Graduate · Fourier incompressibility projection · 113 of 650
Fourier incompressibility projection · Frequency component k₁=3; Frequency component k₂=1.5
Use θ=6π/5 radians. Compute the projected coefficient Pₖv=v−k(k·v)/|k|², its dot product with k, and its norm. Givens: Frequency component k₁=3; Frequency component k₂=1.5.
Try the question, use a hint when you need one, and compare your reasoning with the worked solution. The animation starts with this chapter’s values; changing its controls explores a new case.
01 · Make a prediction
Your practice question
At the nonzero Fourier frequency k=(3,1.5), use the real unit coefficient v=(cos θ,sin θ). Use θ=6π/5 radians. Compute the projected coefficient Pₖv=v−k(k·v)/|k|², its dot product with k, and its norm.
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A starting point
Subtract the part parallel to k. This leaves the coefficient perpendicular to k, as required for this incompressibility condition.
Work through the reasoning
Step 1
Identify the model and target
At the nonzero Fourier frequency k=(3,1.5), use the real unit coefficient v=(cos θ,sin θ). Use θ=6π/5 radians. The governing relation is Pₖv=v−k(k·v)/|k|²; k·Pₖv=0. Subtract the part parallel to k. This leaves the coefficient perpendicular to k, as required for this incompressibility condition.
Step 2
Substitute and calculate
For k=(3,1.5), |k|²=11.25. Subtract k(k·v)/|k|² from v=(-0.809017,-0.587785) to get (0.073311,-0.146621); the check (3)(0.073311)+(1.5)(-0.146621)=0 verifies perpendicularity.
Step 3
Check the mathematical meaning
The denominator |k|²=11.25 is nonzero. The projected norm 0.163928 is at most one, and k·Pₖv=0 up to floating-point roundoff. Projected x: 0.073; Projected y: -0.147; k dot projected vector: 0; Projected norm: 0.164. Decimal values are rounded, so use unrounded intermediate values.
The answer
For k=(3,1.5), |k|²=11.25. Subtract k(k·v)/|k|² from v=(-0.809017,-0.587785) to get (0.073311,-0.146621); the check (3)(0.073311)+(1.5)(-0.146621)=0 verifies perpendicularity. The denominator |k|²=11.25 is nonzero. The projected norm 0.163928 is at most one, and k·Pₖv=0 up to floating-point roundoff. Animation check: Projected x: 0.073; Projected y: -0.147; k dot projected vector: 0; Projected norm: 0.164. 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
At a nonzero Fourier frequency, incompressibility requires the velocity coefficient to be perpendicular to that frequency. The orthogonal projection removes only the longitudinal component. Rotating an input coefficient shows which information is kept and which is discarded without pretending to solve the full fluid dynamics. At the nonzero Fourier frequency k=(3,1.5), use the real unit coefficient v=(cos θ,sin θ). Use θ=6π/5 radians. The requested state occurs at 60% playback; use the exact target stated in the question for your calculation.
Pₖv=v−k(k·v)/|k|²; k·Pₖv=0
03 · Reflect and transfer
Explain what changes and why.
Why does projecting this one Fourier coefficient neither evolve the whole velocity field nor settle the general Navier–Stokes regularity problem?
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.