# Lecture 19: Sound Waves and the Doppler Effect ## Physics I — Mechanics & Thermodynamics --- ## Learning Objectives By the end of this lecture, students will be able to: 1. Describe sound as a longitudinal pressure wave and explain how it propagates 2. Calculate the speed of sound in different media 3. Define and use the decibel scale for sound intensity level 4. Explain and apply the Doppler effect for moving sources and observers 5. Describe beats and calculate beat frequency 6. Analyze standing waves in open and closed pipes --- ## Lecture Content ### I. Nature of Sound Waves Sound is a **longitudinal mechanical wave** consisting of alternating compressions and rarefactions of the medium. Because it is a mechanical wave, sound requires a medium and cannot travel through a vacuum. A vibrating source, such as vocal cords, a speaker cone, or a tuning fork, creates pressure variations that propagate outward. These variations are described by Delta P(x, t) = Delta P_max sin(kx - omega t), where Delta P_max is the pressure amplitude, related to the displacement amplitude by Delta P_max = rho v omega s_max. The frequency of a sound wave determines its **pitch**: higher frequency corresponds to higher pitch. The human hearing range spans approximately 20 Hz to 20,000 Hz. Frequencies below 20 Hz are classified as infrasound, and those above 20,000 Hz as ultrasound. The amplitude of a sound wave determines its **loudness**: larger amplitude produces louder sound. ### II. Speed of Sound The speed of sound depends on the properties of the medium. In a gas, v = sqrt(gamma R T / M), where gamma is the ratio of specific heats (1.4 for diatomic gases like air), T is the absolute temperature in Kelvin, and M is the molar mass. In air at 20 degrees C, v is approximately 343 m/s, and a useful approximation is v approximately equal to 331 + 0.6 T (in m/s, with T in Celsius). Sound travels faster in hotter gases (where molecules move faster), in lighter gases (lower molar mass), and generally faster still in liquids and solids where intermolecular forces are stronger. Water carries sound at approximately 1500 m/s and steel at approximately 5900 m/s. ### III. Sound Intensity and the Decibel Scale **Intensity** (I) is the power delivered per unit area. For a point source radiating equally in all directions, I = P / (4 pi r^2), following the inverse-square law, measured in W/m^2. The **threshold of hearing** is I_0 = 10^-12 W/m^2 at 1000 Hz, while the **threshold of pain** is approximately 1 W/m^2. The enormous range of twelve orders of magnitude between these extremes motivates the use of a **logarithmic scale**. The **sound intensity level** (beta) in decibels (dB) is defined as beta = 10 log_10 (I / I_0). On this scale, 0 dB corresponds to the threshold of hearing, every increase of 10 dB represents a tenfold increase in intensity, every increase of 3 dB approximately doubles the intensity, and 120 dB marks the threshold of pain. Common sound levels include a whisper at roughly 30 dB, normal conversation at 60 dB, a rock concert at 110 dB, and a jet engine at 140 dB. A logarithmic scale showing sound intensity levels from 0 dB to 140 dB. At each 10 dB increment, a common sound source is illustrated: 0 dB (threshold of hearing), 30 dB (whisper), 60 dB (normal conversation), 80 dB (busy traffic), 100 dB (power tools), 120 dB (threshold of pain), 140 dB (jet engine at 30 m). The corresponding intensity in W/m^2 is shown on a parallel axis. Hearing damage zones are color-coded (green for safe, yellow for caution, red for dangerous). ### IV. Standing Waves in Pipes An **open pipe** (open at both ends) has pressure nodes (displacement antinodes) at each end. The resonant wavelengths are lambda_n = 2L/n for n = 1, 2, 3, ..., giving resonant frequencies f_n = n v / (2L) = n f_1. All harmonics are present. A **closed pipe** (closed at one end, open at the other) has a displacement node (pressure antinode) at the closed end and a displacement antinode (pressure node) at the open end. The resonant wavelengths are lambda_n = 4L/n for n = 1, 3, 5, ... (odd integers only), giving resonant frequencies f_n = n v / (4L). Only **odd harmonics** are present, and the fundamental frequency is lower than that of an open pipe of the same length. Panel A: An open pipe of length L showing the first three harmonics (n = 1, 2, 3). Displacement standing wave patterns are drawn inside each pipe, with antinodes at both open ends. Wavelengths and frequencies are labeled. Panel B: A closed pipe (closed at the left end) showing the first three allowed modes (n = 1, 3, 5). A node is at the closed end and an antinode at the open end. Only odd harmonics appear. Wavelengths and frequencies are labeled for each mode. ### V. Beats When two waves of slightly different frequencies f_1 and f_2 overlap, the combined wave oscillates at the average frequency f_avg = (f_1 + f_2) / 2, while the amplitude modulates at the **beat frequency** f_beat = |f_1 - f_2|. Beats are perceived as a periodic variation in loudness. They are useful for tuning musical instruments: adjusting the instrument until the beats disappear indicates that the frequencies match. Beats are only perceptible when the two frequencies are close together, within a few Hz of each other. ### VI. The Doppler Effect The **Doppler effect** is the change in observed frequency that occurs when there is relative motion between a source and an observer. For sound, the general Doppler equation is f' = f [(v + v_o) / (v - v_s)], where f is the source frequency, f' is the observed frequency, v is the speed of sound, v_o is the speed of the observer (positive when moving toward the source), and v_s is the speed of the source (positive when moving toward the observer). When the source approaches the observer, f' > f (higher pitch). When the source recedes, f' < f (lower pitch). The same pattern holds for observer motion: approaching increases the observed frequency, and receding decreases it. The Doppler effect for sound is **asymmetric**: a moving source and a moving observer at the same speed produce different frequency shifts, because the medium defines a preferred reference frame. Applications of the Doppler effect include Doppler ultrasound for measuring blood flow velocity, radar speed guns, echocardiography, and the astronomical redshift of light from receding galaxies. ### VII. Shock Waves and the Sonic Boom When a source moves at the speed of sound (v_s = v), the wavefronts pile up in front of it, creating a **sound barrier**. When the source moves supersonically (v_s > v), it outruns its own sound waves, and a **Mach cone** (shock wave) forms behind it. The half-angle of this cone is sin(theta) = v / v_s = 1 / Ma, where the **Mach number** Ma = v_s / v. The shock wave is experienced as a **sonic boom**, a sudden intense pressure pulse. Contrary to a common misconception, sonic booms are produced continuously as long as the source remains supersonic, not just at the moment of "breaking" the sound barrier. Panel A: A stationary source emitting circular wavefronts evenly spaced in all directions. Panel B: A source moving to the right at subsonic speed — wavefronts are compressed ahead (shorter wavelength, higher frequency) and stretched behind (longer wavelength, lower frequency), illustrating the Doppler effect. Panel C: A source moving at supersonic speed — wavefronts overlap to form a Mach cone (V-shaped envelope). The half-angle theta = arcsin(v/v_s) is labeled. The shock wave front is drawn as a bold V-shape trailing behind the source.