Did Euler or did d'Alembert incorporate initial conditions into the solution to the 1D wave equation?

Question

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.

Background

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

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:

• D'Alembert's Solution, MathWorld
• D'Alembert's formula, English Wikipedia
• 4.6 PDEs, separation of variables, and the heat equation, Jiří Lebl website
• R. L. Herman, d'Alembert Solution of the Wave Equation

Additional sources:

• English translation of what may be Euler's work on this: Treatise on the vibration of chords (Server of the University of Mainz)
• Original in Latin: Euler, Leonhard, De vibratione chordarum exercitatio (1749). Euler Archive - All Works. 119.

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

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: $$ \Psi(x)+\Psi(-x)=\textstyle\int\sigma(x)\,dx. \tag{10} $$

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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.