Category: Part 4: Oscillations & Waves

Problem Statement Solve the oscillation/wave problem: Ultrasonic cleaners operate at $f = 40$ kHz with intensity $I = 5$ W/cm². Estimate whether cavitation occurs in water at room temperature. Pressure amplitude of the ultrasound: $$p_0 = \sqrt{2\rho v I} = \sqrt{2\times10^3\times1500\times5\times10^4} = \sqrt{1.5\times10^{11}} \approx 3.87\times10^5 \ Given Information Mass $m$ and spring constant $k$ (or…

Problem Statement Solve the oscillation/wave problem: A crack in a metal emits acoustic emission (AE) pulses. Two sensors separated by $d = 0.50$ m detect a pulse with time delay $\Delta t = 80$ $\mu$s. Find the crack location. Sound speed in steel: $v \approx 5000$ m/s. Path difference: $$\Delta x = v\Delta t =…

Problem Statement Solve the oscillation/wave problem: Express the following musical intervals in terms of frequency ratios and cents: octave, perfect fifth, major third. A cent is $1/1200$ of an octave, so $n$ cents correspond to frequency ratio $2^{n/1200}$. Interval Just ratio Equal temperament Cents (ET) Octave 2:1 $2^{12/12} = 2$ 1200 Perfect fifth Given Information…

Problem Statement Solve the oscillation/wave problem: An earthquake P-wave travels at $v_P = 8.0$ km/s and arrives 15 s before the S-wave ($v_S = 4.5$ km/s). How far is the earthquake epicenter? Let $d$ = distance to epicenter. Travel time difference: $$\Delta t = \frac{d}{v_S} – \frac{d}{v_P} = d\left(\frac{1}{4.5} – \frac{1}{8.0}\right) = 15 \text{ s…

Problem Statement Solve the oscillation/wave problem: An open organ pipe resonates at 256 Hz (fundamental). It is partially closed by pushing a stopper to reduce effective length to 3/4 of original. Find the new fundamental and the lowest note that it can still produce. Original open pipe of length $L$: $f_1 = v/(2L) = 256$…

Problem Statement Solve the oscillation/wave problem: In an intense sound field, there is a DC (steady) flow of the medium called acoustic streaming. Explain its origin and estimate the streaming velocity. Acoustic streaming (Eckart streaming) arises because sound waves lose momentum to the medium through viscous absorption. The absorbed acoustic momen Given Information Mass $m$…

Problem Statement Solve the oscillation/wave problem: In Kundt’s tube experiment, cork dust forms nodes at every 15 cm in air and every 7.0 cm in an unknown gas. Ratio of wave speeds and gas identification? In Kundt’s tube, standing waves form and cork dust collects at nodes. The node spacing = $\lambda/2$. $$\lambda_{\rm air}/2 =…

Problem Statement Solve the oscillation/wave problem: A standing ultrasound wave in air (frequency $f = 40$ kHz, pressure amplitude $p_0 = 1000$ Pa) can levitate small objects. Find the acoustic radiation pressure. The acoustic radiation pressure (Langevin radiation pressure) on a perfectly reflecting surface: $$P_{\rm rad} = \frac{\langle E\rangle}{V} Given Information Mass $m$ and spring…

Problem Statement Solve the oscillation/wave problem: A supersonic aircraft flies at altitude $h = 10$ km at Mach 2. Find the duration of the sonic boom heard by an observer on the ground. The Mach cone half-angle: $\sin\theta = 1/Ma = 0.5$, $\theta = 30°$. The boom is heard when the Mach cone sweeps past…

Problem Statement Solve the oscillation/wave problem: A sonar transmits 50 kHz pulses in water ($v = 1500$ m/s) at a submarine moving at $u = 20$ m/s toward the sonar. Find the frequency of the echo received back at the sonar. Step 1: Frequency received by submarine (moving source at rest, observer moving toward): $$f_1…