Category: Part 4: Oscillations & Waves

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 $m$…

Problem Statement Solve the oscillation/wave problem: Sound is trapped in a water layer (depth $H$, speed $v_1 = 1500$ m/s) between two slower media ($v_2 = 1450$ m/s). Find the cutoff frequency for the lowest guided mode. Total internal reflection occurs when the grazing angle $\theta v_2$, so the wave is refracted toward normal at…

Problem Statement Solve the oscillation/wave problem: Define spatial coherence length for a sound field and explain its significance for acoustic imaging. The spatial coherence length $L_c$ is the distance over which the pressure at two points is significantly correlated. For a sound field with bandwidth $\Delta f$ centered at $f_0$: $$L_c \sim \frac{v Given Information…

Problem Statement Solve the oscillation/wave problem: Two intense ultrasound beams at $f_1 = 200$ kHz and $f_2 = 202$ kHz are co-propagated in water. A difference-frequency wave at $f_d = 2$ kHz is generated. Why is this useful? The nonlinear term $\partial(\rho v^2)/\partial x^2$ in the wave equation couples the two primary beams and generates…

Problem Statement Solve the oscillation/wave problem: A sound wave passes through a layer of acoustic insulation (thickness $l$, impedance $Z_1$) between two media of impedance $Z_0$. Find the transmission coefficient. For a layer of thickness $l$ and impedance $Z_1$ sandwiched between identical media ($Z_0$): $$T = \frac{1}{1 + \frac{(Z_1^2-Z_0^2)^2}{ Given Information Mass $m$ and spring…

Problem Statement Solve the oscillation/wave problem: A plane wave of frequency 440 Hz (wavelength 78 cm) passes through a circular aperture of diameter $d = 2.0$ m. Describe the diffraction pattern. The relevant parameter is $d/\lambda = 2.0/0.78 \approx 2.6$. Since $d/\lambda \sim$ a few, diffraction is significant but the beam has some directionalit Given…