g(x) = 0 }[/math], the first part disappears, if [math]\displaystyle{ h(x) = 0 }[/math], the second part disappears, and if [math]\displaystyle{ f(x) = 0 }[/math], 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

External links

• An example of solving a nonhomogeneous wave equation from www.exampleproblems.com

https://www.knowledgeablegroup.com/2020/09/equations%20change%20world.html