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d'Alembert's Solution


The method of d'Alembert provides a solution to the one-dimensional wave equation

 (partial^2y)/(partialx^2)=1/(c^2)(partial^2y)/(partialt^2)
(1)

that models vibrations of a string.

The general solution can be obtained by introducing new variables xi=x-ct and eta=x+ct, and applying the chain rule to obtain

partial/(partialx)=(partialxi)/(partialx)partial/(partialxi)+(partialeta)/(partialx)partial/(partialeta)
(2)
=partial/(partialxi)+partial/(partialeta)
(3)
partial/(partialt)=(partialxi)/(partialt)partial/(partialxi)+(partialeta)/(partialt)partial/(partialeta)
(4)
=-cpartial/(partialxi)+cpartial/(partialeta).
(5)

Using (4) and (5) to compute the left and right sides of (3) then gives

(partial^2y)/(partialx^2)=(partial/(partialxi)+partial/(partialeta))((partialy)/(partialxi)+(partialy)/(partialeta))
(6)
=(partial^2y)/(partialxi^2)+2(partial^2y)/(partialxipartialeta)+(partial^2y)/(partialeta^2)
(7)
(partial^2y)/(partialt^2)=(-cpartial/(partialxi)+cpartial/(partialeta))(-c(partialy)/(partialxi)+c(partialy)/(partialeta))
(8)
=c^2(partial^2y)/(partialxi^2)-2c^2(partial^2y)/(partialxipartialeta)+c^2(partial^2y)/(partialeta^2),
(9)

respectively, so plugging in and expanding then gives

 (partial^2y)/(partialxipartialeta)=0.
(10)

This partial differential equation has general solution

y(x,t)=f(xi)+g(eta)
(11)
=f(x-ct)+g(x+ct),
(12)

where f and g are arbitrary functions, with f representing a right-traveling wave and g a left-traveling wave.

The initial value problem for a string located at position y(x,t=0)=y_0(x) as a function of distance along the string x and vertical speed partialy/partialt|_(t=0)=v_0(x) can be found as follows. From the initial condition and (12),

 y_0(x)=f(x)+g(x).
(13)

Taking the derivative with respect to t then gives

v_0(x)=f^'(x)(partial(x-ct))/(partialt)+g^'(x)(partial(x+ct))/(partialt)
(14)
=-cf^'(x)+cg^'(x),
(15)

and integrating gives

 int_a^xv_0(s)ds=-cf(x)+cg(x).
(16)

Solving (13) and (16) simultaneously for f and g immediately gives

f(x)=1/2y_0(x)-1/(2c)int_a^xv_0(s)ds
(17)
g(x)=1/2y_0(x)+1/(2c)int_a^xv_0(s)ds,
(18)

so plugging these into (13) then gives the solution to the wave equation with specified initial conditions as

 y(x,t)=1/2y_0(x-ct)+1/2y_0(x+ct)+1/(2c)int_(x-ct)^(x+ct)v_0(s)ds.
(19)

See also

Wave Equation

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References

Bekefi, G. and Barrett, A. H. Electromagnetic Vibrations, Waves, and Radiation. Cambridge, MA: MIT Press, pp. 161-163, 1987.

Cite this as:

Weisstein, Eric W. "d'Alembert's Solution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/dAlembertsSolution.html

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