Speed of a Travelling Wave

The phase-speed relation v = νλ = ω/k

To measure how fast a wave moves, NCERT fixes attention on a point of constant phase and tracks it through time. For a wave moving in the positive $x$-direction, a fixed phase point satisfies $kx-\omega t = \text{constant}$. Demanding that the phase stay constant as $t$ advances gives the phase speed:

$$v=\frac{dx}{dt}=\frac{\omega}{k}=\frac{2\pi\nu}{2\pi/\lambda}=\nu\lambda=\frac{\lambda}{T}$$

This is the general relation for every progressive wave. In the time taken for one element of the medium to complete a single oscillation (one period $T$), the wave pattern advances exactly one wavelength $\lambda$. The relation is purely kinematic — it links $v$, $\nu$ and $\lambda$ — but it does not by itself tell you what $v$ is. That number is set by the medium.

Transverse wave on a stretched string

The speed of any mechanical wave is set by two competing properties of the medium: the restoring force that pulls a disturbed element back, and the inertia that resists its motion. Speed rises with the restoring force and falls with the inertia. For a transverse wave on a string, the restoring force is the tension $T$, and the inertia is the linear mass density $\mu = m/L$ (mass per unit length).

Dimensional analysis gives $[\,T/\mu\,]=[\text{LT}^{-1}]^2$, and the exact derivation fixes the dimensionless constant at unity:

$$v=\sqrt{\frac{T}{\mu}}\qquad\text{(NCERT Eq. 14.14)}$$

This speed depends only on the string's properties — tension and linear mass density — and not on the wavelength or frequency of the wave riding on it. The source sets $\nu$; the string sets $v$; the wavelength then follows from $\lambda = v/\nu$.

Longitudinal waves: fluids and solids

For a longitudinal wave, the medium is compressed and rarefied along the direction of travel. The restoring property is now an elastic modulus and the inertia is the volume mass density $\rho$. In a fluid (gas or liquid), the relevant modulus is the bulk modulus $B$, defined by $B=-\Delta P/(\Delta V/V)$:

$$v=\sqrt{\frac{B}{\rho}}\qquad\text{(NCERT Eq. 14.19)}$$

A long, thin solid rod is special. Its lateral expansion is negligible, so it carries longitudinal strain rather than pure compression, and the controlling modulus becomes Young's modulus $Y$:

$$v=\sqrt{\frac{Y}{\rho}}\qquad\text{(NCERT Eq. 14.20)}$$

Both share the same template, $v=\sqrt{\text{elasticity}/\text{inertia}}$. Solids and liquids carry sound faster than gases: their densities are larger, but their elastic moduli are larger by a much wider margin, so $v_{\text{gas}}

Newton's formula and Laplace's correction

To find the speed of sound in a gas, the bulk modulus must be expressed in terms of pressure. Newton assumed the compressions and rarefactions happen isothermally — slowly enough for temperature to stay constant. For an isothermal change of an ideal gas, $B=P$, which gives Newton's formula:

$$v=\sqrt{\frac{P}{\rho}}\qquad\text{(Newton, isothermal)}$$

At STP this yields about 280 m s⁻¹, roughly 15% below the measured value of ≈ 331 m s⁻¹. The error is far too large to be experimental. Laplace identified the flaw: sound's pressure changes are far too rapid for heat to flow in and out, so the process is adiabatic, not isothermal. For an adiabatic ideal gas, $B_{\text{ad}}=\gamma P$, where $\gamma = C_p/C_v$. This is the Laplace correction:

$$v=\sqrt{\frac{\gamma P}{\rho}}\qquad\text{(Laplace, adiabatic)}$$

For air, $\gamma=7/5=1.4$. The corrected speed at STP comes out as ≈ 331 m s⁻¹, in close agreement with experiment.

Factors affecting the speed of sound

Rewriting Laplace's formula using the ideal-gas equation $PV=nRT$ and $\rho=M/V$ converts $\sqrt{\gamma P/\rho}$ into a cleaner form:

$$v=\sqrt{\frac{\gamma P}{\rho}}=\sqrt{\frac{\gamma RT}{M}}$$

This single expression reveals every factor that controls the speed of sound in a gas, and just as importantly, the factors that do not.

Speed formulae compared

Every wave speed in this chapter is a ratio of an elastic property to an inertial property under a square root. The table below puts the four cases — and the Newton–Laplace pair — side by side for revision.