Waves, Heat, and Hidden Patterns
Master the art of breaking complex signals into simple waves and solving the equations that describe how heat spreads and strings vibrate. Move beyond basic calculus to see the mathematical harmony in physical systems.
Like Duolingo, but for Waves, Heat, and Hidden Patterns. Tomo turns the whole topic into a game you play five minutes a day, until it actually sticks.
180 levels across 8 sections, about 360 minutes end to end, roughly 72 days at five minutes a day. It moves through Building Complex Shapes from Simple Waves, Visualizing Convergence and Approximation, Predicting Heat Flow in One Dimension, The Physics of Vibrating Strings, Expanding to Two Dimensions, The Rigorous Foundation of Convergence, Advanced Tools for Complex Systems, and Real-World Applications and Edge Cases. It assumes you already know the basics.
Free forever · No credit card · iPhone & Android
Key ideas in Waves, Heat, and Hidden Patterns
- Fourier series only describe patterns that repeat in cycles
- Higher frequency waves provide the detail for sudden changes
- The recipe assumes the shape will look the same in the next interval
- Sharp edges require very fast, small ripples to cancel out the curves
- A single wave just gives a basic smooth hump
- Adding more waves makes the top flatter and the sides steeper
- Frequency determines how many ripples fit in a space
- How adding waves changes the overall shape
- Amplitude determines how much a specific wave pushes the shape up or down
- Relationship between wave height/frequency and the final result
- Sines and cosines are the purest, simplest form of smooth oscillation
- Other shapes are actually just combinations of sines and cosines themselves
- Jagged 'sawtooth' patterns can be reconstructed
- Blocky 'square' patterns can be reconstructed
- Any repeating shape, no matter how complex, has a wave recipe
- A complex signal is just a sum of simpler parts
You've tried the other tabs
Thirty open tabs. Four facts you actually kept.
You watched. You nodded. By Sunday it was gone.
One answer, then back to scrolling.
Eight weeks. You meant to finish. You didn't.
Tomo gives Waves, Heat, and Hidden Patterns the Duolingo treatment: levels, streaks, and quick quizzes that test what you just learned. That game loop is what the tabs above never had, so it's the one you actually finish.
Here's what playing it feels like
A real question from this course. Take your best guess.
If you want to add sharp detail to a smooth wave, what kind of waves should you add to the stack?
Get it right to open this lesson and 179 more in the app.
How to learn Waves, Heat, and Hidden Patterns, step by step
Go in order. 180 short levels across 8 sections, about 6 hours in all, five minutes a day.
- 1
Building Complex Shapes from Simple Waves
- The Core Idea: Summing Sines and Cosines
- Matching the Pattern: Finding the Coefficients
- Handling Jumps: The Challenge of Discontinuous Functions
- Symmetry Shortcuts: Even and Odd Functions
- 2
Visualizing Convergence and Approximation
- The Overshoot Problem at Sharp Corners
- How Many Terms Do You Actually Need?
- Energy in Waves: Summing the Squares
- Approximating Smoothness: Why Differentiability Matters
- The Geometry of Function Spaces
- Orthogonality: Why Perpendicular Functions Work
- 3
Predicting Heat Flow in One Dimension
- Setting the Stage: The Heat Equation Setup
- Separating Variables: Turning One Problem into Two
- Fixed Temperatures: Solving with Boundary Conditions
- Insulated Ends: When the Slope is Zero
- Steady State vs. Transient Behavior
- 4
The Physics of Vibrating Strings
- The Wave Equation: Modeling Tension and Motion
- Standing Waves and Natural Frequencies
- Plucked vs. Struck: How Initial Shapes Change Sound
- Traveling Waves: The Two-Direction Solution
- 5
Expanding to Two Dimensions
- Heat Flow on a Flat Plate
- Vibrating Membranes: The Sound of a Drum
- Laplace’s Equation: Finding Equilibrium
- Rectangular vs. Circular Domains
- 6
The Rigorous Foundation of Convergence
- Pointwise vs. Uniform Convergence
- The Dirichlet Kernel: A Deeper Look at Sums
- When Does the Series Exactly Equal the Function?
- Mean Square Error and Best Approximations
- The Role of Integration in Defining Functions
- 7
Advanced Tools for Complex Systems
- Moving to Complex Exponentials
- The Negative Frequency Concept
- Non-Periodic Patterns: Introducing the Transform
- 8
Real-World Applications and Edge Cases
- Signal Filtering: Removing the Noise
- Heat Diffusion in Non-Uniform Materials
- Quantum Mechanics: Waves as Probability
- The Limits of the Fourier Method
- Modern Computational Shortcuts
8 sections · 36 units · 180 levels. Built to play, not to enroll.
Questions about Waves, Heat, and Hidden Patterns
How do I learn Waves, Heat, and Hidden Patterns?
Go in order: start with Building Complex Shapes from Simple Waves, then Visualizing Convergence and Approximation, then Predicting Heat Flow in One Dimension, then The Physics of Vibrating Strings, and 4 more sections after that. On Tomo that's 180 short levels, about 6 hours in all, played five minutes a day. Start free and play the first level in under a minute.
Is there a Duolingo-style app for Waves, Heat, and Hidden Patterns?
Yes. Tomo works like Duolingo: short daily lessons, streaks and quick quizzes. Waves, Heat, and Hidden Patterns is one of its free courses, and you can type any other topic to get a course of your own.
Is the Waves, Heat, and Hidden Patterns course free?
Yes. Waves, Heat, and Hidden Patterns is playable on Tomo's free plan, which doesn't expire. No credit card, no waitlist. Download the app and start your first lesson.
How long does it take to learn Waves, Heat, and Hidden Patterns?
180 levels across 8 sections, about 6 hours of play in total. It's built for five minutes a day, so you finish it a little at a time instead of all at once.
Do I need any background to start Waves, Heat, and Hidden Patterns?
No. It's pitched at an intermediate level, so it starts where you are and builds up. If you can tap a phone, you can start.
Tomo is in ChatGPT
Tomo is also a ChatGPT plugin: the same playable courses, built and played inside your chat.
Add Tomo to ChatGPT- What to ask
- “Make me a Tomo course on black holes.” “Turn this chat into a Tomo course.”
- What you get
- Quick questions with instant feedback, hearts and stars, ready to play in about a minute.
- Where it goes
- No sign-up to start. Connect your Tomo account and your streak carries on in the app.
You pick the voice
Waves, Heat, and Hidden Patterns is taught in the Professor style: clear, structured, thorough. Want a different feel? In the app you can spin up the same topic in any of Tomo's teaching styles. Same facts, totally different vibe.
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Start Waves, Heat, and Hidden Patterns today.
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