Split a wave into two traveling profiles

Try

A string can be flat at an instant while moving rapidly. Initial displacement alone therefore cannot determine its future. Let uₜₜ=c²uₓₓ on the whole line, with positive speed c, initial displacement f and initial velocity g. Trace the two lines of slopes ±c that reach a selected observation point.

Their intersections with the initial line are x−ct and x+ct. The interval between them will determine which initial data can affect that observation.

Predict

If f is zero and g is the constant 2, should the later displacement remain zero? Predict a spatially constant solution using only the time equation. Then use this prediction to check the coefficient in the velocity integral below.

Experiment

d’Alembert’s formula is

u(x,t)=[f(x−ct)+f(x+ct)]/2 + (1/(2c))∫ from x−ct to x+ct of g(s) ds.

For c=2, f(x)=x² and g(x)=2, evaluate at x=1,t=1.

Contribution	Calculation	Value
Left displacement	f(−1)/2	1/2
Right displacement	f(3)/2	9/2
Velocity	Integral of 2 over [−1,3], divided by 4	2

The total is 7. Changing g to zero removes only the last contribution. Check that the formula at t=0 returns f and that its time derivative at zero returns g.

Understand

Introduce characteristic coordinates ξ=x−ct and η=x+ct. The operator ∂ₜ²−c²∂ₓ² becomes −4c²∂ξ∂η. Hence a sufficiently smooth homogeneous solution has the form A(ξ)+B(η). The initial conditions give A+B=f and −cA′+cB′=g. Solving these relations and integrating produces the formula above; arbitrary integration constants cancel in the combined solution.

The displacement data contribute through the two endpoints, while velocity contributes through the entire interval. This is finite propagation: changing data outside that interval does not alter the value at (x,t). It does not mean only the endpoints matter for both initial functions.

For f in C² and g in C¹, direct differentiation verifies a classical solution. Less regular data can still define weaker solutions, but corners in f can travel along characteristics and prevent classical second derivatives. A plotted kink is not necessarily a numerical failure; it may be part of the exact weak solution.

The speed c appears both in the interval width and in the prefactor of the velocity integral. If c changes, changing only one of those places breaks the initial-velocity condition. Dimensional reasoning helps: integrating velocity over length and dividing by a speed produces displacement.

Apply

Choose a compactly supported initial velocity and zero displacement. Identify exactly when a fixed observation point first receives influence by checking interval intersection with the support. Then calculate the integral for a piecewise constant velocity and state where the resulting solution fails to be classically smooth. Verify the initial condition in a suitable limiting sense.

Continue

Wave energy proves uniqueness and finite dependence without relying on an explicit formula. It also prepares the extension to higher dimensions, where the shape of the dependence region changes.