Category: Part 4: Oscillations & Waves

Problem Statement Solve the oscillation/wave problem: The human vocal tract (modeled as a tube $L = 17.5$ cm, closed at glottis, open at lips) has resonances (formants). Find the first three formant frequencies for neutral vowel /ə/. The vocal tract as a closed–open tube: $f_n = (2n-1)v/(4L)$, $n = 1, 2, 3, \ldots$ $$f_1 =…

Problem Statement Solve the optics problem: Light of wavelength $\lambda_0 = 632$ nm passes through an acousto-optic modulator with ultrasound at $f_s = 80$ MHz. Find the Bragg angle and the frequency of deflected light. In water ($v_s \approx 1500$ m/s), the acoustic wavelength: $$\Lambda = v_s/f_s = 1500/(80\times10^6) = 18.75 \ \mu\text{m} Given Information…

Problem Statement Solve the oscillation/wave problem: Find the pressure amplitude required to initiate cavitation in water at $T = 20°C$ ($p_{\rm vap} = 2.3$ kPa, $p_{\rm atm} = 101$ kPa). Cavitation occurs when the local pressure during the rarefaction half-cycle drops below the vapor pressure of water. The acoustic pressure must lower the total press…

Problem Statement Solve the oscillation/wave problem: Love waves are shear horizontal (SH) surface waves that exist only when a soft layer overlies a stiffer half-space. Why can’t SH waves exist on a homogeneous half-space? For SH polarization (particle motion parallel to surface, perpendicular to propagation), the governing equation in a homogeneous h Given Information Mass…

Problem Statement Solve the oscillation/wave problem: A scanning acoustic microscope uses $f = 1.0$ GHz ultrasound in water. Find the wavelength and diffraction-limited resolution. $$\lambda = \frac{v}{f} = \frac{1500}{10^9} = 1.5\times10^{-6} \text{ m} = 1.5 \ \mu\text{m}$$ Rayleigh diffraction limit: $d_{\min} \approx 0.61\lambda/NA$ where $NA \appro Given Information Mass $m$ and spring constant $k$ (or…

Problem Statement Solve the oscillation/wave problem: Rayleigh waves travel along the surface of an elastic half-space. The wave speed is approximately $v_R \approx 0.92 v_S$ (for Poisson’s ratio $\nu = 0.25$). What are the particle displacement components? Rayleigh waves are polarized in the sagittal plane (containing the wavevector and surface normal Given Information Mass $m$…

Problem Statement Solve the oscillation/wave problem: Estimate the mean free path of acoustic phonons at room temperature in a crystal where the Debye temperature is $\theta_D = 300$ K and the sound speed is $v_s = 3000$ m/s. At $T = \theta_D = 300$ K (room temperature), all phonon modes are excited. The thermal conductivity:…

Problem Statement Solve the oscillation/wave problem: A piezoelectric crystal resonates at $f = 5.0$ MHz. Find the crystal thickness ($E = 80$ GPa, $\rho = 2700$ kg/m³, assuming half-wave resonance). Speed of longitudinal waves in the crystal: $$v = \sqrt{\frac{E}{\rho}} = \sqrt{\frac{80\times10^9}{2700}} = \sqrt{2.96\times10^7} \approx 5443 \text{ m/s Given Information Mass $m$ and spring constant…

Problem Statement Solve the oscillation/wave problem: A driven oscillator in a nonlinear medium has displacement $x = a_1\cos(\omega t) + a_2\cos(2\omega t)$. Under what condition is the second harmonic $a_2$ significant? For a nonlinear spring: $F = -kx – \epsilon x^2$, the $x^2$ term drives the second harmonic when $a_1$ is large or $\epsilon$ is…

Problem Statement Solve the oscillation/wave problem: A wave packet at time $t=0$ has a Gaussian spectral distribution: $A(k) = a\exp[-(k-k_0)^2/2\sigma_k^2]$. Find the packet shape and width at $t=0$ and at later time $t$. At $t=0$: $$\xi(x,0) = \int A(k)e^{ikx}dk = a\sigma_k\sqrt{2\pi}\exp\left(-\frac{x^2\sigma_k^2}{2}\right)e^{ik_0 x}$$ Gaussian in Given Information Mass $m$ and spring constant $k$ (or equivalent), or…