Understanding the Fourier Transform Through Tides

The Fourier transform takes a signal that changes over time and reveals which frequencies are hiding inside it. If you haven't encountered it before, 3Blue1Brown's visual explanation is the best introduction I know of.

Any repeating signal, no matter how complicated, can be built by adding together simple sine waves of different frequencies. A tide chart, a heartbeat, a musical chord. Each is just a sum of pure oscillations. The Fourier transform works in reverse. Given a messy signal, it tells you which sine waves were added together to create it, and how strong each one is.

We've known for centuries, thanks to astronomy, exactly which frequencies should appear in the tides. This makes tide data a perfect test case. We can check whether our Fourier analysis actually works by seeing if the expected peaks show up.

Tides and Their Causes

People have been predicting tides for millennia. Water rises and falls roughly twice a day, and the heights vary with the phases of the Moon. But the details get complicated fast.

Tides come from the gravitational pull of the Moon and Sun. As the Earth rotates, different parts of the surface move through regions of higher and lower gravitational pull, and the water sloshes back and forth. The important thing is that these gravitational influences have very specific periods.

Major Tidal Constituents

The naming convention (M2, S2, K1, etc.) comes from 19th-century tidal analysis. "M" means lunar (Moon), "S" means solar (Sun), and the number indicates roughly how many cycles occur per day.

If the Fourier transform works, we should see peaks at exactly these periods.

Getting the Data

NOAA maintains tide gauges across the US coast and provides free historical data through their API. I pulled 20 years of hourly water level measurements from San Francisco (2004-2023), which comes out to about 175,000 data points.

Here's what a few days of tide data looks like:

You can see the roughly twice-daily pattern, with the two daily high tides often having different heights (that's the "diurnal inequality" caused by K1 and O1).

Zooming out to a month:

Now a longer pattern emerges. The tidal range varies over about a two-week cycle. These are "spring tides" (large range, when Moon and Sun align) and "neap tides" (small range, when they're at right angles).

A First Attempt at the Fourier Transform

The Fourier transform takes our sequence of water heights and outputs an "amplitude" for each possible frequency. High amplitude means that frequency is strongly present in the signal.

Here's the power spectrum (amplitude squared) of our tide data:

There are clear peaks jumping out of the noise. Let's see if they match our predictions.

The Hann Window

There's a subtlety when computing the Fourier transform of real data. The FFT assumes your signal repeats forever, with the end wrapping around to the beginning. But our tide data doesn't line up neatly. The water level at the end of our 20-year chunk has no reason to match the level at the beginning, so there's a discontinuity where the signal "jumps" when it wraps around.

This discontinuity creates problems. Sharp jumps require high frequencies to represent, so the FFT sees frequencies that aren't really in the data. Energy from the true peaks spreads into neighboring frequencies. This is called "spectral leakage."

The solution is to multiply our signal by a smoother window function before computing the FFT. The Hann window (named after Julius von Hann) tapers gradually to zero at both ends.

This smooth tapering has a much cleaner frequency response, and the ripples are greatly reduced.

The top panel shows the spectrum computed without windowing. Notice the elevated "skirt" around each peak where energy has leaked into neighboring frequencies. The bottom panel applies the Hann window first, and the peaks are sharper with a lower noise floor between them.

Windowing doesn't change where the peaks are. The frequencies are still correct. It just makes them easier to see and measure accurately.

Matching Peaks to Physics

Now let's look at the full spectrum with our expected tidal constituents labeled:

Every major peak corresponds to a known gravitational effect. There are also many smaller peaks, so let's zoom in.

Semidiurnal Region (11-14 hours)

Besides the big three (M2, S2, N2), we can see numerous minor constituents:

Diurnal Region (22-28 hours)

Long-Period Signals (days to years)

With 20 years of data, we can also detect very low-frequency signals:

The annual signal includes both gravitational effects and meteorological factors (barometric pressure, wind patterns, river discharge).

Why This Matters

The tidal example demonstrates several key concepts:

1. The Fourier transform reveals hidden periodicity. The raw tide signal looks complex, but it's actually a sum of pure sinusoids at specific frequencies.

2. Windowing reduces spectral leakage. Multiplying by a smooth window before the FFT gives cleaner results.

3. The physics determines the frequencies. Each peak in the spectrum corresponds to a real gravitational phenomenon.

Appendix: Complete Constituent Reference

Semidiurnal Constituents (~12 hours)

These arise from the Earth rotating under the gravitational bulge twice per day.

Diurnal Constituents (~24 hours)

These arise because the Moon and Sun are usually not directly over the equator, creating asymmetry between the two daily tidal bulges.

Long-Period Constituents (days to years)

These are low-frequency oscillations caused by longer-term orbital variations.

Data: NOAA CO-OPS, San Francisco station 9414290. 175,320 hourly observations from 2004-2023.