Category: Part 4: Oscillations & Waves

Problem Statement Solve the oscillation/wave problem: Solve the oscillation/wave problem: In a sound wave, the displacement is $\xi = a\cos(\omega t – kx)$. Find the pressure variation and show that pressure leads displacement by $90°$. Pressure variation from the adiabatic relation and continuity: $$\Delta p = -B\frac{\partial\xi}{\partial x} = -\rho Given Information See problem statement…

Problem Statement Solve the oscillation/wave problem: Solve the oscillation/wave problem: A transverse wave $y = a\cos(\omega t – kx)$ travels along the $x$-axis. Find the particle velocity and relate it to wave velocity and displacement gradient. Particle velocity (velocity of the medium, not the wave): $$v_y = \frac{\partial y}{\partial t} = -a\omega Given Information See…

Problem Statement Solve the oscillation/wave problem: Solve the oscillation/wave problem: Write the equation of a plane harmonic wave traveling in the $+x$ direction. Define all parameters. The general plane harmonic wave: $$\boxed{\xi(x,t) = a\cos\left(\omega t – kx + \phi_0\right) = a\cos\left(\frac{2\pi t}{T} – \frac{2\pi x}{\lambda} + \phi_0\right) Given Information See problem statement for all given…

Problem Statement Solve the oscillation/wave problem: Solve the oscillation/wave problem: Define phase velocity $v_p$ and group velocity $v_g$ for a wave packet. When are they equal? Phase velocity: speed of a wavefront of fixed phase: $$v_p = \frac{\omega}{k}$$ Group velocity: speed of the envelope of a wave packet (speed of energy/information): $$v_g Given Information See…

Problem Statement Solve the oscillation/wave problem: Solve the oscillation/wave problem: Derive the speed of sound in an ideal gas and express it in terms of temperature $T$, molar mass $M$, and $\gamma = C_p/C_V$. Sound propagation is adiabatic (too fast for heat exchange). The adiabatic bulk modulus: $$B_s = \gamma p$$ Wave speed: $$v =…

Problem Statement Solve the oscillation/wave problem: Solve the oscillation/wave problem: Derive the wave equation for longitudinal waves in a thin elastic rod (Young’s modulus $E$, density $\rho$) and find the wave speed. Consider a thin rod element of length $dx$ and cross-section $S$. For displacement $u(x,t)$, the strain is $\partial u/\partial x$ Given Information See…

Problem Statement Solve the oscillation/wave problem: Solve the oscillation/wave problem: A string has linear density $\mu$ (kg/m) and tension $T$. Derive the wave equation and find the wave speed. For a small transverse displacement $y(x,t)$ of an element $dx$, the net transverse force from tension is $T\partial^2 y/\partial x^2\,dx$ (for small angles Given Information See…

Problem Statement Solve the magnetic field/force problem: Solve the magnetic field/force problem: Derive the electromagnetic wave equation from Maxwell’s equations in free space and identify the wave speed. Maxwell’s equations in vacuum (no sources): $$\nabla\times\mathbf{E} = -\frac{\partial\mathbf{B}}{\partial t}, \qquad \nabla\times\mathbf{B} = \mu_0\ep Given Information See problem statement for all given quantities. Physical Concepts & Formulas…

Problem Statement Solve the oscillation/wave problem: Solve the oscillation/wave problem: At $t=0$, an LC circuit has charge $Q_0$ on $C$ and current $I_0$ through $L$ (in the charging direction). Find $Q(t)$ and the amplitude. General solution: $Q(t) = A\cos(\omega_0 t) + B\sin(\omega_0 t)$ With $Q(0) = Q_0$: $A = Q_0$. Current $I = -\dot{Q}$: at…

Problem Statement Solve the oscillation/wave problem: Solve the oscillation/wave problem: A charged capacitor $C_1$ (charge $Q_0$) is suddenly connected to an uncharged capacitor $C_2$ through an inductor $L$. Describe the subsequent charge redistribution. Let $Q(t)$ = charge on $C_1$. The circuit equation: $$L\ddot{Q} + \frac{Q}{C_1} + \frac{Q_0 – Q}{ Given Information See problem statement for…