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Graduate · Heat semigroup modes

Heat semigroup modes: A weak second harmonic

Heat semigroup modes: investigate a weak second harmonic with higher-mode amplitude a = 0.2; higher frequency k = 2.

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

Watch the relationship

Paused
Heat semigroup modes: A weak second harmonic. Time: 0. Fundamental amplitude: 1. Higher-mode amplitude: 0.2. Mean half-square energy: 0.26High frequencies decay faster-1.071.0703.146.28temperature deviationTeal: current heat profile · dashed: initial profile
The domain is periodic, k is an integer at least two, and the equation is the linear heat equation uₜ=0.2uₓₓ. This is an exact two-mode solution. It is not a nonlinear fluid simulation or a claim about general Navier–Stokes behavior.

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.

Time
0
Fundamental amplitude
1
Higher-mode amplitude
0.2
Mean half-square energy
0.26

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Understand what you are seeing

The idea behind the motion.

Fourier sine modes are eigenfunctions of the periodic Laplacian, so heat evolution damps each mode at a rate proportional to frequency squared. A short-wavelength component can disappear much faster than a broad component. Orthogonality also gives an exact expression for the spatially averaged quadratic energy. This investigation starts with Higher-mode amplitude a = 0.2; Higher frequency k = 2. Predict the result before playing, then change one parameter while keeping the other fixed to test the reason for the change.

A relationship to keep

u=e^(−0.2t)sin x+a e^(−0.2k²t)sin(kx)

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

  1. STEP 1

    Set up the mathematical model

    Inspect the initial combination sin x+a sin(kx) on a full 2π-periodic interval. The starting case is “A weak second harmonic.”

  2. STEP 2

    Follow the changing quantity

    Evolve time from zero to three with diffusivity 0.2. Track the two amplitudes separately while their sum changes shape.

  3. STEP 3

    Explain and test the result

    Compare decay rates 0.2 and 0.2k². Check that mean half-square energy equals one quarter of the sum of the squared amplitudes.

Your turn to explain

Make a prediction. Test your reasoning.

Keep Higher-mode amplitude a = 0.2; Higher frequency k = 2. Pause the timeline at 80%. Given time = 2.4, calculate fundamental amplitude, higher-mode amplitude, mean half-square energy. Show the substitution into the displayed formula.

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

Compare your explanation

At t=2.4, the amplitudes are exp(−0.2(2.4))=0.618783 and 0.2exp(−0.2(2)²(2.4))=0.029321. Orthogonality gives mean half-square energy [0.618783²+0.029321²]/4=0.095938. Results: Fundamental amplitude: 0.619; Higher-mode amplitude: 0.029; Mean half-square energy: 0.096. Decimal values are rounded; retain the original parameters when checking.

Work through a full lesson

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Graduate · Calculus

Watch diffusion damp high spatial frequencies faster

Solve a heat equation through its Fourier eigenmodes.