Sound Waves Simulator: Hear Physics in Real Time

✓ Verified Content: All equations and data verified against authoritative sources including physics textbooks, ISO standards, and peer-reviewed acoustics references. See verification log

Introduction

Here's what happens when you actually try this: strike a tuning fork labeled "440 Hz" and hold it near your ear. You hear a pure tone, the A above middle C. Now strike two forks, one at 440 Hz and one at 442 Hz. What you expect is two notes. What you get is one note that pulses, getting louder and softer twice per second. The elegant part is that your ears are doing physics in real time, adding waves together and producing interference patterns you can hear.

Most sound simulations show you silent animations. Waves ripple across screens, equations update, but you never actually hear anything. It's like learning to swim by watching videos of water. What Helmholtz figured out in the 1860s (and what every musician knows instinctively) is that hearing and seeing sound together creates understanding that neither alone can provide [1, 2].

Our simulator generates real audio through your speakers or headphones using the Web Audio API. When you adjust the frequency slider, you hear the pitch change. When you drag the source toward the observer in Doppler mode, you hear the pitch rise, then fall as it passes. Set up two frequencies 4 Hz apart and count the beats: one, two, three, four per second. The universe doesn't care about our intuition, but it does reward direct observation.

How to Use This Simulation

Here's what happens when you actually try this: click Play Sound with your volume at a reasonable level and start adjusting the frequency slider. You'll hear pitch change in real time while watching the waveform respond. Most sound simulators are silent animations. This one makes noise.

Main Controls

Operating Modes

Wave Types

Frequency Presets

Wave Parameters

Beats Mode Controls

If you could slow this down enough to watch, you'd see two waves drifting in and out of phase. When aligned, they add (loud). When opposed, they cancel (soft). Your ears do this math automatically.

Doppler Mode

Drag the red S (source) or blue O (observer) across the canvas. Moving the source toward the observer compresses waves and raises pitch. Moving away stretches waves and lowers pitch. The velocity and frequency shift display in real time.

Keyboard Shortcuts

Tips for Exploration

1. Start with sine waves at 440 Hz: This is concert pitch A4. Orchestras worldwide tune to this frequency. It's your reference point.

2. Switch wave types while playing: Hear how square waves sound buzzier than sine waves. The FFT spectrum shows why: more harmonic peaks.

3. Try the Beats mode with 440 Hz and 444 Hz: Count the pulses. Should be exactly 4 per second. This is how piano tuners work.

4. Use Doppler mode actively: Don't just look at it. Drag the source quickly toward the observer while sound plays. Hear the pitch rise. This is the ambulance effect.

5. Enable your microphone and hum: Watch the FFT identify your pitch. Try whistling (nearly pure sine wave) versus speaking (complex harmonics).

What Is a Sound Wave?

Sound is a longitudinal pressure wave: molecules bunching together (compression) and spreading apart (rarefaction) as energy propagates through a medium [1, 2]. Unlike electromagnetic waves, sound requires matter to travel through. In vacuum, there is no sound, no matter how loud the explosion in that movie.

The wave equation describes how displacement varies in space and time:

y(x,t) = A sin(kx - ωt + φ)

Where A is amplitude, k is wave number (2π/λ), ω is angular frequency (2πf), and φ is phase. If you could slow this down enough to watch individual air molecules, you'd see them oscillating back and forth along the direction of propagation, not up and down like waves on a string.

How the Simulator Works

Four modes let you explore different phenomena:

• Waveform: Basic wave visualization with FFT spectrum
• Beats: Two frequencies creating interference patterns
• Doppler: Draggable source and observer demonstrating frequency shift
• Microphone: Analyze real sounds with FFT

Types of Waveforms

Sine Waves (Pure Tones)

A tuning fork produces something close to a sine wave: a single frequency with no harmonics. The elegant part is that Joseph Fourier proved any periodic wave can be decomposed into a sum of sine waves [4, 5]. A sine wave is therefore the fundamental building block, the hydrogen atom of acoustics.

Strike a tuning fork, and the prongs vibrate at one frequency determined by their mass and stiffness. The resulting sound wave has the mathematical form y(t) = A sin(2πft). Pure. Simple. You'll recognize the sound as "clean" or "hollow" compared to musical instruments.

Square Waves

If you could snap air pressure instantly between high and low states, you'd create a square wave. The universe doesn't care about our intuition about what "should" sound harsh, but square waves do sound buzzy because they contain all odd harmonics (3rd, 5th, 7th, etc.) at amplitudes of 1/n.

The Fourier series is: 4/π × [sin(ωt) + sin(3ωt)/3 + sin(5ωt)/5 + ...]

Early video games used square waves because they're computationally cheap to generate. That "chiptune" sound? Pure odd harmonics.

Triangle Waves

Triangle waves rise and fall linearly, creating a softer sound than square waves. They also contain only odd harmonics, but the amplitudes fall off as 1/n², making higher harmonics much weaker. The result sounds mellower, almost flute-like at higher frequencies.

Sawtooth Waves

Sawtooth waves rise linearly then drop sharply. They contain both odd and even harmonics, making them the richest simple waveform. The sound is bright and buzzy. Bowed string instruments produce waveforms similar to sawtooths because the bow alternately grips and slips across the string.

Key Parameters

Essential Formulas

Wave Equation

y(x,t) = A sin(kx - ωt + φ)

The complete description of a traveling wave. Position x, time t, amplitude A, wave number k, angular frequency ω, and phase φ. If you could freeze time and walk along the wave, you'd see the sine shape. If you could stand still and watch time pass, you'd see oscillation at frequency f.

Wave Speed Relationship

v = fλ = ω/k

The elegant constraint connecting frequency, wavelength, and medium properties. For sound in air at 20°C, v ≈ 343 m/s [1, 7]. Double the frequency, halve the wavelength. The product is constant for a given medium.

Doppler Effect

f_observed = f_source × (v + v_observer)/(v - v_source)

What Christian Doppler described in 1842 to explain why stars might appear different colors depending on their motion [1, 4]. The same physics makes ambulance sirens rise in pitch as they approach. Moving toward the source? Waves bunch up, frequency rises. Moving away? Waves stretch, frequency falls.

Beat Frequency

f_beat = |f₁ - f₂|

When two waves of slightly different frequencies superpose, the amplitude modulates at their difference frequency. Set f₁ = 440 Hz and f₂ = 443 Hz, and you'll hear 3 pulses per second. Piano tuners use this phenomenon: tune until the beats disappear, and the strings match.

Sound Intensity Level

β = 10 log₁₀(I/I₀)

Where I₀ = 10⁻¹² W/m² (threshold of hearing). The decibel scale matches human perception: 10 dB sounds roughly twice as loud. A normal conversation is about 60 dB; a rock concert, 110 dB; a jet engine at 30 meters, 140 dB (pain threshold) [1, 2].

Learning Objectives

After working through this simulation, you will be able to:

1. Hear the difference between waveforms and explain why square waves sound buzzier than sine waves (harmonics)
2. Calculate wavelength from frequency using v = fλ and predict how pitch changes with frequency
3. Experience the Doppler effect by dragging the source and observer, then explain the physics
4. Count beats and verify that beat frequency equals |f₁ - f₂|
5. Analyze real sounds using the microphone and FFT, identifying fundamentals and harmonics
6. Predict how temperature affects sound speed and therefore wavelength

Exploration Activities

Activity 1: Waveform Comparison

Objective: Hear and see why different waveforms sound different

Steps:

1. Set frequency to 440 Hz (concert A, the reference pitch)
2. Select Sine wave and click Play Sound
3. Watch the smooth waveform and note the clean tone
4. Switch to Square wave without stopping
5. Notice the harsher timbre and watch the FFT spectrum light up with harmonics
6. Try Triangle (mellower) and Sawtooth (brightest)

What to observe: The FFT spectrum reveals the secret. Sine waves show one peak. Square waves show peaks at odd harmonics (3×, 5×, 7× the fundamental). Sawtooth shows all harmonics. Your ears hear "timbre"; the physics is harmonic content.

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Activity 2: Doppler Effect

Objective: Hear pitch shift from relative motion

Steps:

1. Switch to Doppler Effect mode
2. Click Play Sound to start a continuous 440 Hz tone
3. Drag the red source (S) rapidly toward the blue observer (O)
4. Listen for the pitch to rise as the source approaches
5. Drag past the observer and away
6. Listen for the pitch to drop below 440 Hz

What to observe: The Observed Frequency display shows the math in real time. At 50 m/s toward the observer, frequency rises about 15%. At 50 m/s away, it drops about 13%. The asymmetry is real: Doppler shift isn't symmetric around the source frequency.

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Activity 3: Beat Frequency

Objective: Hear interference between two close frequencies

Steps:

1. Switch to Beats mode
2. Set Frequency 1 to 440 Hz
3. Set Frequency 2 to 444 Hz
4. Click Play Sound and count the pulses: should be 4 per second
5. Change Frequency 2 to 441 Hz: now 1 beat per second
6. Set both to 440 Hz: beats disappear entirely

What to observe: This is how piano tuners work. Two strings for the same note beat until they're tuned to match. The beat rate tells you how far off you are. The elegant part: your ear is performing real-time wave addition and detecting the amplitude envelope.

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Activity 4: Microphone Analysis

Objective: Analyze real sounds and identify their frequency content

Steps:

1. Switch to Microphone mode
2. Grant microphone permission
3. Hum a steady note near your device
4. Watch the detected pitch and note name
5. Whistle (produces nearly pure sine waves)
6. Speak vowels and watch the complex harmonic structure

What to observe: The FFT display shows that human voice contains a fundamental frequency plus many harmonics. The specific pattern of harmonics is what makes your voice recognizable. Whistling produces something close to a sine wave (one peak). Vowels show complex harmonic fingerprints.

Real-World Applications

1. Musical Instrument Tuning: The A440 standard (ISO 16:1975) gives orchestras a universal reference. Electronic tuners use FFT analysis identical to our microphone mode. Before electronic tuners, musicians relied on beat frequencies between reference forks and their instruments.

2. Medical Doppler Ultrasound: Blood flow velocity is measured by detecting frequency shifts in reflected ultrasound [1]. A typical fetal heart monitor uses 2-3 MHz sound waves. The Doppler shift from moving blood cells is in the audible range, which is why you can hear the heartbeat through the speaker.

3. Police Radar: Same Doppler physics, but with electromagnetic waves instead of sound. A 24 GHz radar beam reflected from a car moving at 100 km/h shifts by about 4.4 kHz, easily measured to determine speed.

4. Noise-Canceling Headphones: Microphones sample ambient noise, DSP generates inverted waveforms, speakers output the anti-noise. When the phases align, destructive interference reduces ambient sound by 20-30 dB. Same physics as beat frequency, engineered for silence.

5. Concert Hall Acoustics: Reverberation time, standing waves, and interference patterns determine whether a hall sounds "warm" or "muddy" [1, 4]. The Royal Albert Hall famously required acoustic diffusers to break up problematic reflections.

Reference Data

Speed of Sound in Different Media

Musical Note Frequencies (Equal Temperament, A4 = 440 Hz)

Human Hearing Range

Challenge Questions

1. Conceptual: A police car siren emits 700 Hz when stationary. As it approaches at 30 m/s, what frequency do you hear? (Use v = 343 m/s.) What frequency as it recedes at 30 m/s? Why aren't these shifts symmetric around 700 Hz?

2. Calculation: An organ pipe produces middle C (262 Hz) at 20°C. What is the wavelength? If the room cools to 0°C, what happens to (a) the speed of sound, (b) the wavelength, and (c) the frequency?

3. Analysis: Two guitar strings beat at 3 Hz. One string is tuned to 330 Hz. What are the two possible frequencies for the other string? How would tightening the second string help you determine which is correct?

4. Application: An ultrasound transducer emits 5 MHz into tissue (v = 1540 m/s). Blood flows toward it at 0.5 m/s. Calculate the Doppler shift. Why is this shift in the audible range even though 5 MHz isn't?

5. Design: A hearing test sweeps from 20 Hz to 20 kHz in 30 seconds. If you want to spend equal time on each octave (there are about 10 octaves), how should frequency vary with time? (Hint: logarithmic, not linear.)

Common Misconceptions

1. Frequency and Amplitude Are the Same Thing

Frequency determines pitch (high or low). Amplitude determines loudness (soft or loud). Double the frequency and pitch rises one octave. Double the amplitude and loudness increases by about 6 dB. These are independent properties.

2. Sound Travels at "The Speed of Sound"

Which speed? In air at 20°C, 343 m/s. At 0°C, 331 m/s. In water, 1500 m/s. In steel, 6000 m/s. The speed of sound is a property of the medium, not a universal constant [1, 7].

3. Doppler Shift Is Symmetric

Moving toward a source at speed v increases frequency more than moving away at speed v decreases it. The formula is asymmetric: approaching a source moving at 0.5v gives infinite frequency (sonic boom); receding at 0.5v gives only half the source frequency.

4. Beats Are at the Sum Frequency

Beats occur at the difference frequency |f₁ - f₂|, not the sum. Two 440 Hz and 445 Hz tones produce 5 beats per second. The perceived pitch is near the average (442.5 Hz), not the sum (885 Hz).

5. Higher Frequency = Louder

Higher frequency means higher pitch, not necessarily louder. Human ears are most sensitive around 2-4 kHz; a 3 kHz tone sounds louder than a 100 Hz tone at the same physical intensity. See the Fletcher-Munson curves for the full picture.

Frequently Asked Questions

Why do beats occur between two close frequencies?

When two waves of frequencies f₁ and f₂ add together, the amplitude varies at frequency |f₁ - f₂| [2, 5]. At instants when both waves are in phase, they add constructively (loud). When 180° out of phase, they partially cancel (quiet). Your ear perceives this as pulsing at the beat frequency.

How does FFT analysis work?

The Fast Fourier Transform decomposes a time-domain signal into its frequency components [3, 5]. Any periodic signal can be represented as a sum of sine waves at different frequencies. The FFT reveals which frequencies are present and their relative amplitudes. Our simulator runs this in real time using Web Audio API's AnalyserNode.

Why does a square wave sound different from a sine wave?

A square wave contains odd harmonics (3×, 5×, 7× the fundamental) at amplitudes of 1/n. These harmonics add "buzz" to the sound. A sine wave has no harmonics, sounding "pure" or "hollow." The FFT display shows this directly: sine has one peak, square has many.

What determines the speed of sound?

Speed depends on the medium's stiffness and density: v = √(B/ρ) for fluids, where B is bulk modulus and ρ is density [1, 7]. In gases, temperature matters because hotter molecules move faster. In air, v ≈ 331 + 0.6T m/s, where T is temperature in Celsius.

Why do ambulance sirens change pitch?

The Doppler effect. As the ambulance approaches, each successive wave crest is emitted from a position closer to you, so crests arrive more frequently (higher pitch). As it recedes, crests are emitted from positions farther away, arriving less frequently (lower pitch). The effect is larger at higher source speeds.

References

1. HyperPhysics - Sound and Hearing: Georgia State University comprehensive acoustics resource. Available at: http://hyperphysics.gsu.edu/hbase/Sound/soucon.html (Free educational)

2. The Physics Classroom - Sound Waves: Interactive lessons on wave properties and behavior. Available at: https://www.physicsclassroom.com/class/sound (Free educational)

3. W3C Web Audio API Specification: Technical standard for web audio implementation. Available at: https://www.w3.org/TR/webaudio/ (Free W3C standard)

4. MIT OpenCourseWare 8.03: Physics III - Vibrations and Waves lecture notes. Available at: https://ocw.mit.edu/courses/8-03sc-physics-iii-vibrations-and-waves-fall-2016/ (CC BY-NC-SA)

5. LibreTexts Physics - Sound: Open-access textbook covering acoustics and wave phenomena. Available at: https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_University_Physics_(OpenStax)/Book%3A_University_Physics_I_-_Mechanics_Sound_Oscillations_and_Waves_(OpenStax)/17%3A_Sound (CC BY)

6. Khan Academy - Sound: Video lessons and practice on sound waves and properties. Available at: https://www.khanacademy.org/science/physics/mechanical-waves-and-sound/sound-topic (Free educational)

7. Engineering Toolbox - Speed of Sound: Reference data for speed of sound in various media. Available at: https://www.engineeringtoolbox.com/speed-sound-d_82.html (Free technical resource)

8. NIST - Physical Constants: Standard reference for physical constants including acoustic properties. Available at: https://physics.nist.gov/cuu/Constants/ (Public domain)

About the Data

Speed of sound values are from Engineering Toolbox and verified against HyperPhysics data [1, 7]. Musical frequencies use equal temperament tuning with A4 = 440 Hz (the internationally recognized standard since 1939, formally adopted as ISO 16 in 1975). Human hearing ranges are based on standard audiometric data from educational sources [1, 2, 6]. The simulation's FFT analysis uses a 2048-sample buffer at 44.1 kHz, giving approximately 21 Hz frequency resolution per the Web Audio API specification [3].

How to Cite

Simulations4All Team. "Sound Waves Simulator." Simulations4All, 2025. Available at: https://simulations4all.com/simulations/sound-waves-simulator

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