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Sine Wave Parameter Explorer

Run the Sine Wave Parameter Explorer MicroSim Fullscreen

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About This MicroSim

Every sine wave in this course is described by the same three numbers:

\[y(t) = A \sin(2\pi f t + \varphi)\]

The trouble is that all three are usually introduced at once, so it is easy to leave with a vague sense that they all "change the wave somehow." They do not. Each one changes exactly one visual property and leaves the other two alone:

Parameter Changes Leaves alone
Amplitude \(A\) Height of the peaks Peak spacing, position
Frequency \(f\) How many cycles fit the window Peak height, y(0) shape
Phase \(\varphi\) Horizontal position of the whole curve Peak height, peak spacing

Each numeral in the on-screen equation is tinted to match the slider that controls it, so you can always see which term you just moved.

The red line at \(t = 0\) with a dot on the curve is where phase makes itself visible. At \(\varphi = 0\) the wave starts at zero and rises. Change nothing but phase and watch that dot climb.

How to Use

  1. Move Amplitude alone. Confirm the peaks get taller but stay in exactly the same horizontal positions.
  2. Return amplitude to 1.0 and move Frequency alone. Count the cycles in the 3-second window at 1 Hz, then at 3 Hz. Confirm peak height never changes.
  3. Return frequency to 1.0 and move Phase alone. The whole curve slides sideways; the dot at \(t=0\) traces out the wave's value at the origin.
  4. Set phase to 90°. Now check Overlay cosine wave. The two curves land on top of each other — that is the whole content of the identity \(\cos\theta = \sin(\theta + 90°)\).

Controls

Control Range Default
Amplitude (A) 0.2 - 2.0 1.0
Frequency (f) 0.5 - 5 Hz 1 Hz
Phase (φ) 0 - 360° 0°
Overlay cosine wave — off

Lesson Plan

Grade Level

Undergraduate (college junior/senior) or advanced high school

Duration

10-15 minutes

Prerequisites

  • Familiarity with the sine function on the unit circle
  • Reading a plot with labeled axes

Learning Objective

Students will be able to demonstrate which visual feature of a plotted wave each of \(A\), \(f\), and \(\varphi\) controls, and calculate \(y(0)\) for a given set of parameters.

Activities

  1. One at a time (6 min): Students vary each slider in isolation and write one sentence per parameter describing what changed and what did not.
  2. Predict y(0) (4 min): For \(A = 1.5\), \(f = 2\), \(\varphi = 90°\), students compute \(y(0)\) by hand, then set the sliders and check the readout.
  3. The cosine identity (4 min): Students use the overlay to explain in their own words why cosine is "sine with a head start."

Assessment

Ask: "Two waves have identical peak heights and identical peak spacing, but one is shifted a third of a cycle to the right. Which parameter differs, and by how many degrees?" (Phase, 120°.)

References

  1. Sine wave — the standard parameterization and its terminology.
  2. Phase (waves) — phase offset, phase difference, and the sine/cosine relationship.