Graduate · Heat semigroup modes · 78 of 650
Heat semigroup modes · Higher-mode amplitude a=0.2; Higher frequency k=3
Use time t=1.8. Find the two Fourier amplitudes and the spatial mean of u²/2. Givens: Higher-mode amplitude a=0.2; Higher frequency k=3.
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01 · Make a prediction
Your practice question
On the 2π-periodic interval solve uₜ=0.2uₓₓ with u(x,0)=sin x+0.2sin(3x). Use time t=1.8. Find the two Fourier amplitudes and the spatial mean of u²/2.
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A starting point
A sine mode with frequency k decays by e^(−0.2k²t). Orthogonality removes the cross term in the spatially averaged energy.
Work through the reasoning
Step 1
Identify the model and target
On the 2π-periodic interval solve uₜ=0.2uₓₓ with u(x,0)=sin x+0.2sin(3x). Use time t=1.8. The governing relation is u=e^(−0.2t)sin x+a e^(−0.2k²t)sin(kx). A sine mode with frequency k decays by e^(−0.2k²t). Orthogonality removes the cross term in the spatially averaged energy.
Step 2
Substitute and calculate
At t=1.8, the amplitudes are exp(−0.2(1.8))=0.697676 and 0.2exp(−0.2(3)²(1.8))=0.007833. Orthogonality gives mean half-square energy [0.697676²+0.007833²]/4=0.121703.
Step 3
Check the mathematical meaning
The higher frequency 3 has decay rate 1.8, larger than 0.2. Mean half-square energy is [0.697676²+0.007833²]/4=0.121703; very small amplitudes remain positive even if an animation readout rounds to zero. Time: 1.8; Fundamental amplitude: 0.698; Higher-mode amplitude: 0.008; Mean half-square energy: 0.122. Decimal values are rounded, so use unrounded intermediate values.
The answer
At t=1.8, the amplitudes are exp(−0.2(1.8))=0.697676 and 0.2exp(−0.2(3)²(1.8))=0.007833. Orthogonality gives mean half-square energy [0.697676²+0.007833²]/4=0.121703. The higher frequency 3 has decay rate 1.8, larger than 0.2. Mean half-square energy is [0.697676²+0.007833²]/4=0.121703; very small amplitudes remain positive even if an animation readout rounds to zero. Animation check: Time: 1.8; Fundamental amplitude: 0.698; Higher-mode amplitude: 0.008; Mean half-square energy: 0.122. Decimal displays are rounded; retain the original parameters and exact π until the final step.
Compare the method as well as the result. A different valid method may reach the same answer. Keep exact values until the last step when the question asks for rounding.
02 · Explore the model
See the mathematical relationship move.
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Watch the relationship
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The mathematical idea
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. On the 2π-periodic interval solve uₜ=0.2uₓₓ with u(x,0)=sin x+0.2sin(3x). Use time t=1.8. The requested state occurs at 60% playback; use the exact target stated in the question for your calculation.
u=e^(−0.2t)sin x+a e^(−0.2k²t)sin(kx)
03 · Reflect and transfer
Explain what changes and why.
Why does doubling a Fourier frequency multiply its decay rate by four, and why is this linear heat calculation not a solution of general nonlinear fluid motion?
This is a distinct guided scenario using a reusable mathematical model. Similar-looking diagrams can represent different given values and conclusions; they are not different mathematical theories.