What Determines Wave Speed

Restoring action competes with inertia

Why does a pulse move faster on a taut rope than on a slack one? A greater tension produces a stronger restoring response when a string is bent. For a given restoring force, a more massive element accelerates less, so greater mass per unit length tends to slow transmission. These are properties of the prepared medium. The source sets how frequently it drives the motion; once speed and frequency are fixed, wavelength follows from λ = v/f. For a small wave on an ideal uniform string, increasing the driving frequency shortens the wavelength rather than making the wave travel faster.

Use F for the string tension in this lesson so that T can continue to mean oscillation period. The source sometimes uses T for tension. Linear mass density μ = m/L is mass per stretched length in kg m⁻¹, unlike the volume density ρ in kg m⁻³. The length and mass must refer to the same string section under the stated conditions.

Matching mass, length and time powers with [v] = L𝕋⁻¹ gives α + β = 0, α − β = 1 and −2α = −1. The last equation gives α = 1/2, hence β = −1/2. Therefore v = C√(F/μ). Dimensional analysis cannot establish C = 1: it only fixes the powers. The detailed Newton-law string calculation, not developed in this source, supplies that constant for the ideal string model.

F/μ has units (kg m s⁻²)/(kg m⁻¹) = m² s⁻², so its square root has the correct speed unit. Quadrupling tension doubles speed; quadrupling μ halves it. Amplitude is absent because the ideal linear model assumes the disturbance does not appreciably alter the tension. Very large slopes, changing tension, bending stiffness or nonuniform string properties can invalidate that simplified model.

Compression and the bulk modulus

For sound in a fluid, the restoring response comes from pressure changes when a small volume is compressed. Bulk modulus B measures the pressure increase required for a given fractional reduction in volume. If pressure rises, volume falls, so the minus sign makes the modulus positive for a stable material. Use sufficiently small changes, or the corresponding local derivative, to characterise the response near equilibrium.

The ratio B/ρ has dimensions (ML⁻¹𝕋⁻²)/(ML⁻³) = L²𝕋⁻². The same stiffness–inertia reasoning as for a string gives √(B/ρ), with the dimensionless factor supplied by the full dynamics. It is the ratio, not density alone, that determines speed. Liquids and solids are denser than gases, but their resistance to compression is generally much greater as well, so many transmit sound faster. This is a trend with exceptions, not a ranking rule based only on the material’s state.

For longitudinal extension and compression in a slender elastic rod, the relevant effective stiffness is Young’s modulus Y. In the elementary long-wavelength rod model, v = √(Y/ρ). This is not a universal formula for all longitudinal waves in an unbounded three-dimensional solid, where additional elastic moduli enter. A transverse wave on a tensioned steel wire, longitudinal sound in a rod, and bulk sound in steel are different mechanical models.

Newton’s isothermal estimate

To estimate B for a gas we need a relation between pressure and volume during the disturbance. Newton assumed the compression and expansion were isothermal: temperature remains constant. For a fixed number N of ideal-gas molecules, PV = NkᴮT. At constant T, PV is constant. For small changes, (P + ΔP)(V + ΔV) − PV = 0. Ignoring the second-order product ΔPΔV gives VΔP + PΔV = 0. Rearranging yields −ΔP/(ΔV/V) = P, so the isothermal bulk modulus equals the equilibrium pressure.

Laplace’s correction: compression is effectively adiabatic

The pressure changes in ordinary sound are rapid enough that neighbouring regions have little time to exchange the heat needed to maintain equal temperature during each cycle. A small compressed element warms and becomes stiffer than in an isothermal compression. Model the local reversible response as adiabatic, PVᵞ = constant, with γ = Cₚ/Cᵥ. For the same gas, γ is dimensionless and greater than one. This approximation describes ordinary small-amplitude sound; thermal conduction and losses can matter in other frequency or length regimes.

Divide by Vᵞ to obtain ΔP + γP(ΔV/V) = 0. Move the volume term to the other side and divide by −ΔV/V: the adiabatic bulk modulus is γP. It is larger than the isothermal modulus by γ. Since speed depends on the square root of modulus, the sound-speed correction factor is √γ, rather than γ.

Using the preceding unrounded density and pressure gives v ≈ 330.48 m s⁻¹. Using standard constants P = 1.01325 × 10⁵ Pa and molar volume 22.414 L gives approximately 331.1 m s⁻¹. Both agree at the precision of the stated textbook inputs with the quoted 331 m s⁻¹. Do not claim an extra decimal place that the rounded data do not support.

Pressure, temperature and humidity

At a fixed temperature and composition, increasing pressure also increases density in the same ratio, leaving P/ρ and therefore the ideal-gas sound speed unchanged. It would be wrong to change P alone while pretending ρ is independent of it. At fixed composition and approximately constant γ, speed grows as the square root of absolute temperature, so temperature ratios must use kelvin. Humid air contains water vapour of molar mass 18 g mol⁻¹, replacing some dry air of average molar mass about 29 g mol⁻¹. At equal temperature and pressure this reduces the mixture’s average molar mass and generally increases sound speed. Humidity is water vapour mixed into air, not suspended liquid droplets.

Changing the medium, not the source rhythm

At a stationary interface with steady media, the reflected and transmitted waves keep the driving frequency: adjacent regions must respond in step at the boundary. Their speeds can differ, so their wavelengths change according to λ = v/f. Increasing the frequency of the source does not by itself increase the speed in the ideal nondispersive medium. Similarly, moving the source does not alter the medium’s rest-frame elastic properties. Source motion can change the frequency detected elsewhere; a full Doppler treatment is outside the supplied chapter.

The frequency-independent speeds used above describe nondispersive models. A pulse is built from a range of sinusoidal components. In a dispersive medium the phase speed depends on frequency, so those components accumulate different relative phases as they propagate. Their superposition generally changes shape: the pulse broadens or distorts. A single pure sinusoid still has a defined wavelength and phase speed, but one universal speed need not describe an entire pulse. This explains why a pulse in a dispersive medium cannot generally be treated as one unchanged function F(x − vt).

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