The Wave Equation and Its Characteristics · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Waves that carry their shape along fixed paths

Mathematics · Differential Equations · ages 22-23
Name ______________________   Date ____________

  1. For u_tt equals 4 u_xx, which lines are the characteristics through the origin?

    • The lines x equals t and x equals negative t
    • The lines x equals 2t and x equals negative 2t
    • The single line x equals 0

  2. A plucked string starts with a sharp corner. Sara says the wave equation carries that corner along unchanged, while the heat equation would round it off immediately. Is Sara right?

    Circle one:   True   False

  3. For u_tt = 4 u_xx, which lines are the characteristics through the origin?

    • The lines x = t and x = -t
    • The lines x = 2t and x = -2t
    • The lines t = 2x and t = -2x
    • The single line x = 0

  4. For u_tt equals u_xx with zero displacement and velocity 2 everywhere, use d'Alembert's formula to find u(0,1).

    Answer: ______________

  5. For u_tt equals u_xx with zero displacement and velocity g(s) equals s, what is u(2,2)?

    • 4
    • 2
    • 0

  6. Tom says the domain of dependence of (x,t) keeps widening as t grows, because the backward characteristics spread apart. Is Tom right?

    Circle one:   True   False

  7. For u_tt = 9 u_xx, one characteristic through (1,2) has the form x - 3t = const. What is that constant?

    Answer: ______________

  8. For u_tt = u_xx, what is the domain of dependence of the point (4,2)?

    • [3, 5]
    • [0, 8]
    • [4, 6]
    • [2, 6]

  9. For u_tt equals u_xx with displacement x cubed and zero velocity, use d'Alembert's formula to find u(1,1).

    Answer: ______________

  10. The domain of dependence of a point keeps widening as time grows, because the backward characteristics spread apart.

    Circle one:   True   False

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Answer key

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

Waves that carry their shape along fixed paths W1-mt_-_QCiBWg1w-s1

  1. The lines x equals 2t and x equals negative 2t · Signals travel at speed 2 here, so the slopes are plus and minus 2.
  2. True · Wave motion transports each half of a kink unchanged along characteristics, while heat flow smooths it at once.
  3. The lines x = 2t and x = -2t · Signals travel at speed 2 here, so the two characteristics through the origin are x = 2t and x = -2t.
  4. 2 · With f zero the formula keeps half the velocity integral over length 2.
  5. 4 · The velocity integrates to 8 over the feet interval, and half of 8 is 4.
  6. True · The feet of the backward characteristics spread apart linearly, so later points sample wider initial intervals.
  7. -5 · Along x - 3t = const through (1,2), the constant is 1 - 6 = -5.
  8. [2, 6] · With speed 1, signals reaching (4,2) started between 4 - 2 and 4 + 2, so the interval is [2, 6].
  9. 4 · Averaging f(0) equals 0 and f(2) equals 8 gives 4.
  10. True · The feet sit at x minus ct and x plus ct, whose separation grows with t.
Worksheet · LightMySky