= Solution
A <wave> propagates an organized disturbance through space. What oscillates depends on the medium: displacement in a string or solid, pressure and density in a fluid, <electric fields> and <magnetic fields> in an <electromagnetic wave>, or a quantum <wavefunction>. The propagation law follows from the restoring mechanism, inertia and relevant <boundary conditions>. A common linear equation is $u_{tt}=c^2\nabla^2u$, whose <plane waves> obey $\omega=c|\mathbf k|$. The resulting <phase velocity> and <group velocity> coincide, so a narrow <wave packet> can travel without dispersive spreading. More generally the material supplies a <dispersion relation> $\omega(\mathbf k)$; the envelope velocity is $\nabla_{\mathbf k}\omega$, while constant phase surfaces move at the corresponding <phase velocity>.
Mechanical waves make the balance of restoring <force> and inertia particularly visible. A flexible stretched string has $c=\sqrt{T/\mu}$: greater <tension> stiffens the string, whereas greater <linear mass density> slows its transverse motion. Small-amplitude <acoustic waves> in a uniform fluid are longitudinal, with $c_s=\sqrt{K_s/\rho}$, where $K_s$ is the adiabatic <bulk modulus>. For an <ideal gas>, $c_s=\sqrt{\gamma p/\rho}$. Pressure changes in rapid sound oscillations are approximately adiabatic because little heat crosses a <wavelength> in one period. Fluids do not support ordinary bulk shear waves, whereas an elastic solid supports both longitudinal and transverse waves. In an isotropic solid their speeds are $c_L=\sqrt{(K+4G/3)/\rho}$ and $c_T=\sqrt{G/\rho}$, where $G$ is the <shear modulus>.
A material can be dispersive even when its equilibrium properties are uniform. In a liquid with a free surface, <gravity> and <surface tension> act as restoring mechanisms. A simple inviscid <capillary-gravity wave> has
$$
\omega^2=\left(gk+\frac{\gamma_s}{\rho}k^3\right)\tanh(kh),
$$
where $h$ is depth and $\gamma_s$ is <surface tension>. Long shallow gravity waves have approximately constant speed $\sqrt{gh}$; deep-water gravity waves have $\omega=\sqrt{gk}$ and <group velocity> half their <phase velocity>. Short capillary waves have a different dispersion dominated by <surface tension>. Bending stiffness similarly changes short-wavelength propagation in a string or plate, giving a contribution proportional to $k^4$ in $\omega^2$. Different restoring mechanisms can therefore control different <wavelength> ranges of the same medium.
<Electromagnetic waves> propagate in <vacuum> without material displacement, because <Maxwell equations> couple changing electric and magnetic fields. They are transverse and travel at $c=1/\sqrt{\epsilon_0\mu_0}$. In a simple transparent isotropic material, the phase speed is $1/\sqrt{\epsilon\mu}=c/n$, where $n$ is the <refractive index>. Real material polarization depends on <frequency>. A <Lorentz oscillator model> for bound charges explains why <permittivity> and <refractive index> become strongly <frequency> dependent near material resonances, accompanied by absorption. In a birefringent medium, different field polarizations have different propagation constants, permitting <wave plates> and mode-dependent delays. A <electrical conductor> additionally supports dissipative currents; its complex <wavenumber> gives attenuation and, for a good conductor, a <skin depth> approximately $\sqrt{2/(\mu\sigma\omega)}$. <Energy> removed from the wave becomes heat rather than disappearing.
At an interface, <frequency> is conserved for stationary media, and tangential <wave phase> must match. These conditions give <Snell's law> for refraction and constrain reflected and transmitted directions. Amplitudes follow from continuity of the appropriate fields or mechanical forces. On strings the relevant <mechanical impedance> is $\sqrt{T\mu}$; for sound it is related to $\rho c_s$, and for electromagnetic waves to $\sqrt{\mu/\epsilon}$. A mismatch gives <wave reflection>, and a lossless boundary conserves the total incident, reflected and transmitted <power>. Multiple reflections can reinforce or cancel, as in a <Fabry–Pérot etalon>. Total internal reflection and other boundary conditions confine waves in <waveguides>. Transverse modes then become discrete, and even an otherwise nondispersive material develops longitudinal cutoffs and <wave dispersion>. A membrane strip, for example, obeys $\omega_n^2=c_m^2[k^2+(n\pi/a)^2]$.
Propagation also involves transport and loss. The <Poynting vector> gives electromagnetic <energy flux>; mechanical <energy> transport is <force> or pressure times velocity, averaged over a cycle. <Wave absorption>, viscosity, <electrical resistance> and imperfect boundaries dissipate <energy>. Scattering redistributes <energy> into other directions and modes, while geometric spreading reduces local intensity without necessarily dissipating anything. These mechanisms should be distinguished when interpreting a decreasing measured amplitude. In strongly absorbing or rapidly varying dispersive regimes, the derivative called <group velocity> is not automatically the speed of a sharp signal front; pulse reshaping and the medium's causal response must be considered.
Finally, quantum <matter waves> illustrate how the same propagation mathematics can describe a different physical quantity. For a free nonrelativistic particle, $\omega=\hbar k^2/(2m)$, so the <group velocity> $\hbar k/m$ equals its particle velocity and differs from the phase speed $\hbar k/(2m)$. In a periodic crystal, <Bloch's theorem> replaces the free dispersion by <energy bands>. Their slope controls wave-packet velocity, $v_g=\hbar^{-1}dE/dk$, and their curvature gives <band effective mass>. Thus mechanical confinement, material response and microscopic periodicity all reshape propagation through their allowed modes and dispersion, while the physical interpretation of the wave and its transported quantities remains essential.
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