Category: Part 4: Oscillations & Waves

Problem Statement Solve the oscillation/wave problem: Solve the oscillation/wave problem: A rectangular cavity of dimensions $a\times b\times c$ resonates in acoustic modes. Find the resonant frequencies. Pressure must satisfy the wave equation with $\nabla p = 0$ at rigid walls (velocity normal to wall = 0). The modes are: $$p_{n_1 n_2 n_3} = A\cos\fr Given…

Problem Statement Solve the oscillation/wave problem: Solve the oscillation/wave problem: An aircraft moves at speed $u > v$ (sound speed). Find the half-angle $\theta$ of the Mach cone and the Mach number $Ma$. The aircraft moves distance $ut$ in time $t$, while the sound emitted from the origin spreads to radius $vt$. The Mach cone…

Problem Statement Solve the oscillation/wave problem: Solve the oscillation/wave problem: Find the resonant frequencies of (a) an open pipe (open at both ends) and (b) a closed pipe (one end open, one closed), each of length $L$. (a) Open pipe: Pressure nodes at both ends (displacement antinodes). Same as fixed-fixed string: $$f_n = \frac{nv}{2L}, \qua…

Problem Statement Solve the oscillation/wave problem: Solve the oscillation/wave problem: Ultrasound at frequency $f = 1.0$ MHz has intensity $I = 10$ W/cm² in water ($\rho = 1000$ kg/m³, $v = 1500$ m/s). Find the displacement amplitude and pressure amplitude. Intensity: $I = \frac{1}{2}\rho v\omega^2 a^2$ $$a = \sqrt{\frac{2I}{\rho v\omega^2}} = \sqrt Given Information See…

Problem Statement Solve the oscillation/wave problem: Solve the oscillation/wave problem: Derive the Doppler formula for the frequency heard when the source moves at speed $v_s$ and the observer moves at $v_o$, with sound speed $v$. Let $f_0$ = source frequency, $v$ = sound speed (in medium at rest). The effective wavelength between source and observer…

Problem Statement Solve the oscillation/wave problem: Solve the oscillation/wave problem: A string of length $L$ fixed at both ends vibrates in standing modes. Find the allowed frequencies (harmonics). Boundary condition: $y = 0$ at $x = 0$ and $x = L$. For $y = A\sin(kx)\cos(\omega t)$ (automatically zero at $x=0$): $$\sin(kL) = 0 \Rightarrow kL…

Problem Statement Solve the oscillation/wave problem: Solve the oscillation/wave problem: A wave on a string (impedance $Z_1 = \mu_1 v_1$) hits a boundary with a string of impedance $Z_2$. Find the reflection and transmission amplitude coefficients. Boundary conditions: continuity of displacement and continuity of force (stress). Let incident amplitude Given Information See problem statement for…

Problem Statement Solve the oscillation/wave problem: Solve the oscillation/wave problem: Show that the superposition of two equal waves traveling in opposite directions produces a standing wave. Find the positions of nodes and antinodes. Forward wave: $y_1 = a\cos(\omega t – kx)$ Backward wave: $y_2 = a\cos(\omega t + kx)$ Sum using product-to-sum for Given Information…

Problem Statement Solve the oscillation/wave problem: Solve the oscillation/wave problem: Show that in a plane wave the time-averaged kinetic and potential energy densities are equal, and find the total energy density. For wave $\xi = a\cos(\omega t – kx)$: Kinetic energy density: $u_k = \frac{1}{2}\rho(\partial\xi/\partial t)^2 = \frac{1}{2}\rho a^2\o Given Information See problem statement for…

Problem Statement Solve the oscillation/wave problem: Solve the oscillation/wave problem: Find the intensity (power per unit area) of a plane harmonic wave in an elastic medium. The intensity is the energy transported per unit time per unit cross-sectional area: $$I = \langle p\cdot v_{\rm particle}\rangle$$ where $p = -E\partial\xi/\partial x$ is the Given Information See…