Traveling Waves

A wave is an organized disturbance that propagates through space or a material medium, transporting energy and momentum without transporting macroscopic matter.

1. General Form of a 1D Traveling Wave

Consider a 1D disturbance $\psi(x, t)$ maintaining its shape as it translates along the x-axis with speed $v$: $$\psi(x, t) = f(x \mp vt)$$ where the minus sign denotes propagation in the $+x$ direction (forward wave) and the plus sign denotes propagation in the $-x$ direction (backward wave).

2. The Classical 1D Wave Equation

Using the chain rule with variables $\xi = x - vt$ and $\eta = x + vt$: $$\frac{\partial \psi}{\partial x} = f'(\xi), \quad \frac{\partial^2 \psi}{\partial x^2} = f''(\xi)$$ $$\frac{\partial \psi}{\partial t} = -v f'(\xi), \quad \frac{\partial^2 \psi}{\partial t^2} = v^2 f''(\xi)$$ Equating second derivatives: $$\frac{\partial^2 \psi}{\partial x^2} = \frac{1}{v^2} \frac{\partial^2 \psi}{\partial t^2}$$ This is the celebrated linear, second-order hyperbolic Classical Wave Equation. Its general solution, established by Jean le Rond d'Alembert (1747), is: $$\psi(x, t) = f(x - vt) + g(x + vt)$$ where $f$ and $g$ are arbitrary twice-differentiable functions determined by initial Cauchy boundary data $\psi(x, 0)$ and $\dot{\psi}(x, 0)$.

3. Harmonic Plane Waves

For sinusoidal disturbances: $$\psi(x, t) = A \cos(k x - \omega t + \phi) = A \cos\left[ \frac{2\pi}{\lambda} (x - v t) + \phi \right]$$ where:

• $A$: Wave amplitude.
• $k = \frac{2\pi}{\lambda}$: Wavenumber (spatial angular frequency in rad/m).
• $\omega = 2\pi f$: Temporal angular frequency in rad/s.
• $v = \frac{\omega}{k} = f \lambda$: Phase speed.
• Complex notation: $\psi(x, t) = \text{Re}\{ A e^{i(kx - \omega t)} \}$.

Consider a flexible, perfectly elastic string of uniform linear mass density $\mu$ (kg/m) stretched under constant equilibrium tension $T$ (N).

1. Derivation from Newton's Second Law

Let the string undergo small transverse vibrations in the xy-plane. Consider an infinitesimal segment located between $x$ and $x + dx$ with mass $dm = \mu \, dx$. The slope of the string at $x$ is $\frac{\partial y}{\partial x} = \tan\theta_1 \approx \sin\theta_1$. The net transverse force $dF_y$ acting on the segment is: $$dF_y = T \sin\theta_2 - T \sin\theta_1 \approx T \left[ \left( \frac{\partial y}{\partial x} \right)_{x+dx} - \left( \frac{\partial y}{\partial x} \right)_x \right] = T \frac{\partial^2 y}{\partial x^2} dx$$ By Newton's second law ($dF_y = dm \, a_y = \mu \, dx \frac{\partial^2 y}{\partial t^2}$): $$T \frac{\partial^2 y}{\partial x^2} dx = \mu \, dx \frac{\partial^2 y}{\partial t^2}$$ $$\frac{\partial^2 y}{\partial x^2} = \frac{\mu}{T} \frac{\partial^2 y}{\partial t^2}$$ Comparing with the general wave equation $\frac{\partial^2 y}{\partial x^2} = \frac{1}{v^2} \frac{\partial^2 y}{\partial t^2}$ immediately yields: $$v = \sqrt{\frac{T}{\mu}}$$ Physical Meaning: Wave propagation speed is governed strictly by the ratio of the medium's elastic restoring property ($T$) to its inertial property ($\mu$). Amplitude and wavelength have zero effect in non-dispersive strings.

Longitudinal waves transmit disturbances via compressive and rarefactive displacements along the axis of propagation.

1. Longitudinal Waves in an Elastic Solid Bar

Consider a thin cylindrical solid rod of cross-sectional area $A$, material density $\rho$, and Young's modulus $Y$. Let $u(x, t)$ denote the longitudinal displacement of a cross-section originally at $x$. An infinitesimal segment of initial length $dx$ experiences longitudinal strain: $$\epsilon = \frac{\partial u}{\partial x}$$ The normal compressive/tensile stress is: $$\sigma = Y \epsilon = Y \frac{\partial u}{\partial x}$$ The net force acting on the element of mass $dm = \rho A \, dx$ is: $$dF = [\sigma(x+dx) - \sigma(x)] A = A \frac{\partial \sigma}{\partial x} dx = A Y \frac{\partial^2 u}{\partial x^2} dx$$ Applying Newton's second law ($dF = dm \frac{\partial^2 u}{\partial t^2}$): $$A Y \frac{\partial^2 u}{\partial x^2} dx = \rho A \, dx \frac{\partial^2 u}{\partial t^2} \implies \frac{\partial^2 u}{\partial x^2} = \frac{\rho}{Y} \frac{\partial^2 u}{\partial t^2}$$ The speed of longitudinal acoustic waves in a thin solid bar is: $$v = \sqrt{\frac{Y}{\rho}}$$ (For steel: $Y \approx 2 \times 10^{11} \text{ Pa}, \rho \approx 7850 \text{ kg/m}^3 \implies v \approx 5050 \text{ m/s}$).

2. Acoustic Plane Waves in Fluid Media

In fluids, shear modulus vanishes, so restoring forces are provided exclusively by the Bulk Modulus ($B$): $$B = -V \frac{dP}{dV} = \rho \frac{dP}{d\rho}$$ Following identical dynamic balance for an acoustic plane wave: $$v = \sqrt{\frac{B}{\rho}}$$

• Newton's Isothermal Formula (1687): Newton assumed sound compressions occurred isothermally ($P V = \text{const} \implies B_{iso} = P$). $$v_{\text{Newton}} = \sqrt{\frac{P}{\rho}} \approx 280 \text{ m/s in air at STP (16\% error!)}$$
• Laplace's Adiabatic Correction (1816): Pierre-Simon Laplace recognized that acoustic compressions and rarefactions happen so rapidly that heat conduction between adjacent regions is negligible. Acoustic cycles are strictly adiabatic ($P V^\gamma = \text{const}$): $$B_{ad} = \gamma P$$ $$v = \sqrt{\frac{\gamma P}{\rho}} = \sqrt{\frac{\gamma R T}{M}}$$ where $\gamma = C_p / C_v \approx 1.40$ for diatomic air ($M = 0.02897 \text{ kg/mol}$). At 20°C (293.15 K): $v = \sqrt{1.40 \times 8.314 \times 293.15 / 0.02897} = 343.2 \text{ m/s}$, matching experimental data perfectly!

Traveling waves transport mechanical energy down the medium as particles oscillate in succession.

1. Energy Density of a Harmonic Transverse Wave

Consider a harmonic wave on a string: $y(x, t) = A \cos(kx - \omega t)$. For an element of mass $dm = \mu \, dx$:

• Kinetic Energy ($dK$): $$dK = \frac{1}{2} dm \left( \frac{\partial y}{\partial t} \right)^2 = \frac{1}{2} \mu \, dx \left[ \omega A \sin(kx - \omega t) \right]^2 = \frac{1}{2}\mu \omega^2 A^2 \sin^2(kx - \omega t) dx$$
• Potential Energy ($dU$): Work done in stretching the string element from $dx$ to $ds = \sqrt{dx^2 + dy^2} \approx dx [1 + \frac{1}{2}(\frac{\partial y}{\partial x})^2]$: $$dU = T (ds - dx) = \frac{1}{2} T \left( \frac{\partial y}{\partial x} \right)^2 dx = \frac{1}{2} T \left[ -k A \sin(kx - \omega t) \right]^2 dx = \frac{1}{2} T k^2 A^2 \sin^2(kx - \omega t) dx$$ Since $v = \omega / k = \sqrt{T / \mu} \implies T k^2 = \mu \omega^2$: $$dU = \frac{1}{2} \mu \omega^2 A^2 \sin^2(kx - \omega t) dx = dK$$

Fundamental Property: In a pure traveling wave, kinetic energy and potential energy are in phase and locally identical at all points! Total energy per unit length (linear energy density): $$u_L = \frac{dE}{dx} = \mu \omega^2 A^2 \sin^2(kx - \omega t)$$ Average energy density over one wavelength: $$\langle u_L \rangle = \frac{1}{2} \mu \omega^2 A^2 \quad (\text{J/m})$$

2. Wave Power and Intensity

The rate at which energy is transmitted through any cross-section is the instantaneous power: $$P(t) = -T \left( \frac{\partial y}{\partial x} \right) \left( \frac{\partial y}{\partial t} \right) = T k \omega A^2 \sin^2(kx - \omega t)$$ Since $T k = \mu v^2 (\omega / v) = \mu v \omega$: $$P(t) = \mu v \omega^2 A^2 \sin^2(kx - \omega t)$$ The time-averaged transmitted power is: $$\langle P \rangle = \frac{1}{2} \mu v \omega^2 A^2 = \langle u_L \rangle v \quad (\text{Watts})$$ For a 3D medium with volume density $\rho$, the wave intensity $I$ (power per unit area) is: $$I = \frac{\langle P \rangle}{\text{Area}} = \frac{1}{2} \rho v \omega^2 A^2 = 2 \pi^2 \rho v f^2 A^2 \quad (\text{W/m}^2)$$ Intensity is strictly proportional to the square of frequency and the square of amplitude ($I \propto f^2 A^2$).

The linear nature of the classical wave equation implies that multiple wave disturbances superpose linearly.

1. The Principle of Superposition

If $\psi_1(x, t)$ and $\psi_2(x, t)$ are individual solutions to the linear wave equation, any linear combination: $$\psi(x, t) = c_1 \psi_1(x, t) + c_2 \psi_2(x, t)$$ is also an exact solution. When two waves pass through the same region, the net displacement is simply the algebraic sum of their separate displacements.

2. Shallow Water Waves in an Open Canal

Consider surface waves propagating along a shallow canal of uniform depth $h$ where wavelength $\lambda \gg h$. The horizontal velocity of water parcels is nearly uniform from bed to surface. The wave speed is governed purely by gravitational restoring forces: $$v = \sqrt{g h}$$ Remarkably, this shallow water wave speed is completely non-dispersive (independent of wavelength $\lambda$). (This explains the immense speed of ocean tsunamis: across an ocean basin of depth $h = 4000 \text{ m}$, speed reaches $v = \sqrt{9.8 \times 4000} \approx 200 \text{ m/s} \approx 720 \text{ km/h}$).

3. Capillary Waves and Ripples

On water surfaces, restoring forces are provided by both gravity ($g$) and surface tension ($\gamma$). Hydrodynamic analysis of Airy wave theory yields the general dispersion relation for surface waves on water of depth $h$: $$\omega^2 = \left( g k + \frac{\gamma}{\rho} k^3 \right) \tanh(k h)$$ For deep water ($k h \gg 1 \implies \tanh(kh) \to 1$): $$v_p^2 = \frac{\omega^2}{k^2} = \frac{g}{k} + \frac{\gamma}{\rho} k = \frac{g \lambda}{2\pi} + \frac{2\pi \gamma}{\rho \lambda}$$ Two asymptotic regimes exist:

• Gravity Waves ($\lambda \gg 1.7\text{ cm}$): Gravity dominates. Phase speed increases with wavelength: $$v_p \approx \sqrt{\frac{g \lambda}{2\pi}}$$
• Ripples / Capillary Waves ($\lambda \ll 1.7\text{ cm}$): Surface tension dominates. Phase speed increases as wavelength gets smaller: $$v_p \approx \sqrt{\frac{2\pi \gamma}{\rho \lambda}}$$

Minimum Phase Speed: Minimizing $v_p(\lambda)$: $$\frac{d(v_p^2)}{d\lambda} = \frac{g}{2\pi} - \frac{2\pi \gamma}{\rho \lambda^2} = 0 \implies \lambda_c = 2\pi \sqrt{\frac{\gamma}{\rho g}}$$ For pure water at 20°C ($\gamma = 0.0728 \text{ N/m}, \rho = 1000 \text{ kg/m}^3$): $$\lambda_c = 2\pi \sqrt{\frac{0.0728}{1000 \times 9.80}} \approx 1.71 \text{ cm}$$ $$v_{p,min} = \left( \frac{4 g \gamma}{\rho} \right)^{1/4} = \left( \frac{4 \times 9.80 \times 0.0728}{1000} \right)^{1/4} \approx 0.231 \text{ m/s} = 23.1 \text{ cm/s}$$ No surface disturbance can propagate across quiet water slower than 23.1 cm/s!

When the wave velocity depends on frequency or wavelength ($\frac{dv}{d\lambda} e 0$), the medium is said to be dispersive.

1. Phase Velocity ($v_p$) vs. Group Velocity ($v_g$)

Consider the superposition of two harmonic waves with slightly different frequencies and wavenumbers: $$\psi(x, t) = A \cos(k_1 x - \omega_1 t) + A \cos(k_2 x - \omega_2 t)$$ Let $k = \frac{k_1 + k_2}{2}, \Delta k = k_1 - k_2$ and $\omega = \frac{\omega_1 + \omega_2}{2}, \Delta \omega = \omega_1 - \omega_2$. Using the trigonometric identity $\cos\alpha + \cos\beta = 2\cos\frac{\alpha-\beta}{2}\cos\frac{\alpha+\beta}{2}$: $$\psi(x, t) = 2 A \cos\left( \frac{\Delta k}{2} x - \frac{\Delta \omega}{2} t \right) \cos(k x - \omega t)$$ This represents a high-frequency carrier wave modulated by a slowly varying envelope:

• Phase Velocity ($v_p$): The speed at which individual crests and troughs of the carrier advance: $$v_p = \frac{\omega}{k}$$
• Group Velocity ($v_g$): The speed at which the modulation envelope (and physical wave energy/information) propagates: $$v_g = \lim_{\Delta k \to 0} \frac{\Delta \omega}{\Delta k} = \frac{d\omega}{dk}$$

2. Rayleigh's Dispersion Relation

Since $\omega = k v_p$: $$v_g = \frac{d(k v_p)}{dk} = v_p + k \frac{dv_p}{dk}$$ Rewriting in terms of wavelength $\lambda = 2\pi / k$ (where $dk = -\frac{2\pi}{\lambda^2} d\lambda$): $$v_g = v_p - \lambda \frac{dv_p}{d\lambda}$$

• Non-dispersive medium ($\frac{dv_p}{d\lambda} = 0$): $v_g = v_p$ (e.g., sound in air, light in vacuum).
• Normal dispersion ($\frac{dv_p}{d\lambda} > 0$): $v_g < v_p$ (e.g., deep-water gravity waves where $v_g = \frac{1}{2} v_p$).
• Anomalous dispersion ($\frac{dv_p}{d\lambda} < 0$): $v_g > v_p$ (e.g., surface ripples where $v_g = \frac{3}{2} v_p$).

3. Fourier Series and Harmonic Wave Packets

Joseph Fourier (1822) proved that any arbitrary periodic function $f(x)$ with period $\lambda$ can be synthesized as an infinite sum of discrete sinusoidal harmonics: $$f(x) = \frac{a_0}{2} + \sum_{n=1}^\infty \left[ a_n \cos(n k x) + b_n \sin(n k x) \right]$$ where the Fourier coefficients are obtained via orthogonality integrals: $$a_n = \frac{2}{\lambda} \int_0^\lambda f(x) \cos(n k x) dx, \quad b_n = \frac{2}{\lambda} \int_0^\lambda f(x) \sin(n k x) dx$$ In a non-dispersive medium, all harmonics travel at the same speed $v$, maintaining the wave pulse shape. In a dispersive medium, each harmonic travels at its own phase speed $v_p(\omega_n)$, causing localized pulses to disperse and broaden over time.