Unit 7: Mechanical Waves

Speed of Wave Motion

Mechanical waves transport energy without permanently displacing the medium. Their speed depends on the medium's elastic and inertial properties. Two fundamental cases are waves on a stretched string and longitudinal waves in a solid.

Waves on a Stretched String

For a string under tension T with linear mass density μ (mass per unit length), the transverse wave speed is derived from balancing the restoring force due to tension with the inertia of the string elements.

v = sqrt(T/μ)

where:

• v – wave speed (m s⁻¹)
• T – tension in the string (N)
• μ – linear mass density (kg m⁻¹)

Application: Musical instruments such as guitars and violins rely on this relation. By adjusting tension (tuning pegs) or using strings of different thickness (changing μ), musicians set the pitch of the notes.

Longitudinal Waves in a Solid Rod

In a bulk solid, longitudinal (compressional) waves travel with a speed determined by the material’s Young’s modulus E and density ρ. The derivation assumes small deformations and Hooke’s law.

v = sqrt(E/ρ)

where:

• E – Young’s modulus (Pa), a measure of stiffness
• ρ – density of the solid (kg m⁻³)

Typical values:

Material	Young’s Modulus E (GPa)	Density ρ (kg m⁻³)	Wave Speed v (m s⁻¹)
Steel	200	7850	≈5050
Aluminium	69	2700	≈5050
Glass (silica)	70	2500	≈5290
Rubber	0.01	1100	≈95

Application: Ultrasonic testing of metals uses this high speed to detect internal flaws; the time‑of‑flight of an ultrasonic pulse reveals discontinuities.

Velocity of Sound in Solid and Liquid

Sound is a longitudinal mechanical wave. In solids the same expression as above applies, while in liquids the restoring force is governed by the bulk modulus B rather than Young’s modulus.

Sound in Solids

For an isotropic solid, the longitudinal sound speed is:

v = sqrt(E/ρ)

Because solids are very stiff (large E) yet not excessively dense, sound travels rapidly—typically several kilometres per second. In steel, v ≈ 5000 m s⁻¹.

Sound in Liquids

Liquids resist compression; the relevant modulus is the bulk modulus B. The speed of sound in a liquid is:

v = sqrt(B/ρ)

where:

• B – bulk modulus (Pa)
• ρ – density (kg m⁻³)

Example values:

Liquid	Bulk Modulus B (GPa)	Density ρ (kg m⁻³)	Speed v (m s⁻¹)
Water (20 °C)	2.2	998	≈1482
Ethanol	1.06	789	≈116076
Mercury	28	13550	≈1450

Application: Sonar systems exploit the predictable speed of sound in seawater (≈1500 m s⁻¹) to determine distances to underwater objects.

Velocity of Sound in Gas

In gases, the wave speed depends on how pressure changes with density during compression and rarefaction. Two historical models—Newton’s isothermal assumption and Laplace’s adiabatic correction—lead to the correct formula.

Newton’s Formula (Isothermal Assumption)

Newton assumed that sound propagation in a gas occurs isothermally (constant temperature), so the pressure‑density relation follows Boyle’s law: P ∝ ρ. The resulting speed is:

v = sqrt(P/ρ)

Using standard atmospheric conditions (P = 1.01×10⁵ Pa, ρ ≈ 1.2 kg m⁻³) gives v ≈ 290 m s⁻¹, far below the measured ~340 m s⁻¹ in air.

Laplace’s Correction (Adiabatic Process)

Laplace recognized that sound oscillations are rapid enough that heat does not have time to flow; the process is adiabatic. For an adiabatic change, P V^γ = const, leading to P ∝ ρ^γ. The corrected speed becomes:

v = sqrt(γ P/ρ)

Using the ideal‑gas law P = ρ R T / M (where R is the universal gas constant, T absolute temperature, M molar mass), we obtain:

v = sqrt(γ R T / M)

where:

• γ = C_p/C_v – ratio of specific heats (adiabatic index)
• R – 8.314 J mol⁻¹ K⁻¹
• T – absolute temperature (K)
• M – molar mass of the gas (kg mol⁻¹)

Example for dry air (γ ≈ 1.4, M ≈ 0.029 kg mol⁻¹) at 300 K:

v = sqrt(1.4 × 8.314 × 300 / 0.029) ≈ 347 m s⁻¹

This matches the measured speed of sound in air at room temperature.

Effect of Temperature, Pressure, and Humidity on Sound Speed in Gases

The derived expression v = sqrt(γ R T / M) shows the dependencies clearly.

Temperature

Speed varies with the square root of absolute temperature:

v ∝ sqrt(T)

Thus, raising the temperature increases molecular speed and hence sound speed. For dry air, an approximate linear relation is:

v (m s⁻¹) ≈ 331 + 0.6 T (°C)

Example: At 0 °C, v ≈ 331 m s⁻¹; at 30 °C, v ≈ 331 + 0.6×30 = 349 m s⁻¹.

Pressure

For an ideal gas, P/ρ = R T / M is independent of pressure; therefore, v does not change with pressure at constant temperature. In real gases, deviations are negligible at ordinary pressures.

Humidity

Water vapor (M ≈ 0.018 kg mol⁻¹) is lighter than dry air (M ≈ 0.029 kg mol⁻¹). Adding vapor lowers the average molar mass of the mixture, increasing v because v ∝ 1/√M. The effect is modest but measurable: at 30 °C, raising relative humidity from 0 % to 100 % increases sound speed by about 1–2 m s⁻¹.

Application: Meteorologists use sound speed measurements to infer atmospheric temperature and humidity profiles.

Summary of Key Formulas

Scenario	Formula	Variables
Transverse wave on string	v = sqrt(T/μ)	T = tension (N), μ = linear mass density (kg m⁻¹)
Longitudinal wave in solid	v = sqrt(E/ρ)	E = Young’s modulus (Pa), ρ = density (kg m⁻³)
Longitudinal wave in liquid	v = sqrt(B/ρ)	B = bulk modulus (Pa), ρ = density (kg m⁻³)
Sound in ideal gas (adiabatic)	v = sqrt(γ P/ρ) = sqrt(γ R T / M)	γ = Cₚ/Cᵥ, R = gas constant, T = temperature (K), M = molar mass (kg mol⁻¹)

These relations form the foundation for designing musical instruments, acoustic engineering, non‑destructive testing, and atmospheric probing.