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📘 How does sound travel as a wave?

Pressure, frequency, and intensity—how sound is a mechanical wave you can measure.

11
lessons
~30 min
to learn
Adults
level
Start the course →

What you’ll learn

  1. Fundamentals of Mechanical WavesDefine mechanical waves and distinguish longitudinal from transverse types using displacement and pressure variables.Mechanical waves require a medium and carry energy without net mass transport. Longitudinal sound waves compress and rarefy air while transverse waves on strings oscillate perpendicular to travel. The relationship between wave speed, wavelength, and frequency is introduced through measured examples that prepare students for the wave equation.
  2. Sound Wave Properties in AirCalculate sound speed, intensity, and acoustic impedance from measurable thermodynamic quantities.Sound speed depends on the square root of the adiabatic bulk modulus divided by density. Intensity follows the inverse-square law in free space while acoustic impedance governs reflection at boundaries. Real laboratory values illustrate how small changes in temperature produce detectable shifts in arrival time.
  3. The One-Dimensional Wave EquationDerive and solve the wave equation for traveling and standing waves on a string with fixed ends.The wave equation emerges from force balance on a differential element. General solutions include traveling waves of arbitrary shape and sinusoidal standing waves fixed by boundary conditions. Students verify that phase velocity equals the square root of tension over mass density using both analytic and numerical checks.
  4. Standing Waves and ResonanceDetermine resonant frequencies and mode shapes for strings, open tubes, and closed tubes.Boundary conditions quantize wavelengths and therefore frequencies. End corrections shift observed resonances away from ideal predictions. Students compare measured spectra with calculated harmonics to quantify damping and radiation losses. Also covers superposition and interference: Linear wave equations permit addition of displacements. Path-length differences translate directly into phase shifts that determine intensity maxima and minima. Worked calculations for two-point sources establish the quantitative foundation
  5. The Doppler EffectDerive and apply the Doppler formulas for moving source, moving observer, and combined cases.Frequency increases when source and observer approach because wavefronts are compressed. The general equation accounts for motion directions through signed velocity terms. Numerical examples illustrate shock-wave formation when source speed exceeds sound speed.
  6. Electromagnetic Waves and LightShow that light satisfies the wave equation with speed determined by permittivity and permeability.Electromagnetic waves are transverse and self-sustaining. The Poynting vector gives instantaneous power flow while time-averaged intensity follows the square of field amplitude. This chapter bridges acoustic wave concepts to optical behavior.
  7. Reflection, Refraction, and Snell’s LawApply boundary conditions to derive Snell’s law and calculate reflection coefficients at dielectric interfaces.Wavefront matching across an interface produces the familiar refraction relation. Impedance mismatch determines reflected versus transmitted amplitudes. Students compute Brewster’s angle and total internal reflection limits using measured indices.
  8. Diffraction and Interference in LightCalculate single-slit and double-slit diffraction patterns using phasor summation.Huygens’ construction explains why light bends around obstacles. The sinc-function intensity profile for a single slit sets the envelope for multi-slit interference. Numerical examples include missing orders in double-slit data.
  9. Polarization of Light WavesRepresent polarization states with Jones calculus and predict transmission through polarizers and retarders.Electric-field orientation defines polarization. Malus’ law governs intensity after a linear polarizer while quarter-wave plates produce circular light. Laboratory measurements confirm the conversion between orthogonal linear states.
  10. Wave Optics versus Ray OpticsIdentify conditions under which wave optics must replace geometric optics and compute Fresnel diffraction.Geometric optics emerges as the short-wavelength limit of wave optics. Diffraction integrals quantify deviations from ray predictions. Students evaluate when lens aberrations or aperture effects require full wave treatment.
  11. Applications in Modern Wave TechnologyConnect wave and optical concepts to fiber communications, ultrasonics, and interferometric sensors.Dispersion, attenuation, and modal structure determine fiber performance. Acoustic impedance matching optimizes ultrasound transducers. Phase-sensitive interferometers achieve picometer displacement sensitivity, illustrating the precision attainable with coherent waves.

Questions this course answers

A loudspeaker cone moves forward and backward, sending a sound wave down a tube. Which description correctly pairs the particle motion with the wave type?

Sound in air is longitudinal; particle displacement occurs along the direction of propagation, alternately compressing and rarefying the medium.

Match each wave variable to the wave type in which it is the primary descriptor.

Transverse waves are described by perpendicular displacement y(x,t); longitudinal waves are equivalently described by parallel displacement s(x,t) or the resulting pressure fluctuation p(x,t).

A plane wave in 20 °C air carries a pressure amplitude of 2 Pa. Estimate its time-averaged intensity.

I = (Δp_max)² / (2Z) with Z = 413 rayls gives 4.84 mW m⁻²; the inverse-square law is not needed because the wave is planar inside the tube.

Dry air at 0 °C and 1 atm has Z = 428 rayls. If temperature rises to 30 °C at constant pressure, what happens to Z?

Density drops as 1/T while speed rises only as sqrt(T), so their product Z falls by roughly 5 % at 30 °C.

Place the following steps in the order they occur when deriving the wave equation from Newton's second law.

The derivation begins with force balance on a finite element, equates it to mass times acceleration, then passes to the continuum limit to produce the PDE whose coefficient is identified as v².

A string has tension 200 N and linear density 0.005 kg/m. Estimate the fundamental frequency for length 2 m.

v = √(T/μ) = √(200/0.005) = 200 m/s, so f₁ = v/(2L) = 200/(4) = 50 Hz. The estimate lies within 20 % of this exact value.

Grounded in trusted sources

  • National Institute of Standards and Technology
  • National Aeronautics and Space Administration
  • National Oceanic and Atmospheric Administration
  • OpenStax University Physics Volume 1 — Sound Waves
  • NIST — acoustics and sound measurement
  • NASA — sound and atmosphere educational resources

Every Wunder lesson is built from real, reputable sources — never invented.

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