My question is: Who is responsible for incorporating the initial conditions into the one dimensional wave equation solution? References or technical information would be appreciated, especially regarding the development of the integral term.


If $\phi(t,x)$ is the general solution to the one dimensional wave equation and if the initial conditions $\phi(0,x)$ and $\phi_t(0,x)$ are given, then the formula for the solution incorporating initial conditions is:

Letting $g(x)=\phi(0,x)$ and $h(x)= \phi_t(0,x)$ (with $c=1$),

$$\phi(t,x)= \frac 12[ g(x-t)+ g(x+t) ]+ \frac12 \int_{x-t}^{x+t} h(y)dy .$$

All of this is usually attributed to d'Alembert, however my "References for Euler / d'Alembert" imply that the solution with initial conditions were contributed by Euler. If so, then the integral term was Euler's work?


References for Euler / d'Alembert:

  • Michela Massimi, Kant and Philosophy of Science Today, p185
  • Amir R. Alexander, Duel at Dawn: Heroes, Martyrs, and the Rise of Modern Mathematics, p63
  • Manuel D. Salas, A Shock-Fitting Primer, p3
  • Hankins, Jean d'Alembert-Science , p48
  • Leonhard Euler: Life, Work and Legacy, edited by Robert E. Bradley, Ed Sandifer, p30

References for d'Alemberts formula:

Additional sources:

This question was first asked on: https://physics.stackexchange.com/q/302304/45664. It was too old to migrate.


1 Answer 1


I doubt D’Alembert had the integral term. Weber’s treatise (1901, p. 213) quotes the formula as “originating with” him, but I think only the general form $\Psi(x+ct)+\Phi(x-ct)$ is meant here.

In fact Riemann (1867, p. 90; 1869, pp. 111-113, 188-201) — relied on by both Weber (p. 205) and the Encyklopädie (1900, p. 557) — attributed the integral term to Euler (1750, likely p. 78).

Euler was using rather different notation, still found in e.g. Forsyth (1929, p. 488). If you want the literal formula, Kline (1972, p. 513) seems to say it was first written by Lagrange (1762, p. 27).

Edit: It seems I put too much faith in Riemann. Here is from Burkhardt (1901, p. 12):

In an immediately subsequent second paper 44), D’Alembert first details the construction of the curve $y= \Psi(x+t)-\Psi(x-t)$ from the curve $y=\Psi(x)$. Then he turns to the more general assumption that at $t=0$ one arbitrarily prescribes not only the initial ordinate $y=\Sigma(x)$ but also the initial velocity $\partial y\,/ \,\partial t=\sigma(x)$ of each point of the string 45). In that case (...) $$ \Psi(x)-\Psi(-x)=\Sigma(x) \tag9 $$ and: $$ \Psi'(x)-\Psi'(-x)=\sigma(x) $$ or: $$ \Psi(x)+\Psi(-x)=\textstyle\int\sigma(x)\,dx. \tag{10} $$

44) p. 220 [1749].

45) nº 23, p. 230. Riemann’s oft-repeated assertion that this case was first treated by Euler must be based on an oversight of this section.

  • $\begingroup$ Thanks, particularly for the references. I am specifically trying to find the original development of the integral term in D'Alemberts Formula--hopefully equations and a drawing showing the geometry and physics. The antique math and German and French are making it hard. $\endgroup$
    – user45664
    Apr 9, 2018 at 20:43
  • $\begingroup$ Here they seem to state on page 2 (equations 2,3,4) that Euler developed the solution with initial conditions; scribd.com/document/32298888/… $\endgroup$
    – user45664
    Apr 12, 2018 at 16:33
  • $\begingroup$ @ Francois Ziegler I have re-opened this question at hsm.stackexchange.com/questions/15932. Hope to see your inputs. Re-opened because of Euler's comments on the velocity IC in the English translation. $\endgroup$
    – user45664
    Nov 23, 2023 at 19:31

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