Category: Part 4: Oscillations & Waves

Problem Statement Solve the oscillation/wave problem: In a large-amplitude sound wave, the compression parts travel faster than the rarefaction parts. Estimate the distance at which a sinusoidal wave develops a shock. In a nonlinear medium, the local sound speed is $c(x,t) = v_0 + \frac{\gamma+1}{2}v_{\rm particle}$, where $v_{\rm particle} = v_0(\xi_x Given Information Mass $m$…

Problem Statement Solve the oscillation/wave problem: Light of frequency $\nu_0$ is scattered by acoustic phonons (frequency $\nu_s$). Find the Brillouin frequency shifts for forward and backward scattering. An acoustic wave (phonon) with frequency $\Omega$ and wavevector $q$ scatters a photon $k_i \to k_s$. Energy and momentum conservation: $$\omega_s Given Information Mass $m$ and spring constant…

Problem Statement Solve the oscillation/wave problem: Two coherent point sources separated by $d = 2\lambda$ emit in phase. Find the number of maxima in the far field. Maxima occur when path difference $\Delta = d\sin\theta = n\lambda$, $n$ integer. With $d = 2\lambda$: $$\sin\theta = \frac{n}{2}, \quad n = 0,\pm1,\pm2$$ For $|\sin\theta| \leq 1$: $|n|…

Problem Statement Solve the oscillation/wave problem: Find the lowest resonant frequency of sound in a spherical cavity of radius $R$. The pressure modes in a sphere satisfy the Helmholtz equation $\nabla^2 p + k^2 p = 0$ with rigid boundary condition $\partial p/\partial r|_{r=R} = 0$. The purely radial (breathing) mode has $p \propto \sin(kr)/r$.…

Problem Statement Solve the oscillation/wave problem: Explain the whispering gallery effect and estimate the condition for sound to travel along the curved wall of a circular room of radius $R$. Sound emitted tangentially near a curved wall can undergo repeated total internal reflection, traveling around the perimeter. This requires: The sound must ski Given Information…

Problem Statement Solve the oscillation/wave problem: A Gaussian pulse $\xi(x,0) = a\exp(-x^2/2\sigma^2)$ travels in a non-dispersive medium. Find the shape at time $t$. In a non-dispersive medium, every Fourier component travels at the same speed $v$, so the pulse shape is preserved. The wave equation $\partial^2\xi/\partial t^2 = v^2\partial^2\xi/\pa Given Information Mass $m$ and spring…

Problem Statement Solve the oscillation/wave problem: In a cubic crystal with elastic constants $c_{11}$, $c_{12}$, $c_{44}$, find the speed of longitudinal waves along [100] and transverse waves along [100]. For propagation along the [100] direction ($\hat{x}$): Longitudinal (P-wave): polarization along $\hat{x}$: $$\rho v_L^2 = c_{11}$$ $$\boxed{v_L Given Information Mass $m$ and spring constant $k$ (or…

Problem Statement Solve the oscillation/wave problem: A steel rod of length $L = 1.0$ m ($E = 200$ GPa, $\rho = 7800$ kg/m³) is struck at one end. Find the fundamental frequency and first three harmonics. Wave speed in rod: $v = \sqrt{E/\rho} = \sqrt{200\times10^9/7800} \approx 5064$ m/s For a free–free rod (free ends =…

Problem Statement Solve the oscillation/wave problem: A circular membrane of radius $a$ under tension $T$ (linear density $\mu$) vibrates. Write the wave speed and the condition for the lowest resonant frequency. Wave speed: $v = \sqrt{T/\mu}$ (surface tension $T$ in N/m, $\mu$ in kg/m²). The modes are given by Bessel functions: $J_m(k_{mn}r/a)\cos(m\t Given Information Mass…

Problem Statement Solve the oscillation/wave problem: To couple ultrasound from a transducer into a water-filled chamber, a matching layer of thickness $\lambda/4$ is used. Find the optimal impedance of the matching layer. A $\lambda/4$ layer of impedance $Z_m$ placed between media of impedances $Z_1$ and $Z_2$ provides perfect matching when: $$Z_m = \ Given Information…