Wave Propagation §

The two terms represent waves traveling in the $+x$ and $-x$ directions at speed $c$. For a string under tension $T$ with linear density $\mu$: $c = \sqrt{T/\mu}$. For sound in a fluid: $c = \sqrt{K/\rho}$ where $K$ is the bulk modulus and $\rho$ is the volume mass density. For electromagnetic waves in vacuum: $c = 1/\sqrt{\mu_0 \varepsilon_0}$.

In 3-D: $\nabla^2 y = (1/c^2)\,\partial^2 y/\partial t^2$. Spherical waves decay as $1/r$; plane waves do not.

Consider a string of linear density $\mu$ under tension $T$. Take a small segment of length $dx$ at position $x$. The vertical forces on the segment come from the tension at each end. For small slopes $\partial y/\partial x \ll 1$, the net vertical force is:

Newton’s second law for the segment (mass $\mu\,dx$):

Dividing by $\mu\,dx$:

The same structure appears in every linear, lossless medium: inertia (mass per unit volume or length) on the left, a restoring mechanism (tension, bulk modulus, elastic modulus) on the right. The wave speed is always the square root of (stiffness / inertia).

Dispersion: In a dispersive medium, $c$ depends on frequency: $\omega = \omega(k)$. The phase velocity is $v_p = \omega/k$; the group velocity is $v_g = d\omega/dk$. When $v_g \neq v_p$, wave packets spread over time.

Examples:

• Deep-water gravity waves: $\omega^2 = gk$ (strongly dispersive).
• Sound in air: $\omega = ck$ (nearly non-dispersive).
• Light in glass: $n(\omega)$ varies (chromatic dispersion).

Two waves of the same frequency and amplitude traveling in opposite directions form a standing wave:

Nodes (zero displacement for all $t$) occur where $\cos(kx) = 0$, i.e. $x = (n+\tfrac{1}{2})\pi/k$ for $n = 0, 1, 2, \ldots$. Antinodes (maximum displacement) occur where $|\cos(kx)| = 1$, i.e. $x = n\pi/k$.

Two waves with slightly different frequencies produce beats at frequency $|f_1 - f_2|$.

Interference is the basis of diffraction gratings, noise-cancelling headphones, and the double-slit experiment.

For a string, the wave impedance is $Z = \sqrt{T\mu}$ (units: N·s/m). At an interface between two strings with impedances $Z_1$ and $Z_2$:

where $r$ is the reflection coefficient (amplitude ratio of reflected to incident wave) and $t$ is the transmission coefficient (amplitude ratio of transmitted to incident wave).

Special cases:

• Fixed end ($Z_2 \to \infty$): $r = -1$, $t = 0$. The wave reflects with a phase inversion (sign flip).
• Free end ($Z_2 \to 0$): $r = +1$, $t = 0$. The wave reflects without phase inversion.
• Matched impedances ($Z_1 = Z_2$): $r = 0$, $t = 1$. No reflection; all energy is transmitted.

The same formalism applies to:

• Acoustics: $Z = \rho c$ (acoustic impedance). Impedance matching prevents echo and maximizes sound transmission.
• Optics: Fresnel equations give reflection/transmission of light at a dielectric interface; the impedance analog is $Z = \sqrt{\mu/\varepsilon}$.
• Electrical transmission lines: $Z_0 = \sqrt{L/C}$ (characteristic impedance). Mismatches cause signal reflections.

Energy conservation requires $r^2 + t^2\,Z_1/Z_2 = 1$ (the transmitted power is scaled by the impedance ratio).

Energy is carried by the wave but the medium’s particles only oscillate locally. The energy density is split equally between kinetic and potential parts, oscillating in phase with the motion.

For sound: $I = p_0^2 / (2\rho c)$ where $p_0$ is the pressure amplitude. Here $\rho$ is the volume mass density of the medium (e.g. 1.2 kg/m³ for air), distinct from the linear density $\mu$ used for a 1-D string.

The distinction: $\mu$ [kg/m] is mass per unit length (1-D string); $\rho$ [kg/m³] is mass per unit volume (3-D fluid or solid). Both play the role of “inertia” in the wave speed formula.

Wave speed: $c = \sqrt{T/\mu} = \sqrt{100/0.01} = 100\;\text{m/s}$.

For a string fixed at both ends, the fundamental has wavelength $\lambda_1 = 2L = 2\;\text{m}$.

Fundamental frequency: $f_1 = c/\lambda_1 = 100/2 = 50\;\text{Hz}$.