Waves

Key formulas and results

• $v = f\lambda$
• Speed on a string: $v = \sqrt{T/\mu}$
• Speed of sound (Laplace): $v = \sqrt{\gamma P/\rho}$
• Open pipe: all harmonics, $f_{n} = \frac{nv}{2L}$; closed pipe: odd harmonics, $f = \frac{(2n - 1)v}{4L}$
• Beat frequency $= |f_{1} - f_{2}|$
• Doppler: $f' = f\,\dfrac{v \pm v_{0}}{v \mp v_{s}}$
• Resonance tube (first two resonating lengths $l_{1}$ and $l_{2}$): $v = 2f(l_{2} - l_{1})$
• Progressive wave: $y = A\sin(\omega t - kx + \phi)$, with $k = \frac{2\pi}{\lambda}$ and phase difference $\Delta\phi = \frac{2\pi}{\lambda}\Delta x$
• Particle velocity in a wave: $v_{\text{particle}} = -v \times \left(\text{slope } \frac{\partial y}{\partial x}\right)$
• Energy density $u = \tfrac{1}{2}\rho A^{2}\omega^{2}$; intensity $I = \tfrac{1}{2}\rho A^{2}\omega^{2}v$, so $I \propto A^{2}$
• Intensity from a point source: $I = \frac{P}{4\pi r^{2}}$, so $I \propto \frac{1}{r^{2}}$
• Superposition: resultant amplitude $A = \sqrt{A_{1}^{2} + A_{2}^{2} + 2A_{1}A_{2}\cos\phi}$; $\frac{I_{\max}}{I_{\min}} = \frac{(\sqrt{I_{1}} + \sqrt{I_{2}})^{2}}{(\sqrt{I_{1}} - \sqrt{I_{2}})^{2}}$
• Sound intensity level: $\beta = 10\log_{10}(I/I_{0})\ \text{dB}$, and $\beta_{2} - \beta_{1} = 10\log_{10}(I_{2}/I_{1})$
• Resonance tube: $\lambda = 2(l_{2} - l_{1})$, $v = 2f(l_{2} - l_{1})$, end correction $e = \frac{l_{2} - 3l_{1}}{2} \approx 0.3d$
• Speed of sound in a gas: $v = \sqrt{\gamma RT/M}$