Wave Equation Solver

The one-dimensional wave equation is the partial differential equation obeyed by any small disturbance travelling along a string, a pipe or a transmission line:

$\frac{\partial^2 u}{\partial t^2} = c^2\,\frac{\partial^2 u}{\partial x^2}$

Symbols and SI units:

• $u(x,t)$ — displacement of the medium, metres (m)
• $x$ — position along the medium, metres (m)
• $t$ — time, seconds (s)
• $c$ — propagation speed, metres per second (m/s); for a string $c = \sqrt{T/\mu}$ with tension $T$ in N and linear density $\mu$ in kg/m

In words: the vertical acceleration of each point is proportional to the local curvature. A sharply curved piece of string is pulled back hard; a straight piece is not pulled at all.

When it applies: small-amplitude disturbances in a uniform, lossless, non-dispersive medium.

The assumption people forget: it is a linear equation derived for small slopes. Large-amplitude waves, damping and dispersion each add terms and break the clean solutions below.

D'Alembert's general solution on an infinite line is any right-moving shape plus any left-moving shape:

$u(x,t) = F(x - ct) + G(x + ct)$

With initial displacement $u(x,0) = \varphi(x)$ and initial velocity $u_t(x,0) = \psi(x)$, this becomes

$u(x,t) = \tfrac{1}{2}\left[\varphi(x-ct) + \varphi(x+ct)\right] + \frac{1}{2c}\int_{x-ct}^{x+ct}\psi(s)\,ds$

Separation of variables handles a finite string of length $L$ fixed at both ends. Writing $u = X(x)T(t)$ gives the modes

$u(x,t) = \sum_{n=1}^{\infty}\sin\!\left(\frac{n\pi x}{L}\right)\left[A_n\cos\frac{n\pi ct}{L} + B_n\sin\frac{n\pi ct}{L}\right]$

whose frequencies are $f_n = nc/(2L)$ in hertz — the harmonic series of a musical string.

The assumption people forget: d'Alembert applies to an unbounded domain. Boundaries reflect waves, and the series form is the one to use.

• Confusing this PDE with $v = f\lambda$ — that algebraic relation describes one sinusoidal wave; the PDE governs every possible waveform.
• Losing the $c^2$ — the constant multiplies the space derivative, and squaring it is what makes the units balance.
• Writing $F(x + ct)$ for the right-moving wave — the minus sign travels right, the plus sign travels left.
• Forgetting the factor $\tfrac{1}{2}$ in d'Alembert — the initial shape splits into two half-height pulses.
• Dropping the chain-rule factor — differentiating $F(x-ct)$ twice in $t$ brings out $c^2$, and omitting it makes the verification fail.
• Solving a bounded problem with d'Alembert — use the sine series once the ends are fixed.