The same buzz from your vocal cords becomes "ah," "ee," or "oo" depending on how waves bounce inside your throat.
Deep in your throat, two flaps of tissue called vocal cords vibrate 100–300 times per second. Each vibration releases a puff of air.
Here's the crucial insight: this buzz sounds nearly identical for every vowel. "Ah" and "ee" start with the same raw material—a series of pressure pulses.
A smooth sine wave contains only one frequency. But a sharp pulse? It's built from dozens of frequencies layered together—like how a sharp corner in an image needs many fine details to draw.
Each bar = one harmonic (1×, 2×, 3×, 4×... the fundamental frequency)
Here's Fourier's insight in action. Watch how adding more harmonics gradually builds up the sharp, asymmetric pulse shape:
Drag to add harmonics. Notice how more harmonics → sharper features.
Your vocal cords produce a harmonically rich signal—the fundamental frequency plus all its integer multiples. This raw material contains all the frequencies your mouth will later shape into vowels.
Before we go further, let's really understand what's physically happening when sound travels through your throat.
Sound is a pressure wave: regions where air molecules bunch together (compression) alternating with regions where they spread apart (rarefaction).
Each molecule stays mostly in place—it just jiggles back and forth, bumping its neighbors. What travels is the pattern of compression, like a stadium wave where people stand then sit, but the "wave" travels around the arena.
The graph shows pressure vs. position. Where molecules bunch together, pressure is high (red). Where they spread apart, pressure is low (teal). Throughout this explainer, we'll use these pressure graphs—but remember: underneath, real molecules are doing a coordinated dance.
Here's where the magic begins. When a wave traveling through a tube hits a boundary—or any place where the tube width changes—something remarkable happens: part of the wave bounces back.
Waves reflect whenever they encounter a change in acoustic impedance. This is the same physics that causes light to reflect at glass/air boundaries, electrical signals to reflect in mismatched cables, and earthquake waves to reflect at rock layer boundaries.
Definition of acoustic impedance
Acoustic impedance Z is defined as the ratio of sound pressure p to volume velocity U:
For a plane wave in a tube
The linearized momentum equation (Euler's equation) relates pressure gradient to particle acceleration:
For a harmonic wave p = p₀ exp[i(ωt − kx)] traveling at speed c = ω/k:
This is the specific acoustic impedance of air: z = ρc ≈ 413 Pa·s/m at room temperature.
Converting to volume velocity
Volume velocity U is particle velocity v times area A:
Therefore:
Physical intuition
A narrow tube has high impedance (hard to push air through). A wide tube has low impedance (easy to push air through). At any change in area, impedance changes—and some wave energy reflects.
Outside the tube, impedance drops (infinite area). Pressure can't build up, so the wave reflects with opposite sign.
At a wall, impedance is infinite (area = 0). Air can't move past, so the wave bounces back unchanged.
Your vocal tract isn't just open or closed—it has varying widths. Your tongue narrows the passage here, your pharynx widens it there. At every area change, some wave energy reflects.
The bigger the area change, the more reflection. This is how your mouth shapes sound—dozens of small reflections interfering with each other.
Step 1: The traveling wave equations
In a tube, sound travels as pressure waves. We can decompose any wave pattern into forward-traveling (p⁺) and backward-traveling (p⁻) components:
The corresponding volume velocity U (flow rate) is:
where A is cross-sectional area, ρ is air density, and c is sound speed.
Step 2: Boundary conditions at the junction
At a junction between tubes of area A₁ and area A₂, two things must be continuous:
Step 3: Set up the equations
Let p⁺ᵢ be the incident wave (known), p⁻ᵣ be the reflected wave, and p⁺ₜ be the transmitted wave. We have:
Step 4: Solve for the reflection coefficient
Define the reflection coefficient k = p⁻ᵣ / p⁺ᵢ. From the flow equation:
Step 5: Interpret the limiting cases
Transmission coefficient
The transmission coefficient t = p⁺ₜ/p⁺ᵢ can be found from pressure continuity:
t = 1 + k = 2A₁/(A₁ + A₂)
Note: Energy is conserved! The reflected and transmitted power sum to the incident power: (A₁/A₂)t² + k² = 1 when accounting for the area change.
When waves bounce back and forth in a tube, they start interfering with each other. Sometimes they add up. Sometimes they cancel.
At certain special frequencies, the forward and backward waves align perfectly to create a standing wave—a pattern that appears to vibrate in place, with fixed points of maximum and zero motion.
Nodes are points that never move—forward and backward waves always cancel there. Antinodes are points of maximum motion—waves always reinforce there.
Imagine sending pulses down a tube. Each pulse bounces back and forth. For energy to build up, the returning pulse must arrive in sync with the next outgoing pulse—they reinforce each other.
Pulses arrive in sync → energy builds up
Pulses out of sync → destructive interference
This only works when the round-trip time matches the driving frequency. For a tube closed at one end and open at the other:
Step 1: Boundary conditions
A tube closed at one end and open at the other has these boundary conditions:
Step 2: What wavelengths fit?
For a standing wave to form, we need a pressure antinode at the closed end and a pressure node at the open end. The simplest wave that satisfies this is 1/4 of a wavelength:
Step 3: Higher harmonics
We can fit more quarter-wavelengths if we add them in pairs (to maintain the boundary conditions). The pattern is:
Step 4: Convert to frequency
The fundamental wave equation relates wavelength, frequency, and wave speed:
Substituting our wavelength expression:
Step 5: Numerical example
For a 17 cm vocal tract with c = 343 m/s:
Note: Only odd harmonics appear! This is characteristic of quarter-wave resonators. The "missing" even harmonics would require a pressure node at the closed end, which violates our boundary condition.
For a 17 cm tube, this gives resonances at approximately:
The vocal tract isn't a simple tube. Its varying cross-section shifts these resonances to different frequencies. These shifted resonances are called formants.
The simple standing wave picture is a good intuition, but formants don't arise from "pure" standing waves. Instead, the vocal tract acts as an acoustic filter with a transfer function—a mathematical description of how strongly each frequency passes through.
The peaks in this transfer function are the formants. They occur near where standing waves would form, but their exact frequencies depend on the detailed shape of the entire tract.
What is acoustic impedance?
Acoustic impedance Z is the ratio of pressure to volume velocity—it measures how much a system "resists" sound flow:
For a plane wave in a tube of area A:
where ρ ≈ 1.2 kg/m³ (air density) and c ≈ 343 m/s (sound speed).
Why does area matter?
Consider the same pressure wave entering tubes of different sizes:
The reflection mechanism
Sound energy reflects at any impedance discontinuity. The reflection coefficient is:
This explains why the vocal tract's varying cross-section creates its characteristic formant pattern:
The transfer function
With multiple reflections bouncing around, certain frequencies build up (constructive interference) while others cancel out (destructive interference). The frequencies that build up are the formants.
Mathematically, we can compute a transfer function H(f) that describes how much each frequency is amplified or suppressed. The peaks of |H(f)| are the formant frequencies.
For a simple two-tube model (pharynx + oral cavity), the formant frequencies can be found by solving:
This transcendental equation shows why formant frequencies depend on both the lengths and areas of the tube sections—not just total length.
Where you narrow the tube determines which frequencies get boosted. A constriction at a pressure node (where pressure fluctuation is minimal) has little effect on that resonance. But a constriction at a pressure antinode (where pressure fluctuation is maximal) strongly affects it.
The perturbation approach
To understand how a small constriction affects formant frequencies, we use perturbation theory: start with a uniform tube's known resonances, then calculate how a small change modifies them.
Energy in a standing wave
In a standing wave, energy oscillates between two forms:
The perturbation formula
When we change the tube area by a small amount δA at position x, the fractional frequency shift is:
where v(x) and p(x) are the velocity and pressure amplitudes at position x.
Physical interpretation
The formula has two competing terms:
Example: constriction near the glottis
At the closed (glottal) end of the vocal tract:
Example: constriction near the lips
At the open (lip) end:
Why F1 and F2 can move independently
Since F1 and F2 have different standing wave patterns, their pressure and velocity maxima are at different positions along the tract. A constriction at one position might be at F1's pressure maximum but F2's velocity maximum—raising F1 while lowering F2. This is the physical basis for the vowel space!
Drag to see how position affects each resonance mode differently
F1 is controlled mainly by tongue height. For "ee," the tongue is high, creating a tight constriction in the oral cavity where F1's velocity is large—this adds effective mass and lowers F1. For "ah," the tongue is low and the jaw is open—no constriction to slow F1 down, so F1 stays high (near its natural ~500 Hz). That's why open vowels have high F1 and close vowels have low F1.
F2 responds to tongue front/back position. A front tongue constriction creates a short front cavity with a high resonance → high F2 ("ee"). A back tongue constriction creates a short back cavity → low F2 ("oo").
Now we can see the complete picture. Your vocal cords produce a rich buzz. Your throat and mouth filter it—amplifying some frequencies, suppressing others.
This is the source-filter model of speech, one of the most elegant insights in acoustics.
This is crucial: harmonics and formants are completely different things.
Determined by pitch. Always at integer multiples: f₀, 2f₀, 3f₀, 4f₀...
Example: 120 Hz pitch → harmonics at 120, 240, 360, 480, 600...
Determined by tongue, lips, jaw. Not integer multiples.
Example: F1 ≈ 700 Hz, F2 ≈ 1200 Hz, F3 ≈ 2500 Hz
What you hear is the harmonics weighted by how close they are to formants. A harmonic near a formant peak gets amplified; harmonics far from peaks get suppressed.
Equally-spaced "picket fence" at pitch multiples
Smooth curve with peaks at F1, F2, F3...
Harmonics near formant peaks are loud; others are quiet
When you change pitch, the harmonic "picket fence" shifts—but the formant peaks stay put (determined by tract shape). Different harmonics get boosted, but the overall envelope shape stays similar. That's why we recognize an "ah" whether sung high or low.
The resonant frequencies of your vocal tract are called formants. Each vowel has a unique pattern—and plotting F1 against F2 creates a "vowel space" that mirrors tongue position.
F3 auto-interpolates. Drag to override, double-click to reset.
3 values: Red, Green, Blue
3 values: F1, F2, F3
Just as colors blend smoothly from red to orange to yellow, vowels blend smoothly from /i/ to /e/ to /æ/. There are no hard boundaries—both exist in continuous 3D spaces where relative values create perception.
Open jaw (like "ah") = large pharynx = high F1. Closed jaw (like "ee") = constricted pharynx = low F1.
Tongue forward (like "ee") = high F2. Tongue back (like "oo") = low F2.
Rounded lips or retroflex tongue = low F3. This is why the American "r" sound has such a distinctive low F3.
Now you're ready for the full physics model. This is a complete physical simulation—LF glottal source driving a 44-section Kelly-Lochbaum waveguide with a nasal cavity, running at audio sample rate. The same approach used in speech synthesis since 1962, here producing real sound from pure physics.
Peaks show formant frequencies—the resonances of your current tract shape
The sound wave leaving the lips—notice how the shape changes with different vowels
This is a full physical simulation of the vocal tract, running at audio sample rate. Sound is generated by the LF glottal model (modelling vocal cord vibration), then filtered by the 44-section waveguide (modelling the air column). The nasal cavity connects via the velum. Narrow constrictions in the tract produce fricative noise. Try opening the velum for nasal vowels, or reducing tenseness for a breathy voice.
From buzzing vocal cords to bouncing pressure waves to resonant standing waves—you've traced the complete journey of sound through your vocal tract.
Your mouth doesn't create sound—it filters it. The same source signal becomes "ah" or "ee" depending on how your resonances amplify different frequencies.
This explains not just vowels but consonants, singing, voice disguises, speech synthesis, and even why helium makes your voice squeaky (it changes the speed of sound, shifting all resonances up).
Next time you speak, remember: you're orchestrating a symphony of wave reflections, constructive interference, and resonance—all without thinking about it.