Sound, Beats, and the Doppler Effect
Three genuinely separate ideas share this unit because they're all things a listener actually experiences with real sound: how loud something is (intensity, on the compressed decibel scale human hearing actually uses), what happens when two close-but-different frequencies overlap (beats — a direct, audible consequence of Unit 301's superposition), and why a sound's pitch shifts when the source or listener is moving (the Doppler effect, which is really just a statement about wave speed being fixed relative to the medium, not relative to either party).
What you'll learn
- Calculate sound intensity level in decibels from intensity, and vice versa.
- Calculate the beat frequency produced by two sound sources of slightly different frequency.
- Apply the Doppler effect formula for a moving source and a stationary observer.
- Apply the Doppler effect formula for a moving observer and a stationary source.
- Determine whether a Doppler-shifted frequency increases or decreases based on the relative motion involved.
- Explain beats and the Doppler effect both as consequences of wave superposition and wave-speed reasoning, respectively.
1. Intensity and the Decibel Scale
Sound intensity (power per unit area, W/m²) spans an enormous range for sounds humans actually encounter — from the threshold of hearing (I₀=10⁻¹² W/m²) to sounds a million times more intense or beyond. The decibel scale compresses this into a manageable logarithmic range: β=10log₁₀(I/I₀).
Because the scale is logarithmic, equal dB *differences* correspond to equal intensity *ratios*, not equal intensity differences — a 10 dB increase always means a 10× intensity increase, whether going from 20 dB to 30 dB or from 80 dB to 90 dB.
2. Beats
When two sound waves of slightly different frequency overlap, their superposition produces a resultant whose amplitude itself oscillates slowly — heard as a periodic loud-soft pulsing called beats, at a rate equal to the difference of the two frequencies: f_beat=|f₁−f₂|. This is a direct, audible demonstration of Unit 301's superposition principle, not a separate physical mechanism.
3. The Doppler Effect
When a sound source and observer move relative to each other, the observer perceives a different frequency than the source emits — not because the wave speed changes (it doesn't; sound always travels at v relative to the medium), but because motion changes either the effective wavelength (moving source) or the rate at which wavefronts are encountered (moving observer).
The two cases use structurally different formulas: f'=fv/(v∓vₛ) for a moving source, f'=f(v±v₀)/v for a moving observer — approaching always raises the perceived frequency, receding always lowers it, in both cases, but the exact size of the shift differs between the two scenarios even at equal speeds.
Key equations
- β = 10 log₁₀(I/I₀) — Converts intensity (W/m², which spans many orders of magnitude for audible sound) to the compressed, human-hearing-matched decibel scale.
- f_beat = |f₁ − f₂| — The beat frequency heard when two sounds of slightly different frequency overlap — audible as a periodic loud-soft pulsing, at a rate equal to the difference of the two frequencies.
- f' = f(v/(v∓vₛ)) — Doppler shift for a moving source and stationary observer.
- f' = f((v±v₀)/v) — Doppler shift for a stationary source and moving observer.