Operadores de coordenadas, modos separados y actualizaciones estables
Introducido| English | Significado |
|---|---|
| separation of variables/ˌsepəˈreɪʃn ɒv ˈveərɪəblz/ | separación de variables |
| stability condition/stəˈbɪlɪti kənˈdɪʃn/ | condición de estabilidad |
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:
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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:
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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.
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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.
¿Qué respuesta se ajusta a este caso?
Utilice la divergencia y el Laplaciano en el sistema de coordenadas seleccionado por la simetría
The field A=k r̂/r² has nonzero divergence at every point of space.
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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.
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Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
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The order follows the stated process; check each stage before the next.