Unit 9: Acoustic Phenomena
Sound Waves and Pressure Amplitude
Sound is a mechanical wave that propagates through a medium by causing local variations in pressure and density. In air, these variations are longitudinal: the particle displacement is parallel to the direction of wave travel.
Pressure Amplitude
The instantaneous pressure at a point can be expressed as the sum of the static atmospheric pressure p₀ and a time‑varying component Δp:
p(x,t) = p₀ + Δp sin(kx − ωt)
Here k = 2π/λ is the wave number, ω = 2πf the angular frequency, λ the wavelength and f the frequency. The pressure amplitude (also called acoustic pressure amplitude) is the maximum magnitude of the varying part:
Δp_max = |Δp|
It represents how far the instantaneous pressure deviates from the ambient value. For a plane wave travelling in a lossless medium, the pressure amplitude is related to the particle velocity amplitude u₀ by the characteristic impedance Z = ρc:
Δp_max = Z·u₀ = ρc·u₀
where ρ is the density of the medium and c the speed of sound.
Example 1 – Calculating Pressure Amplitude from Intensity
The intensity I of a plane wave is given by
I = (Δp_max)² / (2ρc)
Suppose a sound wave in air (ρ = 1.20 kg m⁻³, c = 343 m s⁻¹) has an intensity of 1.0 × 10⁻⁶ W m⁻². Solving for Δp_max:
Δp_max = √(2ρcI) = √(2·1.20·343·1.0×10⁻⁶) ≈ 0.029 Pa
Thus the pressure variation is only about 0.03 Pa, a tiny fraction of atmospheric pressure (~101 kPa).
Characteristics of Sound
Intensity
Intensity is the average power transmitted per unit area perpendicular to the direction of propagation:
I = P / A
where P is the acoustic power (watts) and A the area (square metres). For a spherical wave radiating from a point source, intensity falls off with the square of the distance:
I(r) = P / (4πr²)
Example 2 – Intensity of a 60 dB Sound
The reference intensity I₀ = 1.0×10⁻¹² W m⁻² corresponds to the threshold of hearing. A sound level of L = 60 dB relates to intensity via
L = 10 log₁₀(I / I₀)
Re‑arranging:
I = I₀·10^{L/10} = 1.0×10⁻¹²·10^{6} = 1.0×10⁻⁶ W m⁻²
Thus a 60 dB conversation carries about one microwatt per square metre.
Loudness
Loudness is a perceptual quantity; it grows approximately logarithmically with intensity. The phon scale matches loudness to a 1 kHz tone, while the decibel (dB) scale measures intensity ratio.
The relationship between loudness level L_N (in phons) and intensity is
L_N = 10 log₁₀(I / I₀)(for a 1 kHz reference)
Because the ear’s sensitivity varies with frequency, equal‑loudness contours (Fletcher‑Munson curves) show that a 1 kHz tone at 40 dB sounds as loud as a 100 Hz tone at about 50 dB.
Example 3 – Doubling Intensity
If the intensity of a sound is doubled, the change in decibel level is
ΔL = 10 log₁₀(2I / I) = 10 log₁₀(2) ≈ 3.01 dB
Thus a perceptible increase of roughly 3 dB corresponds to a doubling of acoustic power.
Quality (Timbre)
Timbre enables us to distinguish two sounds that have the same loudness and pitch but are produced by different instruments or voices. It depends on the spectral content: the relative amplitudes of the fundamental frequency and its harmonics, as well as the attack, decay, and envelope of the waveform.
For instance, a violin and a flute playing the same A₄ (440 Hz) produce different waveforms because the violin’s spectrum contains strong odd‑and‑even harmonics, whereas the flute’s spectrum is richer in the fundamental and weaker in higher harmonics.
Pitch
Pitch is the perceptual correlate of frequency. Higher frequency → higher pitch. The just‑noticeable difference (JND) in pitch is about 0.5 % for mid‑frequency tones.
The fundamental frequency f of a wave travelling at speed c with wavelength λ is given by
f = c / λ
In air at 20 °C, c ≈ 343 m s⁻¹. A wavelength of 1 m therefore corresponds to a pitch of about 343 Hz (roughly F₄).
Example 4 – Musical Note Frequencies
- Middle C (C₄):
f ≈ 261.63 Hz - Concert A (A₄):
f = 440.00 Hz - High C (C₅):
f ≈ 523.25 Hz
Doppler's Effect
Concept
The Doppler effect describes the change in observed frequency (f') of a wave when there is relative motion between the source and the observer. For sound waves in a medium, the speed of sound c is constant relative to the medium.
Source Moving, Observer Stationary
If the source moves with speed v_s toward a stationary observer, the wavelength in front of the source is compressed:
λ' = (c − v_s) / f
The observed frequency becomes
f' = c / λ' = f·c / (c − v_s)
When the source moves away, the sign reverses:
f' = f·c / (c + v_s)
These two cases are commonly written as
f' = f·c / (c ∓ v_s)(minus for approach, plus for recession)
Example 5 – Ambulance Siren
An ambulance siren emits a frequency f₀ = 800 Hz. The speed of sound is c = 340 m s⁻¹. If the ambulance approaches a stationary listener at v_s = 30 m s⁻¹:
f' = 800·340 / (340 − 30) = 800·340 / 310 ≈ 877 Hz
After it passes and moves away at the same speed:
f' = 800·340 / (340 + 30) = 800·340 / 370 ≈ 735 Hz
The listener hears a pitch that rises as the ambulance nears and falls as it recedes.
Observer Moving, Source Stationary
If the observer moves with speed v_o toward a stationary source, the relative speed of the wavefronts meeting the observer is c + v_o. The observed frequency is
f' = f·(c + v_o) / c
When moving away:
f' = f·(c − v_o) / c
Compact form:
f' = f·(c ± v_o) / c(plus for approach, minus for recession)
Example 6 – Listener on a Moving Train
A train whistle emits f₀ = 500 Hz. A passenger on the train moves toward the source at v_o = 15 m s⁻¹ (the whistle is fixed on the train). Since both source and observer share the same motion, there is no Doppler shift; however, if the whistle is on the platform and the passenger moves toward it:
f' = 500·(340 + 15) / 340 = 500·355 / 340 ≈ 522 Hz
The passenger perceives a higher pitch.
Both Source and Observer Moving
When both are in motion, the general formula combines the two effects:
f' = f·(c ± v_o) / (c ∓ v_s)
The upper signs apply when the observer moves toward the source or the source moves toward the observer (i.e., decreasing the effective path length); the lower signs apply for recession.
Example 7 – Police Car and Moving Pedestrian
A police car siren emits f₀ = 1000 Hz. The car moves toward a pedestrian at v_s = 20 m s⁻¹, while the pedestrian walks toward the car at v_o = 2 m s⁻¹. Using the formula for approach (both minus in denominator, plus in numerator):
f' = 1000·(340 + 2) / (340 − 20) = 1000·342 / 320 ≈ 1069 Hz
If the pedestrian walks away (v_o = −2 m s⁻¹) while the car still approaches:
f' = 1000·(340 − 2) / (340 − 20) = 1000·338 / 320 ≈ 1056 Hz
Applications of the Doppler Effect
- Radar Speed Guns: Emit radio waves; the frequency shift of the reflected wave from a moving vehicle yields its speed.
- Medical Ultrasound (Doppler Flow): Measures blood flow velocity by detecting frequency shifts of ultrasound reflected from moving red blood cells.
- Astronomy – Redshift/Blueshift: Light from receding galaxies is shifted to longer wavelengths (redshift), allowing determination of cosmic expansion rates.
- Sonar and Navigation: Used to determine the speed of underwater objects or submarines