Physics

Chapter 17 Sound

Imron Rosyadi

Chapter Overview

• 17.1 Sound Waves
• 17.2 Speed of Sound
• 17.3 Sound Intensity
• 17.4 Normal Modes of a Standing Sound Wave
• 17.5 Sources of Musical Sound
• 17.6 Beats
• 17.7 The Doppler Effect
• 17.8 Shock Waves

17.1 Sound Waves

Learning Objectives

By the end of this section, you will be able to:

• Explain the difference between sound and hearing
• Describe sound as a wave
• List the equations used to model sound waves
• Describe compression and rarefactions as they relate to sound

This section introduces the fundamental concept of sound as a physical phenomenon and distinguishes it from the perception of hearing. We’ll explore how sound behaves as a wave and introduce the mathematical models used to describe it, along with key terms like compressions and rarefactions.

Sound vs. Hearing

Sound

• A disturbance of matter transmitted from its source outward.
• On an atomic scale, it’s an ordered disturbance of atoms.
• Often a periodic wave, causing atoms to undergo simple harmonic motion.
• Can induce oscillations and resonance effects.

Hearing

• The perception of sound.
• Analogous to seeing, which is the perception of visible light.
• Involves complex biological processes in the ear.

Sound is the physical event—the vibration and transfer of energy through a medium. Hearing is how our brain interprets that physical event. A good analogy is the difference between light (the electromagnetic wave) and seeing (our brain’s interpretation).

Sound as a Wave

• Speakers produce sound waves by oscillating a cone.
• This motion causes vibrations in surrounding air molecules.
• Energy is transferred as slight compressions (high-pressure) and rarefactions (low-pressure) in the air.

Note

Sound waves in air and most fluids are longitudinal waves because fluids have almost no shear strength.

When a speaker cone moves forward, it pushes air molecules together, creating a region of higher pressure (compression). When it moves backward, it leaves a region where air molecules are spread out, creating lower pressure (rarefaction). These compressions and rarefactions propagate outward as a sound wave.

Visualizing Sound Waves

Figure 17.3 (a) Gauge pressure vs. distance from a speaker. (b) Displacement of air molecules vs. position.

Figure (a) shows how gauge pressure oscillates sinusoidally, with crests aligning with compressions and troughs with rarefactions. Figure (b) illustrates the displacement of air molecules, which is zero at compressions and rarefactions, and maximal where pressure changes are steepest. This reinforces that sound is a longitudinal wave.

Modeling Sound Waves: Pressure and Displacement

1. Pressure Wave Model \[ \Delta P = \Delta P_{\text{max}}\sin(kx \mp \omega t + \phi) \] - \(\Delta P\): Change in pressure from average pressure - \(\Delta P_{\text{max}}\): Maximum change in pressure (pressure amplitude) - \(k = \frac{2\pi}{\lambda}\): Wave number - \(\omega = \frac{2\pi}{T} = 2\pi f\): Angular frequency - \(\phi\): Initial phase

This equation describes how the pressure in the medium varies as the sound wave propagates. It’s a sinusoidal function, similar to other wave equations we’ve studied. The maximum pressure change, \(\Delta P_{\text{max}}\), is directly related to the loudness of the sound.

Modeling Sound Waves: Displacement

2. Displacement Wave Model \[ s(x,t) = s_{\text{max}}\cos(kx \mp \omega t + \phi) \] - \(s\): Displacement of air molecules from equilibrium - \(s_{\text{max}}\): Maximum displacement (displacement amplitude)

Note

The energy of a sound wave spreads over a larger area as it moves away from the source, causing intensity to decrease. Energy is also lost to thermal energy through air viscosity and heat transfer.

This equation describes the actual back-and-forth motion of the individual air molecules. Notice the cosine function; when pressure is at a maximum (compression), the displacement of molecules tends to be zero at that exact point. The key takeaway is that sound has characteristics like wavelength, frequency, amplitude, and speed, just like any other wave.

17.2 Speed of Sound

Learning Objectives

By the end of this section, you will be able to:

• Explain the relationship between wavelength and frequency of sound
• Determine the speed of sound in different media
• Derive the equation for the speed of sound in air
• Determine the speed of sound in air for a given temperature

In this section, we’ll delve into how fast sound travels and what factors influence its speed. We’ll connect frequency and wavelength to speed, explore sound speed in various materials, and derive the specific formula for sound in air, emphasizing its temperature dependence.

Observing the Speed of Sound

Figure 17.4: We see fireworks before hearing them, showing sound travels slower than light.

Tip

To estimate distance to a lightning strike: Count seconds between flash and thunder. Every 5 seconds is approximately 1 mile.

The observation of fireworks or lightning clearly demonstrates that sound travels at a finite speed, and it’s much slower than the speed of light. This everyday experience highlights the importance of understanding sound’s propagation speed.

Wave Properties and Speed

• The velocity of any wave relates to its frequency (\(f\)) and wavelength (\(\lambda\)): \[ v = f\lambda \]
• Wavelength (\(\lambda\)): Distance between sequential identical points in a wave (e.g., between compressions).
• Frequency (\(f\)): Number of waves passing a point per unit time (same as source frequency).

Figure 17.5: A sound wave from a tuning fork, showing speed, frequency, and wavelength.

This fundamental equation, \(v = f\lambda\), is crucial for all types of waves, including sound. It shows an inverse relationship between frequency and wavelength for a constant wave speed. If frequency increases, wavelength must decrease, and vice versa.

Speed of Sound in Various Media

• Speed of sound varies greatly depending on the medium.
• Depends on:
  • Elastic property (restoring force/stiffness) \(\rightarrow\) Faster in more rigid media.
  • Inertial property (density) \(\rightarrow\) Slower in denser media. \[ v = \sqrt{\frac{\text{elastic property}}{\text{inertial property}}} \]

Note

Sound waves satisfy the general wave equation: \[ \frac{\partial^2 y(x,t)}{\partial x^2} = \frac{1}{v^2}\frac{\partial^2 y(x,t)}{\partial t^2} \]

The general formula for mechanical wave speed highlights that stiffer, less compressible materials tend to transmit sound faster, while denser materials tend to slow it down. This is why sound travels faster in solids than in liquids, and faster in liquids than in gases.

Specific Formulas for Speed of Sound

• In a fluid: \[ v = \sqrt{\frac{\beta}{\rho}} \] where \(\beta\) is the bulk modulus and \(\rho\) is the density.
• In a solid: \[ v = \sqrt{\frac{Y}{\rho}} \] where \(Y\) is Young’s modulus and \(\rho\) is the density.
• In an ideal gas: \[ v = \sqrt{\frac{\gamma RT_K}{M}} \] where \(\gamma\) is the adiabatic index, \(R\) is the gas constant, \(T_K\) is absolute temperature (Kelvins), and \(M\) is molar mass.

These specific equations apply the general principle to different states of matter. The bulk modulus for fluids and Young’s modulus for solids represent their elastic properties. For gases, the adiabatic index and temperature play a critical role, as gas pressure and density are highly temperature-dependent.

Speed of Sound in Air

• For air at sea level, the speed of sound depends on temperature: \[ v = (331 \, \text{m/s}) \sqrt{1 + \frac{T_C}{273^\circ \text{C}}} = (331 \, \text{m/s}) \sqrt{\frac{T_K}{273 \, \text{K}}} \] where \(T_C\) is temperature in Celsius and \(T_K\) is absolute temperature in Kelvins.

Tip

The speed of sound in air is nearly independent of frequency. This is why all instruments in a band arrive in cadence regardless of distance. \[ v = f\lambda \] Since \(v\) is constant in a given medium, higher frequency means smaller wavelength.

This equation for air is highly practical. It shows a direct relationship between temperature and sound speed: hotter air means faster sound. Although not a strong dependence, it’s significant enough for precision applications like musical instrument tuning. The independence of speed from frequency is crucial for our perception of music.

Calculating Wavelengths (Example 17.1)

Question: Calculate the wavelengths of sounds at 20 Hz and 20,000 Hz in 30.0°C air.

Strategy:

1. Find the speed of sound \(v\) in 30.0°C air using \(v = (331 \, \text{m/s}) \sqrt{\frac{T_K}{273 \, \text{K}}}\).
2. Use \(f = v/\lambda\) to calculate wavelengths.

This example demonstrates a practical application of the formula for the speed of sound in air. It’s important to convert the temperature to Kelvins first. Then, using the constant speed of sound, we can see how the very different frequencies of the audible range correspond to significantly different wavelengths.

Calculating Wavelengths (Solution)

1. Convert temperature to Kelvin: \(T_K = 30.0^\circ \text{C} + 273.15 = 303.15 \, \text{K}\).
2. Calculate speed of sound: \[ v = (331 \, \text{m/s}) \sqrt{\frac{303.15 \, \text{K}}{273 \, \text{K}}} = 348.7 \, \text{m/s} \]
3. Calculate maximum wavelength (for \(f = 20 \, \text{Hz}\)): \[ \lambda_{\text{max}} = \frac{v}{f} = \frac{348.7 \, \text{m/s}}{20 \, \text{Hz}} = 17 \, \text{m} \]
4. Calculate minimum wavelength (for \(f = 20,000 \, \text{Hz}\)): \[ \lambda_{\text{min}} = \frac{v}{f} = \frac{348.7 \, \text{m/s}}{20,000 \, \text{Hz}} = 0.017 \, \text{m} = 1.7 \, \text{cm} \]

The results show that low-frequency sounds have very long wavelengths (17 meters!), while high-frequency sounds have very short wavelengths (1.7 centimeters). This wide range in wavelengths is due to the constant speed of sound in a given medium.

Seismic Waves

• Earthquakes produce both longitudinal (P-waves) and transverse (S-waves).
• Speeds depend on the rigidity of the medium.
  • Granite’s bulk modulus > its shear modulus.
  • P-waves (compressional) are faster than S-waves (shear).
• P-waves: 4 to 7 km/s.
• S-waves: 2 to 5 km/s.

Figure 17.11: P and S waves from an earthquake and shadow regions.

Seismic waves are a powerful natural example of how sound waves behave in solids. The different speeds of P-waves and S-waves are used by seismologists to pinpoint earthquake epicenters and, importantly, to infer the structure of Earth’s interior, including the liquid outer core which cannot support S-waves, creating shadow regions. This is a brilliant application of wave physics to understand inaccessible regions.

17.3 Sound Intensity

Learning Objectives

By the end of this section, you will be able to:

• Define the term intensity
• Explain the concept of sound intensity level
• Describe how the human ear translates sound

In this section, we move from the speed of sound to its loudness, introducing the concept of intensity and the decibel scale. We’ll also briefly touch upon the amazing sensitivity and range of human hearing.

Sound Intensity

• Intensity (\(I\)): Power (\(P\)) per unit area (\(A\)) carried by a wave. \[ I = \frac{P}{A} \]
• SI unit: W/m².
• For a spherical sound wave with no energy loss, intensity decreases with the square of the distance: \[ I_2 = I_1 \left(\frac{r_1}{r_2}\right)^2 \]
• The intensity of a sound wave is proportional to the square of the pressure variation (\(\Delta P_{\text{max}}\)) and inversely proportional to the medium’s density (\(\rho\)) and sound speed (\(v\)): \[ I = \frac{(\Delta P_{\text{max}})^2}{2\rho v} \]

Intensity is a measure of how much energy a sound wave carries through a given area per second. This inverse square relationship for spherical waves is very important: doubling your distance from a sound source reduces the intensity to one-fourth. The second formula shows the physical properties that contribute to this intensity: higher pressure amplitude means more intense sound, while denser or faster media can reduce it.

Human Hearing and Sound Intensity Levels

• Human ear’s intensity range is vast:
  • Threshold of hearing (\(I_0\)): \(10^{-12} \, \text{W/m}^2\)
  • Threshold of pain (\(I_{\text{pain}}\)): \(1 \, \text{W/m}^2\)
• Due to this large range, sound intensity level (\(\beta\)) is used, measured in decibels (dB): \[ \beta \, (\text{dB}) = 10 \log_{10}\left(\frac{I}{I_0}\right) \] where \(I_0 = 10^{-12} \, \text{W/m}^2\) is the reference intensity at 1.00 kHz.

Important

Every factor of 10 in intensity corresponds to a 10 dB increase.

Example: A 90-dB sound is \(10^3\) (1000) times as intense as a 60-dB sound.

The decibel scale is a logarithmic scale because human hearing perceives loudness logarithmically. This compresses the enormous range of sound intensities into a more manageable scale. It’s crucial to understand that a 10 dB difference means a factor of 10 in intensity, and a 3 dB difference corresponds to a doubling of intensity.

Sound Intensity Levels (Table 17.2)

\(\beta\) (dB)	\(I\) (W/m²)	Example/Effect
0	\(10^{-12}\)	Threshold of hearing at 1000 Hz
10	\(10^{-11}\)	Rustle of leaves
60	\(10^{-6}\)	Normal conversation
100	\(10^{-2}\)	Noisy factory, siren at 30 m; damage from 8 h per day exposure
120	1	Loud rock concert; threshold of pain
160	\(10^{4}\)	Bursting of eardrums

This table gives a good sense of how various common sounds correspond to different decibel levels and intensities. Notice how rapidly the intensity increases for relatively small increases in decibels, highlighting the power of the logarithmic scale.

Hearing and Pitch

• Pitch: Perception of frequency.
  • Humans have excellent relative pitch; can discriminate \(\approx 0.3\%\) frequency difference.
• Loudness: Perception of intensity.
  • About 1 dB difference is discernible; 3 dB is easily noticed.
  • Loudness is also affected by frequency (ear is less sensitive at extremes).
• Timbre (Tone Quality): Perception of combinations of frequencies and intensities.
  • Gives instruments (and voices) their distinctive characteristics.
• Phon: Unit to express loudness numerically (perception unit).
  • Differs from decibels (physical intensity unit).
  • At 1000 Hz, phons = decibels numerically.

Pitch, loudness, and timbre are the three main subjective attributes of sound. Pitch is directly tied to frequency, loudness to intensity (though not solely), and timbre to the complex mix of overtones. The phon scale attempts to quantify loudness based on human perception across different frequencies, showing how our ears are not equally sensitive to all frequencies.

Equal-Loudness Curves

Figure 17.15: Equal-loudness curves, showing loudness in phons vs. intensity level (dB) and frequency (Hz).

These equal-loudness curves illustrate a critical aspect of psychoacoustics: the human ear’s sensitivity varies with frequency. To perceive a low-frequency sound (like 100 Hz) as equally loud as a mid-frequency sound (like 1000 Hz), the low-frequency sound needs to have a significantly higher intensity level (dB). This is why bass Boost functions exist on stereos.

17.4 Normal Modes of a Standing Sound Wave

Learning Objectives

By the end of this section, you will be able to:

• Explain the mechanism behind sound-reducing headphones
• Describe resonance in a tube closed at one end and open at the other end
• Describe resonance in a tube open at both ends

This section dives into the fascinating world of wave interference and resonance, specifically as it applies to sound waves forming standing waves in tubes. These principles are fundamental to understanding musical instruments and noise cancellation technology.

Interference of Sound Waves

• Interference: Hallmark of waves; constructive and destructive.
• Two identical waves can become out of phase if they travel different path lengths.
• Constructive Interference: Occurs when path length difference is an integer multiple of the wavelength (\(n\lambda\)).
• Destructive Interference: Occurs when path length difference is an odd multiple of a half-wavelength (\(n\lambda/2\), where \(n=1,3,5,...\)).

Figure 17.17: Interference patterns from two speakers due to path length differences.

When sound waves from two sources overlap, their amplitudes add or subtract depending on their phase relationship. If they arrive in phase, they reinforce each other (constructive interference), leading to louder sound. If they arrive 180 degrees out of phase, they cancel each other (destructive interference), leading to quieter sound or silence. This principle is key to applications like noise-canceling headphones.

Noise Reduction through Destructive Interference

Figure 17.18: Noise-canceling headphones use destructive interference.

Tip

Noise-canceling headphones generate a sound wave that is 180° out of phase with incoming ambient noise.

This results in destructive interference, significantly reducing perceived noise levels (by 30 dB or more).

Active noise cancellation is a remarkable application of destructive interference. By analyzing incoming noise and generating an “anti-noise” signal, these headphones effectively cancel out unwanted sounds. This technology is used in headphones, aircraft cabins, and even some cars to create quieter environments.

Resonance in a Tube Closed at One End

• Sound waves can resonate in hollow tubes.
• Boundary Conditions (Closed-End Tube):
  • Closed end: Node (minimal air molecule oscillation).
  • Open end: Antinode (maximal air molecule oscillation).
• Fundamental Frequency (First Harmonic):
  • Wavelength (\(\lambda_1\)) = 4L (where L is tube length).
  • Frequency (\(f_1\)) = \(v/4L\).
• Only odd harmonics are produced: \(f_n = n \frac{v}{4L}\), where \(n = 1, 3, 5, \dots\)

Figure 17.22: Fundamental and first three overtones (1st, 3rd, 5th, 7th harmonics) for a tube closed at one end.

For a tube closed at one end, the wave pattern must have a node at the closed end and an antinode at the open end. This constraint means that only specific wavelengths, which are odd multiples of a quarter-wavelength, can form stable standing waves. This unique set of harmonics gives instruments like clarinets and some organ pipes their distinct sound.

Resonance in a Tube Open at Both Ends

• Boundary Conditions (Open-End Tube):
  • Both ends: Antinodes (maximal air molecule oscillation).
• Fundamental Frequency (First Harmonic):
  • Wavelength (\(\lambda_1\)) = 2L (where L is tube length).
  • Frequency (\(f_1\)) = \(v/2L\).
• All harmonics (both odd and even multiples) are produced: \[ f_n = n \frac{v}{2L}, \quad \text{where } n = 1, 2, 3, \dots \]

Figure 17.23: Fundamental and first three overtones for a tube open at both ends.

Unlike the closed-end tube, a tube open at both ends has antinodes at both ends. This symmetrical boundary condition allows for all integer multiples of the fundamental frequency to resonate, similar to a string fixed at both ends. Instruments like flutes and many organ pipes utilize this configuration, resulting in a richer, fuller sound with both odd and even harmonics present.

17.5 Sources of Musical Sound

Learning Objectives

By the end of this section, you will be able to:

• Describe the resonant frequencies in instruments that can be modeled as a tube with symmetrical boundary conditions
• Describe the resonant frequencies in instruments that can be modeled as a tube with anti-symmetrical boundary conditions

Building on our understanding of standing waves in tubes, this section explores how these principles are applied in musical instruments. We’ll differentiate between instruments that behave like open-ended pipes versus closed-end pipes and discuss how they produce their characteristic sounds.

Musical Instruments as Resonating Tubes

• Many instruments (woodwinds, brass, pipe organs) can be modeled as resonating tubes.
  • Open at both ends: Flutes, many organ pipes.
  • Closed at one end: Clarinets, some organ pipes.
• Players vary tube length (finger holes, slides) to change frequencies.
• End Correction: For open tubes, antinodes occur slightly outside the physical opening (approx. \(0.6 \times \text{radius}\)).

Note

The specific combination of fundamental and overtones (their mix of intensities) gives each instrument its unique timbre.

The design of musical instruments ingeniously leverages the physics of standing waves. By changing the effective length of an air column, musicians can produce different notes. The concept of “timbre” is crucial here, as it explains why a flute and a clarinet playing the same note (same fundamental frequency) sound so different – it’s all about the unique mix and relative intensity of their overtones.

Voice as a Resonating Cavity

Figure 17.26: The human vocal tract acts as a resonant air column.

Tip

Our speech is determined by shaping the throat and mouth cavity, and positioning the tongue, to adjust the fundamental and combination of overtones.

The human vocal system is a complex and highly adaptable resonator. Our vocal cords produce a buzzing sound, and the throat and mouth cavity then act like a variable-length tube, filtering and amplifying certain frequencies (formants) to produce different vowel sounds and speech. This is why variations in vocal tract anatomy lead to distinct individual voices.

Resonance in Stringed Instruments

Figure 17.27: Violins and guitars use sounding boxes for resonance.

Note

String instruments use sounding boxes to amplify and enrich the sound from vibrating strings. The string’s vibrations are coupled to the sounding box, causing the air inside to resonate and produce overtones, which define the instrument’s timbre.

Even though a string is the primary vibrator in these instruments, the sounding box is critical. Without it, the sound from a vibrating string would be very faint. The sounding box resonates with the string’s vibrations, transferring energy more efficiently to the air and generating a rich spectrum of overtones that give the instrument its characteristic voice.

17.6 Beats

Learning Objectives

By the end of this section, you will be able to:

• Determine the beat frequency produced by two sound waves that differ in frequency
• Describe how beats are produced by musical instruments

This section introduces the phenomenon of “beats,” a fascinating result of wave superposition when two sound waves of slightly different frequencies interfere. Understanding beats is crucial for tasks like tuning musical instruments.

What are Beats?

• Beats: A phenomenon of constructive and destructive interference when two sound waves with slightly different frequencies overlap.
• Results in a periodic variation in loudness.
• If two sound waves are: \(y_1 = A\cos(k_1x - 2\pi f_1 t)\) \(y_2 = A\cos(k_2x - 2\pi f_2 t)\)
• The superposition results in a wave with a varying amplitude, whose envelope oscillates at the beat frequency.
• Beat Frequency (\(f_{\text{beat}}\)): The absolute difference between the two frequencies. \[ f_{\text{beat}} = |f_2 - f_1| \]

Figure 17.29: Beats produced by two sound waves that differ in frequency.

When two sound waves with slightly different frequencies are played simultaneously, at some moments their crests align (constructive interference), making the sound louder. At other moments, a crest from one wave aligns with a trough from the other (destructive interference), making the sound softer. This periodic change in loudness is what we perceive as “beats.” The beat frequency tells us how many times per second this loudness variation occurs.

Application of Beats: Piano Tuning

• Piano tuners use beats to accurately tune instruments.
• A tuning fork (known frequency) is struck, and a corresponding note is played on the piano.
• If the piano string’s frequency is slightly off, beats will be heard.
• As the string is tightened or loosened, its frequency approaches the tuning fork’s frequency, and the beat frequency decreases.
• When no beats are heard (\(f_{\text{beat}} = 0\)), the frequencies are identical, and the piano is in tune.

This is a classic and very practical application of beats. Tuners don’t need a precise frequency counter; they just need to listen for the absence of beats. When the beat frequency is very low (e.g., one beat every few seconds), the piano is very close to being in tune. When the beat frequency is zero, it’s perfectly in tune.

17.7 The Doppler Effect

Learning Objectives

By the end of this section, you will be able to:

• Explain the change in observed frequency as a moving source of sound approaches or departs from a stationary observer
• Explain the change in observed frequency as an observer moves toward or away from a stationary source of sound

The Doppler effect is one of the most recognizable phenomena involving sound, responsible for the changing pitch of a siren as it passes by. In this section, we will explore the physics behind this frequency shift for both moving sources and moving observers.

The Doppler Effect: Overview

• Doppler Effect: Alteration in the observed frequency of a sound due to relative motion between the source and the observer.
• Doppler Shift: The actual change in frequency.
• Named after Christian Johann Doppler (1803–1853).
• Key Observation:
  • Source or observer moving toward each other \(\rightarrow\) Higher observed frequency.
  • Source or observer moving away from each other \(\rightarrow\) Lower observed frequency.
  • Greater relative speed \(\rightarrow\) Greater shift.

The Doppler effect is ubiquitous, from everyday experiences like passing ambulances to sophisticated scientific applications. It’s not just about sound; it applies to any type of wave, including light and water waves. The core idea is that relative motion compresses or stretches the wavefronts, altering the frequency at which they arrive at the observer.

Visualizing the Doppler Effect

Figure 17.30: Sound waves emitted by stationary and moving sources/observers.

Note

1. Stationary source, stationary observer: \(f_o = f_s\).
2. Moving source: Wavelength is compressed ahead, stretched behind.
3. Moving observer: Observer moves through waves faster (approaching) or slower (receding).

Panel (a) is the baseline: no relative motion, so observed frequency equals source frequency. Panel (b) shows the wave compression ahead of a moving source and rarefaction behind it, leading to higher and lower frequencies respectively. Panel (c) illustrates that the same effect occurs when the observer moves relative to a stationary source – the observer encounters wave crests more frequently when approaching and less frequently when receding.

Doppler Shift Formula

The general formula for the observed frequency (\(f_o\)) when both the source and observer may be moving is:

\[ f_o = f_s \left( \frac{v \pm v_o}{v \mp v_s} \right) \]

Where:

• \(f_o\): Observed frequency
• \(f_s\): Source frequency
• \(v\): Speed of sound in the medium
• \(v_o\): Speed of the observer
• \(v_s\): Speed of the source

Important

• Top signs (+ in numerator, - in denominator) for approaching (source toward observer, observer toward source).
• Bottom signs (- in numerator, + in denominator) for departing (source away from observer, observer away from source).

This single equation elegantly covers all four scenarios of the Doppler effect: source moving, observer moving, both moving, and their relative directions. It’s crucial to correctly choose the signs based on whether the observer or source is approaching or departing to get the correct frequency shift.

Applications of the Doppler Effect

• Medical Diagnostics: Ultrasound for measuring blood velocity.
• Law Enforcement: Radar (microwaves) for measuring car velocities.
• Meteorology: “Doppler Radar” tracks storm cloud motion, rain/snow velocity and direction.
• Astronomy: Determines relative velocities of stars and galaxies (redshift for receding galaxies, blueshift for approaching).
  • Led to estimation of universe’s age (\(\approx 14\) billion years).

The Doppler effect is not just an academic concept; it has profound practical applications across many fields. Its use in astronomy to measure galactic speeds has been instrumental in shaping our understanding of the universe’s expansion and age.

17.8 Shock Waves

Learning Objectives

By the end of this section, you will be able to:

• Explain the mechanism behind sonic booms
• Describe the difference between sonic booms and shock waves
• Describe a bow wake

In this final section, we extend the Doppler effect to extreme cases: when a sound source moves at or above the speed of sound. This leads to the spectacular phenomena of shock waves and sonic booms.

High Velocity and the Doppler Effect

• As a source’s speed (\(v_s\)) approaches the speed of sound (\(v\)), the observed frequency (\(f_o\)) for an approaching source approaches infinity: \[ f_o = f_s \left( \frac{v}{v - v_s} \right) \]
• At \(v_s = v\), the denominator becomes zero. This means all successive wave crests pile up at the same point.

Figure 17.35: Wave compression as source speed approaches speed of sound.

When a sound source travels at the speed of sound, it keeps pace with the sound waves it produces. All the wavefronts “stack up” at the front of the source, leading to a massive buildup of pressure. This intense constructive interference is the precursor to a shock wave.

Shock Waves and Sonic Booms

• Shock Wave: A constructive interference of sound created by an object moving faster than sound.
• Forms a cone in three dimensions.
• Sonic Boom: The sound heard when a shock wave sweeps across the ground.
  • Not just when “breaking the sound barrier” but continuously as long as the object is supersonic.
• Mach Number (\(M\)): Speed of source (\(v_s\)) divided by speed of sound (\(v\)). \[ M = \frac{v_s}{v} \]
• The angle (\(\theta\)) of the shock wave cone is related to Mach number: \[ \sin\theta = \frac{v}{v_s} = \frac{1}{M} \]

Figure 17.36: Shock wave created by a supersonic source.

When a jet goes supersonic, it outruns its own sound. The continuous production of sound waves and their subsequent constructive interference along a conical front creates a powerful pressure disturbance – the shock wave. The sonic boom is the loud crack heard on the ground when this shock wave passes overhead. Larger objects, like space shuttles, can create multiple shock waves, leading to multiple booms.

Bow Wakes and Cerenkov Radiation

• Bow Wake: A broader phenomenon where the wave source moves faster than the wave propagation speed.
  • Familiar V-shaped wake from a duck or boat moving through water.
• Cerenkov Radiation: A “bow wake” of light created when subatomic particles travel through a medium (like water) faster than the speed of light in that medium (which is less than \(c\)).
  • Produces a characteristic blue glow.

Figure 17.38: Duck creating a bow wake.

Figure 17.39: Blue glow of Cerenkov radiation in a reactor pool.

Shock waves are a specific type of bow wake. The concept applies to any wave phenomenon where the source exceeds the wave’s speed in that medium. Cerenkov radiation is a fascinating example from particle physics, showing that even light can have a “wake” if the particle producing it moves faster than the light itself in that particular medium.

Key Takeaways

1. Sound as a Wave: Sound is a longitudinal mechanical wave involving compressions and rarefactions.
2. Speed of Sound: Depends on the medium’s elasticity and density. In air, it depends on temperature.
3. Intensity & Loudness: Intensity (W/m²) is power per area. Loudness is human perception, measured in decibels (dB), a logarithmic scale.
4. Interference & Resonance: Sound waves interfere constructively and destructively. Resonance in tubes (open/closed) creates standing waves and determines instrument timbre.
5. Beats: Produced by the superposition of two slightly different frequencies, heard as periodic variations in loudness.
6. Doppler Effect: Change in observed frequency due to relative motion between source and observer. Higher frequency approaching, lower frequency receding.
7. Shock Waves: Formed when a source moves faster than the speed of sound, creating a “sonic boom.” A type of bow wake.

Key Equations

Equation	Description
\(\Delta P = \Delta P_{\text{max}}\sin(kx \mp \omega t + \phi)\)	Pressure wave model for sound
\(s(x,t) = s_{\text{max}}\cos(kx \mp \omega t + \phi)\)	Displacement wave model for sound
\(v = f\lambda\)	Wave speed, frequency, wavelength relation
\(v = \sqrt{\frac{\gamma RT_K}{M}}\)	Speed of sound in an ideal gas
\(v = (331 \, \text{m/s}) \sqrt{\frac{T_K}{273 \, \text{K}}}\)	Speed of sound in air (temperature-dependent)
\(I = \frac{P}{A} = \frac{(\Delta P_{\text{max}})^2}{2\rho v}\)	Sound intensity
\(\beta \, (\text{dB}) = 10 \log_{10}\left(\frac{I}{I_0}\right)\)	Sound intensity level in decibels
\(f_n = n \frac{v}{4L}, n=1,3,5,\dots\)	Resonant frequencies for tube closed at one end
\(f_n = n \frac{v}{2L}, n=1,2,3,\dots\)	Resonant frequencies for tube open at both ends
\(f_{\text{beat}} = |f_2 - f_1|\)	Beat frequency
\(f_o = f_s \left( \frac{v \pm v_o}{v \mp v_s} \right)\)	Doppler effect for moving source and observer
\(\sin\theta = \frac{1}{M}\)	Angle of shock wave cone (M = Mach number)

Key Terms

Term	Definition
Sound	A disturbance of matter that is transmitted from its source outward.
Hearing	The perception of sound.
Compression	A region in a longitudinal wave where particles are closest together (high pressure).
Rarefaction	A region in a longitudinal wave where particles are furthest apart (low pressure).
Intensity (I)	The power per unit area carried by a wave (W/m²).
Sound Intensity Level (\(\beta\))	A logarithmic measure of intensity relative to a reference, expressed in decibels (dB).
Pitch	The perception of the frequency of a sound.
Loudness	The perception of the intensity of a sound.
Timbre	The tone quality or characteristic sound of an instrument or voice, determined by the mix of overtones.
Node	A point in a standing wave where the displacement or pressure is always zero.
Antinode	A point in a standing wave where the displacement or pressure is maximum.
Fundamental Frequency	The lowest resonant frequency of a vibrating system.
Overtones	Higher resonant frequencies beyond the fundamental.
Harmonics	Resonant frequencies that are integral multiples of the fundamental frequency.
Beats	Periodic variations in loudness produced by the superposition of two sound waves of slightly different frequencies.
Beat Frequency	The absolute difference between the frequencies of two interfering waves.
Doppler Effect	The apparent change in frequency of a wave due to the relative motion between the source and the observer.
Doppler Shift	The actual change in frequency caused by the Doppler effect.
Shock Wave	A constructive interference of sound created by an object moving faster than the speed of sound.
Sonic Boom	The loud sound heard when a shock wave sweeps across a listener.
Mach Number (M)	The ratio of the speed of an object to the speed of sound in the surrounding medium.
Bow Wake	A V-shaped wave pattern created by an object moving faster than the waves it produces in a medium.
Cerenkov Radiation	A type of electromagnetic radiation emitted when a charged particle passes through a dielectric medium at a speed greater than the phase velocity of light in that medium.