Coordinate operators, separated modes and stable updates
Introduced| English |
|---|
| separation of variables/ˌsepəˈreɪʃn ɒv ˈveərɪəblz/ |
| stability condition/stəˈbɪlɪti kənˈdɪʃn/ |
A decision before an answer
- Writing the Laplacian of a radial function as f″ drops a term; the operator belongs to the coordinates, not the symbol.
- Your goal: Use the divergence and Laplacian in the coordinate system selected by the symmetry.
Match operator to symmetry
- Choose coordinates by symmetry before writing operators. For a radial function in spherical coordinates, ∇²f=f″+(2/r)f′, and the divergence of a radial field A=A_r r̂ is (1/r²)d(r²A_r)/dr.
- Check: A=k r̂/r² has divergence (1/r²)dk/dr=0 away from the origin, and f=1/r satisfies ∇²f=f″+2f′/r=2/r³−2/r³=0 away from the origin. Cartesian forms apply only to Cartesian dependences.
For a radial function f(r) in spherical coordinates, the Laplacian is:
Spherical radial dependence gives ∇²f=(1/r²)d(r²f′)/dr=f″+(2/r)f′.
Separate the modes
- Separation of variables turns a boundary-value problem into ordinary modes. For a string fixed at both ends, write u=X(x)T(t); the wave equation gives X″/X=(1/c²)T″/T=−k².
- The fixed ends force X=sin(nπx/L), and the allowed frequencies are ω_n=nπc/L, f_n=nc/(2L). Each mode must satisfy the boundary conditions before any sum over modes.
The explicit Euler update y_(n+1)=(1−10×0.25)y_n for y′=−10y:
The factor is −1.5; its magnitude exceeds 1 and its sign alternates.
Bound the step
- An explicit Euler update for y′=−λy is y_{n+1}=(1−λh)y_n, bounded when |1−λh|≤1, i.e. 0<h≤2/λ for λ>0; asymptotic decay needs 0<h<2/λ.
- With λ=10 /s, h must not exceed 0.2 s; h=0.25 s multiplies by −1.5 each step and diverges with alternating sign. Stability is a property of the recurrence, not of the true solution.
Radial check: f=1/r gives f″+2f′/r=0 away from the origin. A string with L=0.5 m and c=100 m/s has f₁=c/(2L)=100 Hz. Euler stability for λ=10 /s needs h≤0.2 s; refining y′=−4y with y(0)=8 at t=1 s, from h=0.2 to 0.1 to 0.05 gives 0.00256, 0.0483, 0.0922, approaching the exact 0.1465.
A string fixed at both ends has L=0.5 m and wave speed 100 m/s. Its fundamental frequency is ____ Hz.
f₁=c/(2L)=100/1=100 Hz.
Verify the answer
- Check a numerical answer three ways before trusting it: units of every term (λh must be dimensionless), convergence under refinement (halve h and compare; explicit Euler error is first order), and a limiting case with a known exact answer.
- For y′=−4y with y0=8 and h=0.2, five steps give 8×0.2⁵=0.00256 versus exact 8e^(−4)≈0.1465: stable but inaccurate, so refine.
Applying the Cartesian Laplacian to a radial function, summing modes that violate the boundary conditions, or calling a bounded-looking run converged without refining the step. Match operator, boundaries and stability condition to the actual problem.
Which answer fits this case?
Use the divergence and Laplacian in the coordinate system selected by the symmetry
The field A=k r̂/r² has nonzero divergence at every point of space.
Its divergence is zero away from the origin; all source strength sits at the origin as a distribution.
Keep the distinctions
- separation of variables 分离变量法 — Solving a partial differential equation by writing the unknown as a product of single-variable factors.
- stability condition 稳定性条件 — The step-size restriction that keeps a numerical recurrence from amplifying errors.
- Use the divergence and Laplacian in the coordinate system selected by the symmetry.
- Solve a simple separated boundary-value mode problem.
- Check convergence, units and stability of an original numerical update.
Match each term with its precise meaning in this lesson.
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
The order follows the stated process; check each stage before the next.