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

References for D'Alemberts formula:

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


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.

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  • $\begingroup$ 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 '18 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 '18 at 16:33

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