Wave Motion, Speed and Classification
Progressive Waves
A progressive wave transfers energy away from a source of disturbance without transporting matter permanently — particles oscillate about their equilibrium positions.
Transverse Waves: Particle displacement is perpendicular to the wave's direction of travel (e.g., waves on a rope, light).
Longitudinal Waves: Particle displacement is parallel to the wave's direction of travel (e.g., sound in air, compression in a spring).
Solids vs Fluids: Both wave types exist in solids, but fluids support only longitudinal waves — sound in air is always longitudinal.
Pulse vs Periodic: A single disturbance creates a pulse; continuous rhythmic vibrations from a source (like a mass-spring system executing SHM) create periodic waves.
The Nature of Sound Waves
Sound is a mechanical wave that propagates as a longitudinal wave through a medium via successive compression and rarefaction cycles.
$$f = \frac{1}{T}$$
Relates the number of cycles per second to the time duration of one cycle.
$f$=Frequency(Hertz (Hz))
$T$=Period(Seconds (s))
$T \to 0$
→Frequency becomes infinitely high.
Longitudinal Motion: Particles oscillate parallel to the direction of energy transfer, creating alternating high- and low-pressure zones.
Pressure Variations: Compression regions have high pressure/density; rarefaction regions have low pressure/density.
Human Hearing Range: 20 Hz to 20,000 Hz. Below 20 Hz is infrasound; above 20,000 Hz is ultrasound.
The Wave Equation for Sound
The speed of sound ($v$) equals the product of its frequency and its wavelength, linking how fast energy propagates to the wave's spatial and temporal properties.
$$v = f\lambda$$
A crest moves one wavelength $\lambda$ in one period $T$, giving $v = \lambda / T = f\lambda$.
$v$=Wave speed(m/s)
$f$=Frequency(Hz)
$\lambda$=Wavelength(m)
Constant $v$ (same medium)
→$f$ and $\lambda$ are inversely proportional: $f \propto 1/\lambda$.
Wave crosses into a new medium
→Frequency stays constant (set by source); speed and wavelength both change.
Wavelength Definition: Distance between two consecutive identical points (e.g., compression to compression).
Inverse Proportionality Shortcut: For a fixed medium, doubling $f$ halves $\lambda$.
Phase Relationships
Two points on a wave are described by their phase difference, which determines whether they vibrate in unison or opposition.
$$\phi = \frac{2\pi x}{\lambda}$$
The phase lag of a point at distance $x$ from a reference point, measured in radians.
$\phi$=Phase difference(radians)
$x$=Separation between the two points(m)
$\lambda$=Wavelength(m)
$x = \lambda, 2\lambda, 3\lambda, \dots$
→Points are in phase ($\phi = 2\pi n$) — identical displacement and velocity at all times.
$x = \lambda/2, 3\lambda/2, \dots$
→Points are completely out of phase ($\phi = \pi + 2\pi n$) — opposite displacements at all times.
In Phase: Separated by whole multiples of $\lambda$; always have identical displacements.
Out of Phase: Separated by odd multiples of $\lambda/2$; when one is at max displacement, the other is at min.
Speed of Sound in Air — Newton vs Laplace
Newton assumed sound propagation in air is an isothermal process and derived the speed using the gas pressure as the elastic modulus.
$$v = \sqrt{\frac{P}{\rho}}$$
Newton's formula: treats compressions/rarefactions as isothermal (constant temperature). Predicts 280 m/s — about 16% too low.
$P$=Atmospheric pressure(Pa)
$\rho$=Density of air(kg/m³)
At STP ($P = 1.013 \times 10^5$ Pa, $\rho = 1.29$ kg/m³)
→$v = \sqrt{1.013 \times 10^5 / 1.29} \approx 280$ m/s (experimental value is 332 m/s).
Newton's Assumption: Temperature remains constant during compressions and rarefactions (isothermal process, Boyle's law: $PV = \text{const}$).
The Error: Newton's value of 280 m/s is significantly lower than the measured 332 m/s.
Why It Fails: Compressions happen too rapidly for heat to dissipate — the process is actually adiabatic, not isothermal.
Laplace corrected Newton by recognizing that sound compressions are adiabatic — heat doesn't escape fast enough — replacing $P$ with $\gamma P$.
$$v = \sqrt{\frac{\gamma P}{\rho}}$$
Laplace's corrected formula. The factor $\gamma$ accounts for the adiabatic nature of rapid compressions/rarefactions.
$\gamma$=Ratio of specific heats ($C_p / C_v$)(dimensionless)
$P$=Atmospheric pressure(Pa)
$\rho$=Density of gas(kg/m³)
Diatomic gas (air, O₂, N₂)
→$\gamma = 1.40$
Monatomic gas (He, Ne, Ar)
→$\gamma = 1.67$
Polyatomic gas (CO₂)
→$\gamma = 1.29$
The Correction Factor: Laplace multiplied Newton's elastic modulus by $\gamma$: $E = \gamma P$ instead of $E = P$.
Result for Air: $v = \sqrt{1.4} \times 280 \approx 333$ m/s, matching the experimental value of 332 m/s.
Dimensional Check: $[P/\rho] = \text{Pa}/(\text{kg/m}^3) = \text{m}^2/\text{s}^2$, so the square root gives m/s.
Factors Affecting Sound Speed in Gases
Changing the pressure of a gas at constant temperature does not change the speed of sound, because density changes proportionally with pressure.
Why No Effect: From $v = \sqrt{\gamma P / \rho}$, if $P$ doubles at constant $T$, then $\rho$ also doubles (ideal gas law), so $P/\rho$ stays the same.
At the same temperature and pressure, the speed of sound is inversely proportional to the square root of the gas density.
$$v \propto \frac{1}{\sqrt{\rho}}$$
Lighter Gas = Faster Sound: Sound in hydrogen ($\rho_H$) is 4× faster than in oxygen ($\rho_O = 16\rho_H$), since $\sqrt{16} = 4$.
Helium Voice Effect: Helium is ~7× less dense than air, so sound travels ~2.7× faster, shifting resonant frequencies of the vocal tract upward.
The speed of sound in a gas is proportional to the square root of its absolute temperature, and increases by approximately 0.61 m/s per °C rise.
$$\frac{v_t}{v_0} = \sqrt{\frac{T}{T_0}}$$
Relates the speed at temperature $t$ °C to the speed at 0 °C using absolute temperatures.
$v_t$=Speed at temperature $t$ °C(m/s)
$v_0$=Speed at 0 °C (332 m/s for air)(m/s)
$T$=Absolute temperature ($t + 273$ K)(K)
$T_0$=273 K (0 °C)(K)
Small temperature changes
→Use the linear approximation: $v_t \approx v_0 + 0.61t$
$v$ doubles
→Absolute temperature must quadruple ($v \propto \sqrt{T}$, so $2v \Rightarrow 4T$).
Linear Approximation: For moderate temperatures, $v_t \approx 332 + 0.61t$ m/s (derived via binomial expansion of $\sqrt{1 + t/273}$).
Proportionality Shortcut: $v \propto \sqrt{T}$ means to double the speed you need 4× the absolute temperature.
The speed of sound is determined by the elasticity and density of the medium, moving faster through stiffer materials.
$$v = \sqrt{\frac{E}{\rho}}$$
General formula: the elastic modulus $E$ represents resistance to compression, while $\rho$ represents inertia.
$E$=Elastic modulus ([bulk modulus] for fluids, Young's modulus for solids)(Pa)
$\rho$=Mass density of the medium(kg/m³)
Solid (e.g., steel)
→Very high elasticity dominates over higher density → fastest speed (~5000 m/s).
Gas (e.g., air)
→High compressibility (low $E$) → slowest speed (~340 m/s).
Phase Comparison: $v_{\text{solid}} > v_{\text{liquid}} > v_{\text{gas}}$ because the increase in elasticity from gas to solid far outweighs the increase in density.
Typical Sound Speeds
•
Air (STP): 332 m/s
•
Water (20 °C): 1483 m/s
•
Iron: 5130 m/s
•
Hydrogen (STP): 1286 m/s
•
Helium (STP): 972 m/s
Principle of Superposition
When two or more waves overlap in a medium, the resultant displacement at any point is the algebraic sum of the individual displacements — this is the superposition principle.
$$Y = y_1 + y_2 + \cdots + y_n$$
Constructive Interference: When waves are in phase, displacements add — the resultant amplitude is larger.
Destructive Interference: When waves are completely out of phase ($y_1 = -y_2$), the resultant displacement is zero.
Waves Pass Through Each Other: After overlapping, each wave continues unchanged in shape and direction.
Three Key Phenomena: Superposition leads to (1) interference, (2) beats, and (3) stationary waves.
Beats
When two sound waves of slightly different frequencies are heard together, the resulting sound alternates between loud and soft — these periodic intensity fluctuations are called beats.
$$f_{\text{beat}} = |f_1 - f_2|$$
The number of loudness maxima per second equals the absolute difference of the two source frequencies.
$f_{\text{beat}}$=Beat frequency (beats per second)(Hz)
$f_1, f_2$=Frequencies of the two sources(Hz)
$|f_1 - f_2| > 10$ Hz
→Beats become too rapid for the human ear to distinguish; a rough sound is heard instead.
$f_1 = f_2$
→No beats — a steady tone is heard (this is the goal when tuning instruments).
Mechanism: At instants when both waves are in phase, constructive interference produces a loud sound; when out of phase, destructive interference produces silence.
Tuning Application: Musicians use beats to tune instruments — adjust the string until beats vanish ($f_{\text{beat}} = 0$).
Loading a Tuning Fork: Adding wax lowers the fork's frequency. If beat frequency increases after loading, the fork originally had a frequency lower than the reference.
Reflection of Waves
When a wave reaches the boundary between two media, part of it is reflected back with the same frequency and wavelength, but the phase may change depending on boundary type.
Fixed End (Rarer to Denser): Reflected wave undergoes a 180° phase change — an incident crest returns as a trough.
Free End (Denser to Rarer): Reflected wave has no phase change — an incident crest returns as a crest.
Echo: A practical consequence of sound reflection from a distant surface.
Reverberation: When the reflecting surface is close (round-trip time < 0.1 s), the reflected sound blends with the original rather than being heard as a separate echo.
An echo occurs when sound reflects off a surface; the total distance traveled is twice the distance to the obstacle.
$$d = \frac{v \times t}{2}$$
Calculates the distance to a reflector based on the round-trip time of a sound pulse.
$d$=Distance to obstacle(m)
$v$=Speed of sound(m/s)
$t$=Total round-trip time(s)
$t < 0.1$ s
→Human ear perceives reverberation instead of a distinct echo.
Sonar/Ultrasound: Application of echoes for underwater depth measurement and medical imaging.