From Coordinates to Fields: the Lagrangian Density · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

From beads on a string to the field itself

Science · Forces & Motion · ages 22-24
Name ______________________   Date ____________

  1. To solve the wave equation on 0 ≤ x ≤ L, L > 0, with fixed ends by the odd-extension d'Alembert method, always extend the initial displacement outside as an odd function about both endpoints with period 2L, and extend the initial velocity the same way, or set it to zero only when released from rest.

    Circle one:   True   False

  2. For a stretched string with mass per length μ, tension T, and small transverse displacement u(x, t), which Lagrangian density gives the wave equation?

    • ℒ = ½μ u_t² − ½T u_x²
    • ℒ = ½μ u_t² + ½T u_x²
    • ℒ = ½μ u_x² − ½T u_t²

  3. d'Alembert's solution is u of x and t equals f of x minus c t plus g of x plus c t. What does this tell you about the motion?

    • Each point of the string moves in a circle with radius set by c
    • The shape stays put while its height grows and shrinks
    • Two shapes sliding opposite at speed c, added

  4. To solve the wave equation on the interval 0 ≤ x ≤ L, L > 0, with fixed ends by the odd-extension d'Alembert method, you always extend the initial displacement outside as an odd function about both endpoints with period 2L, and you also extend the initial velocity the same way, or explicitly set it to zero when the string is released from rest.

    Circle one:   True   False

  5. A string has mass per length μ = 2 kg/m and tension T = 8 N. Its wave speed is c = √(T/μ). Type c in m/s.

    Answer: ______________

  6. A bump with shape f starts centered at 0 and slides as f(x − ct) with c = 3 m/s. Where is the bump centered at t = 2 s?

    • At x equals 6
    • At x equals 2
    • At x equals 0, unmoved

  7. A string of length L is fixed at both ends. Why do the allowed spatial shapes use sin(nπx/L) with whole numbers n?

    • They are the only sine functions that can be differentiated twice
    • Cosine functions are banned from every wave problem
    • They are the only sines vanishing at both ends

  8. A string has mass per length μ = 0.5 kg/m and tension T = 18 N. What is its wave speed c = √(T/μ), in m/s?

    • 3 m/s
    • 6 m/s
    • 9 m/s

  9. A student models the full vibrating string with a single ordinary differential equation in time only. What is wrong with this?

    • Nothing: a single normal mode ansatz reduces the string dynamics to an ordinary equation in time, so it is fine
    • The full field bends across space too, so its equation must be partial in space and time
    • Wave problems never admit equations at all

  10. As a formal nondimensional check on the whole line, a string starts from rest with shape f(x) = x² and wave speed c = 2. Its motion is u(x, t) = ½f(x − ct) + ½f(x + ct). Type u(2, 3).

    Answer: ______________

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Answer key

For grown-ups. Fold this page away before handing over the rest.

From beads on a string to the field itself W1-mt_fRZN3ZByjr-s1

  1. True · Within the stated method, the displacement extension is always required; the velocity must be extended too, or set to zero only when release from rest is given. Odd extension about both endpoints with period 2L keeps the ends fixed at every moment.
  2. ℒ = ½μ u_t² − ½T u_x² · The density ℒ = ½μ u_t² − ½T u_x² gives ∂ℒ/∂u_t = μ u_t and ∂ℒ/∂u_x = −T u_x, so the field equation is μ u_tt − T u_xx = 0.
  3. Two shapes sliding opposite at speed c, added · f of x minus c t slides one way, g of x plus c t slides the other, and they add.
  4. True · Within the stated odd-extension d'Alembert method, the displacement extension is always required; the velocity must be extended too, or explicitly set to zero only when release from rest is given. Odd extension about both endpoints with period 2L keeps the ends fixed.
  5. 2 · c = √(8/2) = √4 = 2 m/s.
  6. At x equals 6 · The shape rides along at speed c, so after 2 s it has covered 6 m.
  7. They are the only sines vanishing at both ends · Fixed ends demand zero displacement there, and exactly these sines vanish at both ends; the full time-dependent solutions A sin(nπx/L) cos(ωₙt + φ) are standing waves, which are valid.
  8. 6 m/s · c = √(T/μ) = √(18/0.5) = √36 = 6 m/s.
  9. The full field bends across space too, so its equation must be partial in space and time · For the unrestricted full field, bending across space forces a partial equation in space and time; a single ordinary equation fits a chosen normal mode or another explicitly constrained ansatz.
  10. 40 · Sampling points are −4 and 8, giving ½(16 + 64) = 40; this is a formal nondimensional check, not a physical small-slope string.
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