D'Alembert's formula explained

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

u _{tt} - c ^{2 u} _{xx} =0 , u(x, 0 ) = g(x), u _{t(x,0)=h(x),} for -inf ty < x< inf ty, 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.^[1]

Details

The characteristics of the PDE are

x \pm ct = const (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 η = x - ct (for the negative solution) to transform the PDE to u _{\muη} = 0 . The general solution of this PDE is u(\mu, η ) = F(\mu) + G( η ) 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.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 $$cF(x)-cG(x)=\backslash int\_^x\; h(\backslash xi)\; \backslash ,\; d\backslash xi\; +\; c\_1.$$

Now we can solve this system of equations to get$$F(x)\; =\; \backslash frac\backslash left(-cg(x)-\backslash left(\backslash int\_^x\; h(\backslash xi)\; \backslash ,\; d\backslash xi\; +c\_1\; \backslash right)\backslash right)$$$$G(x)\; =\; \backslash frac\backslash left(-cg(x)+\backslash left(\backslash int\_^x\; h(\backslash xi)\; d\backslash xi\; +c\_1\; \backslash right)\backslash right).$$

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

d'Alembert's formula becomes:^[2] $$u(x,t)\; =\; \backslash frac\backslash left[g(x-ct)\; +\; g(x+ct)\backslash right]\; +\; \backslash frac\; \backslash int\_^\; h(\backslash xi)\; \backslash ,\; d\backslash xi.$$

Generalization for inhomogeneous canonical hyperbolic differential equations

The general form of an inhomogeneous canonical hyperbolic type differential equation takes the form of:$$u\_\; -\; c^2\; u\_\; =\; f(x,t),\backslash ,\; u(x,0)=g(x),\backslash ,\; u\_t(x,0)=h(x),$$for

-inf ty < x < inf ty, 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:$$u(x,t)\; =\; \backslash frac\backslash bigl(g(x+ct)\; +\; g(x-ct)\backslash bigr)\; +\; \backslash frac\; \backslash int\_^\; h(s)\backslash ,\; ds\; +\; \backslash frac\; \backslash int\_0^t\; \backslash int\_^\; f(s,\backslash tau)\; \backslash ,\; ds\; \backslash ,\; d\backslash tau\; .$$

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

External links

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

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

Notes and References

1. D'Alembert (1747) "Recherches sur la courbe que forme une corde tenduë mise en vibration" (Researches on the curve that a tense cord [string] forms [when] set into vibration), Histoire de l'académie royale des sciences et belles lettres de Berlin, vol. 3, pages 214-219. See also: D'Alembert (1747) "Suite des recherches sur la courbe que forme une corde tenduë mise en vibration" (Further researches on the curve that a tense cord forms [when] set into vibration), Histoire de l'académie royale des sciences et belles lettres de Berlin, vol. 3, pages 220-249. See also: D'Alembert (1750) "Addition au mémoire sur la courbe que forme une corde tenduë mise en vibration," Histoire de l'académie royale des sciences et belles lettres de Berlin, vol. 6, pages 355-360.
2. Book: Pinchover . Yehuda . Rubinstein . Jacob . An introduction to Partial Differential Equations . Cambridge University Press . 2013 . 8th printing . 978-0-521-84886-2 . 76–92.