Category: Part 4: Oscillations & Waves

Problem Statement Solve the oscillation/wave problem: Problem 4.191 — Waves: Underwater Acoustics — SOFAR Channel See problem statement for all given quantities. This problem applies fundamental physics principles to the scenario described. The solution requires identifying the relevant conservation laws and equations of motion, then solving systematic Given Information Mass $m$ and spring constant $k$…

Problem Statement Solve the oscillation/wave problem: Sound waves in a solid produce temperature fluctuations. Estimate the thermoelastic attenuation coefficient for longitudinal waves. When sound compresses a region, the temperature rises (adiabatic). If the period is long enough for heat conduction, the temperature equalizes — this is irreversible an Given Information Mass $m$ and spring constant…

Problem Statement Solve the oscillation/wave problem: Describe how acoustic waves in a stadium-shaped billiard can exhibit chaotic behavior. A Sinai billiard or stadium billiard has ergodic ray dynamics — a ray starting at almost any direction will eventually visit every region of the billiard and every direction of propagation. In the wave (quantum) c Given…

Problem Statement Solve the oscillation/wave problem: In an anisotropic elastic solid, the three wave modes for a given propagation direction are generally quasi-P and two quasi-S waves, not pure modes. Why? The Christoffel equation for propagation direction $\hat{n}$ is: $$\det(\Lambda_{ik} – \rho v^2\delta_{ik}) = 0$$ $$\Lambda_{ik} = c_{ijkl}n_j n_l Given Information Mass $m$ and spring…

Problem Statement Solve the oscillation/wave problem: A rigid sphere of radius $a \ll \lambda$ is placed in a standing wave field. Find the acoustic radiation force on the sphere. The time-averaged acoustic radiation force (Gorkov potential) on a small rigid sphere in a standing wave $p = p_0\cos(kx)\cos(\omega t)$: $$F = -\nabla U_{\rm Gorkov}$$ $$U_{…

Problem Statement Solve the oscillation/wave problem: Write the nonlinear Schrödinger equation (NLS) for the complex envelope of a dispersive wave packet with cubic nonlinearity and identify its solutions. For a wave packet near carrier $k_0$, $\omega_0$ with envelope $A(x,t)$, in a medium with dispersion $\omega = d^2\omega/dk^2$ and nonlinearity $\ga Given Information Mass $m$ and…

Problem Statement Solve the oscillation/wave problem: When a plate is bowed, sand collects at the nodes forming Chladni figures. For a square plate vibrating in its $m \times n$ mode, describe the nodal pattern. For a square plate of side $a$ with free edges, the flexural modes can be approximated by products of beam modes.…

Problem Statement Solve the oscillation/wave problem: A carrier wave $A_c\cos(\omega_c t)$ is amplitude-modulated by a signal $m(t) = m_0\cos(\omega_m t)$. Write the AM wave and find its frequency components. AM wave: $x(t) = A_c[1 + m_0\cos(\omega_m t)]\cos(\omega_c t)$ Expanding: $$x(t) = A_c\cos(\omega_c t) + \frac{A_c m_0}{2}\cos[(\omega_c+\omega_m Given Information Mass $m$ and spring constant $k$ (or…

Problem Statement Solve the oscillation/wave problem: A lossless transmission line has inductance per length $\mathcal{L}$ and capacitance per length $\mathcal{C}$. Derive the wave equation and find the phase velocity. Telegrapher’s equations for voltage $V(x,t)$ and current $I(x,t)$: $$\frac{\partial V}{\partial x} = -\mathcal{L}\frac{\partial I}{\par Given Information Mass $m$ and spring constant $k$ (or equivalent), or wave…

Problem Statement Solve the oscillation/wave problem: A phononic crystal is a periodic array of elastic inclusions. Describe the condition for a complete phononic bandgap and its use. Bragg bandgap condition: when the acoustic wavelength matches twice the periodicity $a$ of the crystal (Bragg condition), destructive interference creates a stop band: $$ Given Information Mass $m$…