Category: Part 4: Oscillations & Waves

Problem Statement Solve the oscillation/wave problem: Solve the oscillation/wave problem: Sound at 1000 Hz attenuates at $\alpha = 1.0\times10^{-3}$ dB/m in air. Find the intensity level at 1.0 km from a point source of sound power $W = 1.0$ W. At distance $r$, without absorption: $I_0 = W/(4\pi r^2) = 1.0/(4\pi\times10^6) = 7.96\times10^{-8}$ W/m² $$\…

Problem Statement Solve the oscillation/wave problem: Solve the oscillation/wave problem: Derive the wave equation for longitudinal pressure waves in a compressible fluid (bulk modulus $B$, density $\rho_0$) and find the wave speed. For small displacement $\xi(x,t)$ from equilibrium, the continuity equation gives density variation: $$\rho = \rho_0\left Given Information See problem statement for all given…

Problem Statement Solve the oscillation/wave problem: Solve the oscillation/wave problem: A plane wave of wavelength $\lambda$ passes through a slit of width $d$. Find the angles of the first diffraction minima. Using Huygens’ principle: divide the slit into pairs of elements separated by $d/2$. For the first minimum, these pairs cancel: $$\frac{d}{2}\ Given Information See…

Problem Statement Solve the oscillation/wave problem: Solve the oscillation/wave problem: A square wave of amplitude $A$ and period $T$ is expressed as a Fourier series. Write the first few terms and discuss the overtone content. A square wave (period $T$, amplitude $A$, zero mean): $$f(t) = \frac{4A}{\pi}\left[\sin(\omega t) + \frac{1}{3}\sin(3\omega Given Information See problem statement…

Problem Statement Solve the oscillation/wave problem: Solve the oscillation/wave problem: Define reverberation time $RT_{60}$ and state Sabine’s formula. How should a concert hall be designed? Reverberation time ($RT_{60}$): the time for the sound intensity level to decrease by 60 dB after the source stops. Sabine’s formula: $$RT_{60} = \frac{0.161 V}{ Given Information See problem statement…

Problem Statement Solve the oscillation/wave problem: Solve the oscillation/wave problem: An open pipe resonates at frequency $f_1$ (fundamental). When one end is closed, it resonates at $f_1’$. Show $f_1’/f_1 = 1/2$ for an ideal pipe, and explain the role of end corrections. Ideal open pipe (both ends free): $f_1 = v/(2L)$ Ideal closed pipe (one…

Problem Statement Solve the oscillation/wave problem: Solve the oscillation/wave problem: Define acoustic impedance $Z = \rho v$ and use it to find the intensity in terms of pressure amplitude $p_0$. The specific acoustic impedance (resistance) of a medium: $$Z = \rho v \quad \text{(units: Pa·s/m = Rayl)}$$ Values: Air: $Z_{\rm air} \approx 415$ Rayl; Given…

Problem Statement Solve the oscillation/wave problem: Solve the oscillation/wave problem: A Helmholtz resonator has volume $V$, neck length $l$, and neck cross-section $S$. Find the resonant frequency. The air in the neck (mass $m = \rho_0 Sl$) oscillates like a piston. When displaced by $x$, the volume change $\Delta V = Sx$ causes a pressure…

Problem Statement Solve the oscillation/wave problem: Solve the oscillation/wave problem: A guitar string ($L = 0.65$ m, $\mu = 5.0$ g/m) is tuned to $A_4 = 440$ Hz. Find the tension required. How does pressing a fret at $L/9$ from the nut affect the pitch? Fundamental frequency: $f_1 = v/(2L) = \sqrt{T/\mu}/(2L)$ $$T = (2Lf_1)^2\mu…

Problem Statement Solve the oscillation/wave problem: Solve the oscillation/wave problem: Source at rest emits frequency $f_0$. The medium (wind) moves at speed $w$ toward the observer. Observer at rest. Find the observed frequency. The sound travels at speed $v$ relative to the medium. The medium moves at speed $w$ toward the observer. The effective s…