Fourier Analysis and the Wave Packet · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Waves built from waves

Science · Waves, Light & Sound · ages 21-22
Name ______________________   Date ____________

  1. For linear waves, how do their disturbances combine when they overlap?

    • They cancel no matter what
    • They add point by point
    • The taller one erases the smaller

  2. What terms form a Fourier series for a piecewise smooth periodic signal?

    • A mean plus harmonic sine and cosine terms
    • A single base-frequency sine for every periodic signal
    • A product of random waves with unrelated frequencies

  3. At a square-wave jump, the Fourier series tends to the midpoint, while nearby Gibbs overshoot persists as more terms are included.

    Circle one:   True   False

  4. A sharp-edged pulse passes through an ideal low-pass channel with a low cutoff. Its edges become slower and show ringing. What explains the lost edge sharpness?

    • It removes high-frequency edge detail
    • It shifts every signal component to a lower frequency
    • It replaces all input components with a single frequency

  5. A pulse is compressed in time to half its duration without changing its shape. Using the same measure of frequency spread, how does the spread change?

    • It is halved
    • It stays the same
    • It is doubled

  6. Two identical sinusoidal waves travel in the same direction in a linear medium and are exactly in phase. What results when they overlap?

    • Twice the amplitude, with the same wavelength
    • Zero amplitude everywhere, with no travelling wave left
    • The original amplitude, with the wavelength halved

  7. A student claims that one sine wave alone exactly equals a square wave. Which observation rules out that claim?

    • Sine waves at different frequencies cannot be added
    • A sine is smooth; a square wave has jumps
    • A square wave can contain no frequency components

  8. At an agreed accuracy, a pulse needs components up to 5 kHz. An ideal low-pass channel has a 7 kHz cutoff. The pulse's duration is halved without changing its shape. At the same accuracy, what does the channel now do?

    • Passes all needed components because the band shrinks to 2.5 kHz
    • Passes all needed components because the band stays at 5 kHz
    • Cuts needed components above 7 kHz from a 10 kHz band
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Answer key

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

Waves built from waves W1-mt_1ZaojWUZEw-s1

  1. They add point by point · Linear-wave superposition adds the disturbances at the same position and time, including their signs.
  2. A mean plus harmonic sine and cosine terms · A Fourier series includes the mean and sine and cosine terms at whole multiples of the base frequency.
  3. True · At the jump the series tends to the midpoint of the two limiting values. Nearby overshoot persists, but its affected region narrows.
  4. It removes high-frequency edge detail · The cutoff removes high-frequency components that help form sharp edges. The transitions become slower, and an ideal cutoff also produces ringing near jumps.
  5. It is doubled · For fixed-shape time scaling, dividing duration by 2 multiplies frequency spread by 2.
  6. Twice the amplitude, with the same wavelength · The waves have matching crests and troughs. Point-by-point addition doubles their amplitude without changing their wavelength.
  7. A sine is smooth; a square wave has jumps · A single sine wave is smooth and cannot have a square wave's jumps. The square-wave Fourier representation uses infinitely many harmonics.
  8. Cuts needed components above 7 kHz from a 10 kHz band · Halving duration doubles the required upper frequency from 5 to 10 kHz. Since 10 kHz exceeds the 7 kHz cutoff, some needed components are removed.
Worksheet · LightMySky