The Wave Equation and Its Characteristics · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.
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Waves that carry their shape along fixed paths

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

  1. For u_tt = 25 u_xx, what is the signal speed c?

    • 25
    • 5
    • 625
    • 1/5

  2. For u_tt = u_xx with u(x,0) = 0 and u_t(x,0) = 1, use d'Alembert's formula to find u(5,3).

    Answer: ______________

  3. For u_tt equals u_xx with zero displacement and velocity 4 everywhere, use d'Alembert's formula to find u(3,2).

    Answer: ______________

  4. A triangular pulse starts on an infinite string with speed 1. After 3 seconds, each of the two travelling copies still has its sharp peak.

    Circle one:   True   False

  5. For u_tt equals u_xx with zero displacement and velocity g(s) equals 2s, what is u(1,1)?

    • 4
    • 2
    • 0

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

    Answer: ______________

  7. For u_tt = 9 u_xx, what is the domain of dependence of the point (0, 1)?

    • [-1, 1]
    • [-3, 3]
    • [-9, 9]
    • [0, 3]

  8. For u_tt = u_xx with u(x,0) = x² and u_t(x,0) = 0, use d'Alembert's formula to find u(0,2).

    Answer: ______________

  9. For u_tt equals u_xx with displacement x squared and zero velocity, use d'Alembert's formula to find u(3,2).

    Answer: ______________

  10. For u_tt equals 4 u_xx, the domain of dependence of the point (0,t) has length 4t.

    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. 5 · The equation is u_tt = c² u_xx, so c² = 25 and c = 5.
  2. 3 · Half the integral of 1 from 2 to 8 is half of 6, which is 3.
  3. 8 · With f zero, u(3,2) is half the integral of 4 from 1 to 5, which is half of 16.
  4. True · The wave equation splits the pulse into left and right copies that keep their shape, corners included.
  5. 2 · u(1,1) is half the integral of 2s from 0 to 2: half of 4, which is 2.
  6. -5 · Along x - 3t = const through (1,2), the constant is 1 - 6 = -5.
  7. [-3, 3] · c = 3, so the feet are 0 - 3 and 0 + 3.
  8. 4 · The average of f at -2 and 2: (4 + 4)/2 = 4.
  9. 13 · The average of f at the feet 1 and 5: (1 + 25)/2 = 13.
  10. True · c is 2, so the feet are at -2t and 2t, and the interval between them has length 4t.
Worksheet · LightMySky