Why do I hear a pulsating sound instead of two distinct pitches?

Your ear perceives the rapid oscillations at the average frequency as a single pitch. However, the constructive and destructive interference of the two waves causes the amplitude (loudness) to vary periodically. This variation in loudness is the pulsation you hear, not the individual frequencies themselves. The rate of this pulsation is the beat frequency.

How are beats used to tune a guitar string?

When a reference note (e.g., from a tuning fork) and a guitar string are played together, beats are heard if their frequencies differ. A musician adjusts the string's tension until the beat frequency slows down and finally stops (becomes zero). This indicates the two frequencies are now identical, meaning the string is in tune with the reference.

Does the simulator show what each ear would hear separately?

No. This model simplifies the sound as a single pressure wave at a point in space, representing the combined signal before it reaches a listener. In a real scenario with two separate speakers, the interference pattern (and thus the perception of beats) can depend on the listener's position due to phase differences from path lengths, a complexity not included here.

What happens if the two frequencies are exactly the same?

If f₁ = f₂, then the beat frequency f_beat = |f₁ - f₂| is zero. This means there is no amplitude modulation—the waves interfere constructively everywhere, producing a steady tone with constant amplitude. This is the condition for perfect tuning.

Beat Frequency

When two sound waves of nearly identical frequencies interfere, they produce a phenomenon known as beats. This simulator visualizes and sonifies this effect. The core principle is the superposition of two traveling waves, described by the equations y₁ = A sin(2πf₁t) and y₂ = A sin(2πf₂t). Their sum, y_total = y₁ + y₂, can be rewritten using a trigonometric identity as y_total = [2A cos(2π((f₁ - f₂)/2)t)] sin(2π((f₁ + f₂)/2)t). This result represents a wave oscillating at the average frequency, f_avg = (f₁ + f₂)/2, whose amplitude is modulated by a slowly varying envelope at the beat frequency, f_beat = |f₁ - f₂|. The audible 'waxing and waning' of intensity occurs at this beat frequency. The simulator simplifies the real world by assuming ideal, one-dimensional plane waves with perfect sinusoidal shapes and no damping. It also ignores the complexities of sound propagation in air and the directional characteristics of real sources. By adjusting the individual frequencies and their amplitude, students can directly explore the relationship between frequency difference and beat rate, observe the waveform envelope, and learn how beats are used in practical applications like tuning musical instruments.

Who it's for: High school and introductory university physics students studying wave superposition and sound, as well as music students learning the practical skill of instrumental tuning.

Key terms

Beat Frequency
Superposition Principle
Wave Interference
Amplitude Modulation
Frequency
Sine Wave
Envelope
Tuning

How it works

When two tones are close in frequency, their superposition is a carrier at the average frequency modulated by a slowly varying amplitude. The loudness pulses at the beat frequency |f₁−f₂| — you hear ‘wah-wah’ beats. The canvas scrolls time; dashed lines show the envelope.

Key equations

s = 2 cos(πΔf t) sin(2πfₐᵥₑ t), fₐᵥₑ = (f₁+f₂)/2, Δf = f₁−f₂

Beat frequency (loudness maxima per second) ≈ |Δf| when amplitudes match.

Frequently asked questions

Why do I hear a pulsating sound instead of two distinct pitches?: Your ear perceives the rapid oscillations at the average frequency as a single pitch. However, the constructive and destructive interference of the two waves causes the amplitude (loudness) to vary periodically. This variation in loudness is the pulsation you hear, not the individual frequencies themselves. The rate of this pulsation is the beat frequency.
How are beats used to tune a guitar string?: When a reference note (e.g., from a tuning fork) and a guitar string are played together, beats are heard if their frequencies differ. A musician adjusts the string's tension until the beat frequency slows down and finally stops (becomes zero). This indicates the two frequencies are now identical, meaning the string is in tune with the reference.
Does the simulator show what each ear would hear separately?: No. This model simplifies the sound as a single pressure wave at a point in space, representing the combined signal before it reaches a listener. In a real scenario with two separate speakers, the interference pattern (and thus the perception of beats) can depend on the listener's position due to phase differences from path lengths, a complexity not included here.
What happens if the two frequencies are exactly the same?: If f₁ = f₂, then the beat frequency f_beat = |f₁ - f₂| is zero. This means there is no amplitude modulation—the waves interfere constructively everywhere, producing a steady tone with constant amplitude. This is the condition for perfect tuning.

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