Enforce boundary conditions through modes and reflection

Try

A string fixed at both ends cannot translate a pulse straight through its endpoints. Reflections and standing-wave modes enforce the boundary condition. On [0,π], sine modes vanish at both ends, so they provide a natural basis for displacement and velocity data.

Compare this with heat on the same interval: the spatial eigenfunctions are identical, but the time equations differ. Sharing a basis does not mean sharing damping behavior.

Predict

For wave speed c=2 and spatial mode sin(3x), predict whether its angular frequency is 6 or 18. Then decide how the initial velocity coefficient enters a sine-in-time term. Differentiate that term at time zero before assigning its coefficient.

Experiment

Take initial displacement 2sin x and initial velocity 6sin(3x), with c=2. The solution is 2cos(2t)sin x + sin(6t)sin(3x).

Spatial mode	Frequency	Cosine coefficient	Sine coefficient
sin x	2	2	0
sin(3x)	6	0	6/6=1

Differentiate at t=0 to recover the velocity. A common error is to put 6 directly in the last column, which would produce initial velocity 36sin(3x). Check the PDE for each product and then use linearity to combine them.

Understand

For eigenpairs −Xₙ″=λₙXₙ, a modal amplitude satisfies aₙ″+c²λₙaₙ=0. For positive λₙ, its frequency is ωₙ=c√λₙ, giving aₙ=Aₙcos(ωₙt)+(Gₙ/ωₙ)sin(ωₙt). Here Aₙ and Gₙ are projections of initial displacement and velocity. The division by frequency is essential.

For finite sums, direct verification is sufficient. Infinite expansions need convergence hypotheses: energy-space data naturally control sums of λₙAₙ² and Gₙ², with normalization factors from the eigenfunctions. Classical pointwise derivatives require stronger decay. The existence of a convergent displacement series alone does not justify twice differentiating it.

Neumann boundaries admit a zero mode. Its amplitude solves a₀″=0, hence A₀+G₀t. Dividing by ω₀=0 would be invalid. A spatially constant velocity produces uniform drift rather than an oscillation, which is consistent with zero spatial restoring force.

On a half-line with homogeneous Dirichlet data, extend the initial displacement and velocity oddly across the endpoint and apply d’Alembert’s formula. Oddness is preserved, forcing the value at the endpoint to remain zero. Even reflection instead enforces a zero derivative when regularity permits. Corner compatibility determines whether the reflected solution is classical there.

Undamped modal amplitudes oscillate rather than decay. Boundary reflections redistribute the shape, while total energy remains conserved under the homogeneous conditions. Displacement cancellation at one point does not demonstrate dissipation.

Apply

Construct a two-mode fixed-end solution with nonzero displacement and velocity in both modes. Verify all four coefficients by evaluating u and uₜ at zero. Then repeat the structural argument for insulated or free endpoints, explicitly writing the zero-mode term. Explain which coefficient represents uniform translation and why it does not fit the positive-frequency formula.

Continue

In three dimensions, waves depend on spherical averages. Finite propagation survives, but the sharper statement about influence concentrated on a sphere is dimension-dependent.