Math With AmarA C A D E M Y
All math animations

High school · Trigonometry

Build a traveling sine wave

Separate amplitude, spatial frequency, and phase by watching a wave travel to the right.

Predict what will happen, press Play, then pause and explain what changed. Every control also works without playback.

Watch the relationship

Paused
Build a traveling sine wave. Amplitude: 1.5. Wavelength: 3.142 x-units. Phase: 0 rad. Sensor y at x = π/2: 0y = A sin(kx − φ)0π/2π3π/22π-202xAmber sensor stays at x = π/2.
The x-axis is a dimensionless spatial coordinate from 0 to 2π; y runs from −2.5 to 2.5. One playback changes phase by 2π. No physical time units or propagation speed are asserted.

Starts paused. Play once, pause anywhere, or use Step to inspect the mathematics. Playback stops when this panel leaves the screen.

Make it your experiment

Change one value. Notice what follows.

The controls adjust the model. Numbers below describe the current frame. Decimals are rounded.

Amplitude
1.5
Wavelength
3.142 x-units
Phase
0 rad
Sensor y at x = π/2
0

HD animation studio

From experiment to screen.

Present a crisp Canvas scene, save a full-HD image, or capture your model as a silent video.

Understand what you are seeing

The idea behind the motion.

Amplitude sets a wave's maximum displacement; spatial frequency counts cycles in a chosen horizontal interval. Changing phase slides the entire wave without changing its height or wavelength. A fixed sensor can oscillate even though the wave pattern travels across the graph.

A relationship to keep

y(x, φ) = A sin(kx − φ); wavelength = 2π/k

Read the symbols alongside the explanation. A diagram shows the relationship; the assumptions tell you when it applies.

  1. STEP 1

    Set the height

    Raise the amplitude slider. Every crest reaches +A and every trough reaches −A. The graph's vertical scale stays fixed, so the height change is visible rather than hidden by automatic rescaling.

  2. STEP 2

    Count complete cycles

    Pause at the beginning and count waves between x = 0 and x = 2π. Increasing k fits more cycles into this interval; the wavelength shrinks to 2π/k.

  3. STEP 3

    Watch one fixed location

    The amber sensor stays at x = π/2 while its height changes. Playback increases phase from 0 to 2π. The minus sign in kx − φ makes the wave move toward increasing x.

Your turn to explain

Make a prediction. Test your reasoning.

If A = 2 and k = 3, what are the maximum displacement and wavelength?

Use the values specified in the question. Reset restores the initial values for this investigation.

Compare your explanation

The maximum displacement is 2 and the wavelength is 2π/3. The crest-to-trough height is 4, which is twice the amplitude.

Work through a full lesson

Connect the animation to a worked example and practice questions.

Grade 11 · Trigonometry

Separate horizontal shift from angular frequency

Read a factored sinusoidal model's amplitude, period, and phase shift.

Intermediate · Calculus

Why does a faster oscillation change the derivative?

Use the chain rule with a sine model while keeping radians, amplitude, and instantaneous motion distinct.