Waves, Acoustics and Signal Phenomena

Introduction and Context

Waves are organized disturbances that transfer energy and information without requiring the permanent transport of matter. Mechanical waves, such as sound on air, require a material medium. Electromagnetic waves, such as radio and light, can propagate through vacuum. In electrical engineering, wave concepts connect physical phenomena with transmission lines, antennas, acoustics, sensing, modulation, filtering, and digital communication.

A wave is described by measurable quantities such as amplitude, frequency, wavelength, phase, propagation speed, and energy or power. Understanding the relationships among these quantities is essential for analyzing interference, resonance, signal spectra, acoustic systems, and communication channels.

Learning Objectives

After studying this chapter, students should be able to:

1. Formulate and interpret one-dimensional wave equations.
2. Explain superposition, interference, beats, and standing waves.
3. Distinguish phase velocity from group velocity and explain dispersion.
4. Apply the Doppler effect to moving sources and observers.
5. Represent periodic and non-periodic signals using Fourier series and transforms.
6. Relate acoustic pressure, intensity, sound level, and power.
7. Explain the generation and applications of ultrasonics.
8. Select practical methods for noise measurement and control.
9. Connect physical wave behavior with signal and communication systems.

1. Fundamental Description of Waves

A progressive sinusoidal wave traveling in the positive x-direction may be written as

Here, A is the amplitude, k is the wave number in rad m⁻¹, ω is angular frequency in rad s⁻¹, φ is the initial phase, x is position in metres, and t is time in seconds. The corresponding wavelength and frequency relationships are

where λ is wavelength in metres, f is frequency in hertz, and v is phase velocity in m s⁻¹. These equations assume a sinusoidal wave in a uniform, time-invariant medium. The negative sign before ωt indicates propagation in the positive x-direction; changing it to a plus sign describes propagation in the negative x-direction.

The phase of a wave is the argument of the cosine function. Points having the same phase differ in position by an integer multiple of the wavelength. A phase difference of π radians corresponds to opposite displacement for equal amplitudes.

The Wave Equation

For many small-amplitude disturbances in a uniform medium, the displacement satisfies the one-dimensional wave equation:

The quantity y(x,t) is displacement or another wave variable, and v is propagation speed. This equation assumes linear behavior, negligible energy loss, and a uniform medium. Both F(x − vt) and G(x + vt) are general traveling-wave solutions:

The first term travels in the positive x-direction and the second in the negative x-direction. A sinusoid is only one possible waveform; pulses, square waves, and modulated signals can also satisfy the equation under suitable conditions.

For a stretched string, the wave speed is

where T is tension in newtons and μ is linear mass density in kg m⁻¹. This result assumes a flexible string, small transverse displacement, and uniform tension. Its units are √(N/(kg m⁻¹)) = m s⁻¹.

For sound in an ideal gas, the speed is approximately

where γ is the ratio of specific heats, p is absolute pressure, and ρ is mass density. The formula assumes small adiabatic pressure variations. In air near room temperature, sound speed is approximately 343 m s⁻¹, though it varies with temperature, humidity, and composition.

2. Superposition and Interference

In a linear medium, the total response to several waves is the algebraic sum of their individual responses:

This principle is called superposition. It applies when the response is linear—for example, when amplitudes are sufficiently small that material parameters do not change significantly with displacement. Superposition does not mean that waves permanently alter one another; after crossing, ideal waves continue independently.

Interference of Two Sinusoids

Consider two waves of the same frequency and amplitude arriving at a point with phase difference ∆φ. Their sum has amplitude

where A is the amplitude of each wave. Constructive interference occurs when ∆φ = 2mπ, and destructive interference occurs when ∆φ = (2m + 1)π, where m is any integer.

For waves produced by two coherent sources, path difference ∆r creates phase difference

For equal-phase sources, maxima occur when ∆r = mλ and minima when ∆r = (m + 1/2)λ. These conditions require coherence and approximately equal amplitudes. If the sources are incoherent, the phase varies randomly and stable interference fringes do not form.

Beats

When two waves have slightly different frequencies, their superposition produces periodic amplitude variation. For frequencies f₁ and f₂, the beat frequency is

For example, two tuning tones at 440 Hz and 444 Hz produce four beats per second. The perceived pitch is close to the average frequency, while loudness rises and falls at the beat frequency. Beat measurements are useful for tuning, frequency comparison, and sensor systems.

3. Reflection, Transmission and Standing Waves

When a wave reaches a boundary, part may be reflected and part transmitted. The proportions depend on the impedances of the two media. Wave impedance is the ratio of a suitable wave-driving quantity to its associated flow quantity. For a string, characteristic impedance is

where T is tension and μ is linear density. At a fixed end, a transverse pulse reflects with phase reversal; at a free end, it reflects without phase reversal. In electrical transmission lines, impedance mismatch similarly produces reflections and standing-wave patterns.

Two oppositely traveling waves of equal frequency and amplitude form a standing wave:

The locations where displacement is always zero are nodes, satisfying sin(kx) = 0. Locations of maximum oscillation are antinodes. A standing wave transports no net average energy along the ideal medium, although energy oscillates locally between kinetic and potential forms.

Waves on a String Fixed at Both Ends

For a string of length L fixed at both ends, allowed wavelengths and frequencies are

The fundamental frequency is f₁ = v/(2L), and higher modes are harmonics at integer multiples of the fundamental. This ideal result assumes fixed boundary conditions, uniform tension, and negligible damping.

For an open pipe of length L, the same harmonic relation approximately applies. For a pipe closed at one end, only odd harmonics occur:

End corrections are often required in practical acoustic pipes because the displacement antinode lies slightly outside the physical opening.

Figure: A standing-wave mode with fixed-end nodes and intervening antinodes.

4. Dispersion and Group Velocity

A medium is nondispersive if all frequency components travel at the same phase velocity. It is dispersive if phase velocity depends on frequency. In a dispersive medium, a pulse changes shape because its spectral components separate during propagation.

For a component with angular frequency ω and wave number k, phase velocity is

The group velocity, which often describes the motion of a pulse envelope or information-bearing modulation, is

These quantities are equal in a nondispersive medium but generally differ in a dispersive medium. In communication systems, dispersion can cause intersymbol interference: neighboring digital pulses spread into one another. Optical fiber systems therefore use carefully selected wavelengths, dispersion compensation, equalization, or coherent detection.

Dispersion must be distinguished from attenuation. Attenuation reduces amplitude with distance, whereas dispersion changes the relative timing and shape of frequency components. Real channels may exhibit both.

5. Doppler Effect

The Doppler effect is the observed frequency shift caused by relative motion between a source and an observer. For sound in a stationary medium, with source and observer moving along the line joining them, the observed frequency is

Here, f is emitted frequency, f′ is observed frequency, v is sound speed relative to the medium, v_o is observer velocity taken as positive when moving toward the source, and v_s is source velocity taken as positive when moving toward the observer. The formula assumes subsonic motion and a stationary, uniform medium.

If both source and observer are stationary, no Doppler shift occurs. If the observer moves toward a stationary source, the numerator increases. If the source moves toward a stationary observer, the denominator decreases. For electromagnetic waves, the relativistic Doppler formula is required at velocities that are a significant fraction of the speed of light.

Doppler principles are used in radar speed measurement, medical ultrasound, sonar, astronomy, and motion sensors. In radar, the frequency shift is related to radial velocity; the sign and magnitude depend on the system convention and whether a reflected two-way path is involved.

6. Fourier Representation of Signals

A periodic signal can be represented as a sum of harmonically related sinusoids. The trigonometric Fourier series is

where ω₀ = 2π/T is the fundamental angular frequency and T is the period. For a signal defined over one period,

The coefficients describe the average value and the cosine and sine content of each harmonic. These expressions assume a periodic, integrable signal; common engineering signals satisfy the required convergence conditions.

A compact complex form is

with

For a non-periodic signal, the Fourier transform is

and the inverse transform is

The transform convention shown uses frequency f in hertz. X(f) gives complex amplitude versus frequency, including magnitude and phase. Fourier analysis is central to filters, spectrum analyzers, modulation, sampling, vibration diagnostics, and noise measurement.

A narrow pulse in time contains a broad range of frequencies, while a long-duration sinusoid has a narrow spectrum. This time–frequency trade-off is fundamental: improving temporal localization generally reduces frequency localization.

Worked Example: Harmonics of a String

A string has length 0.80 m, tension 64 N, and linear density 0.010 kg m⁻¹. Its speed is

The fundamental frequency is

The third harmonic is f₃ = 3f₁ = 150 Hz. The calculation assumes ideal fixed ends and neglects stiffness and damping.

7. Acoustics: Pressure, Intensity and Sound Level

Sound is a longitudinal mechanical wave consisting of small pressure and particle-velocity variations. The instantaneous acoustic pressure is the deviation from static atmospheric pressure. For a plane progressive wave in a lossless medium,

where I is time-averaged intensity in W m⁻², p_rms is root-mean-square acoustic pressure in Pa, ρ is medium density in kg m⁻³, and v is sound speed in m s⁻¹. The product ρv is the characteristic acoustic impedance of the medium. This relation assumes a plane traveling wave; it is not generally valid in a near field or for a standing wave.

For a sinusoidal pressure p(t) = p_max cos(ωt),

Sound intensity level is

where I₀ = 10^−12 W m⁻² is the conventional reference intensity in air. Sound pressure level is

where p₀ = 20 μPa is the conventional reference pressure in air. The factor 20 appears because intensity is proportional to pressure squared. Sound pressure level and intensity level are numerically similar only under specified plane-wave conditions and common reference conventions.

A 10 dB increase in intensity level corresponds to ten times the intensity, not twice the intensity. A 3 dB increase approximately doubles intensity. Human perception is frequency-dependent, so A-weighted levels, written dB(A), apply a standardized frequency weighting for many environmental-noise assessments. Weighting does not change the physical pressure; it changes the measurement emphasis.

For an approximately point source radiating uniformly in a free field,

where P is acoustic power in watts and r is distance from the source. Thus, doubling distance reduces intensity to one quarter and reduces level by approximately 6 dB. This inverse-square law fails near reflecting surfaces, in enclosed rooms, for directional sources, and in the acoustic near field.

8. Ultrasonics

Ultrasound refers to sound above the nominal human hearing limit of about 20 kHz. The exact upper limit depends on age and individual hearing. Ultrasound is generated using piezoelectric or magnetostrictive transducers. A piezoelectric element changes dimensions when an electric field is applied and generates a voltage when mechanically stressed.

Important applications include:

• medical imaging and Doppler blood-flow measurement;
• nondestructive testing for cracks and voids;
• distance measurement and industrial level sensing;
• ultrasonic cleaning and welding;
• sonar and underwater communication;
• process monitoring and flow measurement.

Ultrasonic imaging relies on reflections caused by differences in acoustic impedance. Time of flight is used to estimate depth:

where d is reflector depth, v is wave speed in the material, and t is round-trip travel time. The factor 2 applies to pulse-echo systems because the wave travels to the reflector and back. Accurate measurement requires calibration of material velocity, probe delay, coupling, and geometry.

Ultrasound can produce heating, cavitation, and mechanical effects. Exposure limits, equipment shielding, operator training, and medical safety standards are therefore important. A transducer should not be treated as harmless merely because the sound is inaudible.

9. Noise and Noise Control

Noise is unwanted sound or unwanted random electrical fluctuation. Acoustic noise may be continuous, intermittent, impulsive, tonal, broadband, airborne, or structure-borne. Electrical noise may arise from thermal motion, shot processes, electromagnetic interference, switching circuits, or mechanical vibration.

Thermal noise voltage in a resistor over bandwidth B has mean-square value

where k is Boltzmann’s constant, T is absolute temperature in kelvin, and R is resistance in ohms. The formula assumes a resistor in thermal equilibrium over the stated bandwidth. It illustrates why larger bandwidth and higher temperature increase noise power.

Effective noise control follows a hierarchy:

1. Control at the source: reduce machine speed, balance rotating parts, isolate vibration, select quieter equipment, and maintain bearings.
2. Interrupt the transmission path: use barriers, enclosures, absorptive materials, resilient mounts, and acoustic isolation.
3. Protect the receiver: use hearing protection, quiet rooms, shielding, filtering, and signal processing.

Source control is generally preferred because it protects many workers or listeners simultaneously. Absorption reduces reflections inside a room but may not prevent sound transmission through a wall. Mass barriers block airborne sound, while resilient mounts address structure-borne vibration. Sealing gaps is often as important as increasing wall thickness.

A professional noise survey should specify measurement position, microphone calibration, frequency weighting, time weighting, environmental conditions, and uncertainty. Measurements should not be compared without confirming that the same metric and bandwidth were used. Occupational exposure decisions must follow applicable national regulations and institutional safety procedures.

10. Waves in Communication and Signal Systems

A communication system maps information onto a physical waveform. In analog modulation, a carrier’s amplitude, frequency, or phase is varied. In digital communication, symbols are represented by controlled changes in amplitude, phase, frequency, or combinations of these quantities. The channel may be a wire, optical fiber, radio path, or acoustic medium.

Wave concepts appear directly in engineering practice:

• Interference: unwanted signals can add destructively or constructively.
• Reflection: impedance mismatch creates echoes and standing waves in cables and transmission lines.
• Dispersion: frequency-dependent delay broadens pulses and causes intersymbol interference.
• Fourier analysis: filters and channels are naturally described in the frequency domain.
• Doppler shift: motion changes received frequency in wireless and radar links.
• Noise: random disturbances limit detection and data reliability.
• Resonance: filters, antennas, cavities, and sensors respond strongly near selected frequencies.

The signal-to-noise ratio is commonly defined as

or in decibels,

where P_s and P_n are signal and noise powers measured over the same bandwidth and at the same reference point. If amplitudes are compared for equal impedance, the equivalent voltage expression is 20 log₁₀(V_s/V_n). Mixing power and amplitude formulas without considering impedance is a common engineering error.

Illustrative Case: Ultrasonic Leak Detection in a Compressed-Air Plant

Background. A manufacturing plant experiences rising compressor energy use, but no obvious pressure failure is visible. Small leaks produce turbulent flow with ultrasonic components that may be difficult to hear in a noisy factory.

Problem. Maintenance staff must locate leaks without shutting down the entire compressed-air network.

Analysis. A directional ultrasonic detector is used to scan joints, valves, and hoses. The detector converts high-frequency acoustic energy into an audible or displayed signal. Measurements are recorded with distance, operating pressure, detector setting, and background conditions. A strong localized signal that changes as the probe is moved suggests a leak, but reflections from metal surfaces can create misleading maxima.

Approach and solution. The team first calibrates the detector and establishes a baseline away from suspected sources. It then scans systematically, confirms each location from several angles, applies a temporary seal where safe, and repeats the measurement. Leak size and economic priority are estimated using plant-specific pressure and flow data rather than relying solely on sound level.

Outcome. Several leaking fittings are repaired during scheduled maintenance. The result is reduced compressor loading and a measurable decline in ultrasonic emission. The method does not replace pressure testing or safety isolation; it is a rapid screening and localization technique.

Learning points. Signal amplitude alone is not a complete diagnosis. Geometry, reflections, frequency response, calibration, and operating conditions influence the reading. Engineering decisions should combine wave measurements with physical inspection, uncertainty assessment, and safety procedures.

11. Limitations, Ethics and Contemporary Developments

Ideal wave equations neglect effects that matter in practice: attenuation, nonlinear distortion, boundary irregularity, turbulence, temperature gradients, finite sensor bandwidth, and imperfect calibration. Numerical models can include these effects, but their predictions depend on material properties and boundary conditions. Engineers should report assumptions and uncertainty rather than presenting simulated results as exact measurements.

Acoustic monitoring can raise privacy concerns, particularly when sensors record speech or operate in workplaces and public spaces. Systems should collect only necessary data, protect recordings, control access, and comply with privacy and workplace regulations. Medical ultrasound requires informed clinical use, validated equipment, and adherence to exposure guidelines.

Modern signal processing uses fast Fourier transforms, adaptive filtering, beamforming, machine learning, and distributed sensor networks. These tools can classify machinery faults, separate speech from noise, and detect unusual vibration patterns. However, AI-based classification may fail when operating conditions differ from training data. Human oversight, representative validation data, uncertainty reporting, and cybersecurity remain essential.

Best practice is to begin with a physical model, define measurable quantities, calibrate instruments, verify units, inspect spectral and time-domain evidence, and compare results against an independent method. A sophisticated algorithm cannot compensate for a poorly placed sensor or an invalid wave assumption.

Chapter Summary

Waves transfer energy and information and are described by amplitude, phase, frequency, wavelength, and velocity. The linear wave equation leads to traveling-wave solutions, while superposition explains interference and beats. Reflection at boundaries can produce standing waves with nodes and antinodes; boundary conditions determine allowed resonant frequencies.

Phase velocity describes individual frequency components, whereas group velocity often describes pulse or envelope motion. Dispersion causes pulse spreading and is important in communication channels. The Doppler effect explains frequency shifts caused by relative motion and supports radar, sonar, medical, and sensing applications.

Fourier series and transforms represent signals in terms of frequency components, providing the foundation for spectral analysis, filtering, and modulation. Acoustic intensity is related to RMS pressure through acoustic impedance, while decibel levels provide logarithmic measures of intensity or pressure. Ultrasonics support imaging, inspection, measurement, and industrial processing. Noise control is most effective when applied at the source, followed by path control and receiver protection. Practical engineering requires calibration, appropriate assumptions, uncertainty evaluation, safety, and ethical handling of recorded or sensed information.

Key Terms

Wave: A propagating disturbance that transfers energy or information.

Amplitude: Maximum magnitude of a wave variable relative to equilibrium.

Wavelength: Spatial period of a wave.

Frequency: Number of cycles completed per second.

Phase: Relative position within a cycle, expressed as an angle.

Wave number: Spatial angular frequency, equal to 2π/λ.

Superposition: Addition of responses in a linear system.

Interference: Result of superposing waves with a defined phase relationship.

Beat: Amplitude variation caused by nearby frequencies.

Standing wave: Pattern formed by oppositely traveling waves of equal frequency.

Node: Point of zero displacement in an ideal standing wave.

Antinode: Point of maximum displacement in an ideal standing wave.

Dispersion: Frequency dependence of propagation velocity.

Doppler effect: Observed frequency shift caused by relative motion.

Fourier series: Harmonic representation of a periodic signal.

Fourier transform: Frequency-domain representation of a non-periodic signal.

Acoustic intensity: Time-averaged acoustic power per unit area.

Sound pressure level: Logarithmic level based on RMS acoustic pressure.

Ultrasound: Sound above the usual upper limit of human hearing.

Signal-to-noise ratio: Ratio of signal power to noise power.

Review Questions

Short Answer Questions

1. What physical quantities determine the wavelength of a sinusoidal wave?
2. State the condition for constructive interference between two coherent waves.
3. Distinguish a node from an antinode in a standing wave.
4. Why does a 10 dB increase in intensity level represent a tenfold intensity increase?
5. Name two applications of ultrasonic waves.

Long Answer Questions

1. Derive the allowed frequencies for a string fixed at both ends and explain the role of boundary conditions.
2. Explain phase velocity, group velocity, and dispersion, including their significance in digital communication.
3. Describe Fourier-series representation and explain the physical meaning of Fourier coefficients.
4. Explain acoustic intensity, RMS pressure, sound pressure level, and the limitations of inverse-square spreading.
5. Discuss the hierarchy of noise control and compare source control, path control, and receiver protection.

Analytical / Application Questions

1. A string has length 1.2 m, tension 90 N, and linear density 0.010 kg m⁻¹. Calculate its wave speed, fundamental frequency, and third-harmonic frequency. State your assumptions.
2. A sound source radiates uniformly in a free field. If intensity is 0.80 W m⁻² at 2 m, estimate the intensity at 8 m and explain why the result may differ indoors.
3. An ultrasonic pulse returns from a flaw after 40 μs in a material where sound speed is 5,000 m s⁻¹. Calculate the flaw depth using pulse-echo reasoning and identify one calibration issue that could affect the result.

Multiple Choice Questions with Answers

Question 1

A sinusoidal wave has frequency 200 Hz and wavelength 1.5 m. What is its phase velocity?

A. 133 m s⁻¹

B. 300 m s⁻¹

C. 500 m s⁻¹

D. 1,333 m s⁻¹

Correct Answer: B. 300 m s⁻¹

Question 2

Which condition produces destructive interference for two equal-amplitude coherent waves?

A. Phase difference 0

B. Phase difference π/2

C. Phase difference π

D. Path difference 2λ

Correct Answer: C. Phase difference π

Question 3

A string fixed at both ends has length L. Which expression gives its fundamental frequency?

A. v/L

B. v/(4L)

C. 2v/L

D. v/(2L)

Correct Answer: D. v/(2L)

Question 4

The beat frequency produced by 440 Hz and 444 Hz tones is

A. 2 Hz

B. 4 Hz

C. 440 Hz

D. 884 Hz

Correct Answer: B. 4 Hz

Question 5

In a dispersive medium, which quantity generally depends on frequency?

A. Phase velocity

B. The definition of wavelength

C. The unit of time

D. The value of π

Correct Answer: A. Phase velocity

Question 6

For a plane progressive acoustic wave, intensity is proportional to

A. RMS pressure

B. The square of RMS pressure

C. The inverse of frequency only

D. Wavelength squared only

Correct Answer: B. The square of RMS pressure

Question 7

A 10 dB increase in intensity level corresponds approximately to

A. Twice the intensity

B. Four times the intensity

C. Ten times the intensity

D. One-tenth the intensity

Correct Answer: C. Ten times the intensity

Question 8

For a point source in a free field, increasing distance from 2 m to 4 m changes intensity to approximately

A. Four times its original value

B. Twice its original value

C. One-half its original value

D. One-quarter its original value

Correct Answer: D. One-quarter its original value

Question 9

Which mathematical tool represents a periodic signal as harmonically related sinusoids?

A. Fourier series

B. Doppler equation

C. Inverse-square law

D. Wave impedance equation

Correct Answer: A. Fourier series

Question 10

In a pulse-echo ultrasonic measurement, why is depth calculated using d = vt/2?

A. The pulse frequency is doubled

B. The wave travels to the reflector and back

C. The material has two densities

D. The transducer emits two frequencies

Correct Answer: B. The wave travels to the reflector and back