Capillary wave

Zhu, Miao, Xu and Cao's 2007 experiment drove a triangular metal frame electromagnetically with a sinusoidal generator. In a roughly $3\,\mathrm{cm}$-deep, temperature-regulated pool, it produced nearly plane <capillary waves>. A $632.8\,\mathrm{nm}$ <laser> illuminated the surface obliquely; the moving ripples acted as a reflection <diffraction grating>. One split beam monitored laser stability, while a <CCD> recorded <diffraction orders> on a screen $6.76\,\mathrm m$ away. An air-supported optical table reduced ambient vibration. https://www.eg.bucknell.edu/~phys310/skills/papers/paperX_full.pdf[Zhu et al., Measurement of the dispersion relation of capillary waves by laser diffraction, American Journal of Physics 75 (2007), §II.A, p. 896; DOI 10.1119/1.2750379]{ref}

\Image[/capillary-waves-diffraction.png]
{title=Drive a <capillary wave> at known <frequency> and measure its <wavelength> by <laser> <diffraction>}
{description=An original schematic of excitation and optical detection, with ray angles and surface displacement exaggerated for readability. The right panel calculates the deep-water <dispersion relation> for the stipulated parameters in <capillary length>; the $100$–$290\,\mathrm{Hz}$ band identifies a drive range to compare with the documented experiment. No diffraction intensities, measured points or fitted <surface tensions> are plotted.}

The <frequency> sweep was $100$–$290\,\mathrm{Hz}$ in $10\,\mathrm{Hz}$ steps, using distilled <water> and kerosene at $20\,{}^\circ\mathrm C$. For each setting they averaged at least three measurements of first-order spot separation. Known optical geometry and the <grating equation> yielded $k=2\pi/\lambda$; generator readings yielded $\omega=2\pi f$. Agreement with the predicted log–log slope $3/2$ tests the capillary law, and its intercept gives <surface tension> if <mass density> is known. The authors identify surface contamination as a major error source; noncontact optical detection reduces additional disturbance. https://www.eg.bucknell.edu/~phys310/skills/papers/paperX_full.pdf[Zhu et al., §§II.B–III, equations (6)–(8), pp. 897–898]{ref}

The optical measurement can be understood from the <grating equation>. Let $\lambda_L$ be the <laser> <wavelength>, $\alpha$ the incident grazing <angle> above the mean surface, and $\beta_m$ the outgoing grazing <angle> of <diffraction order> $m$, with the sign of $m$ chosen so that
$$
\lambda\left(\cos\alpha-\cos\beta_m\right)=m\lambda_L.
$$
Thus calibrated spot positions determine outgoing <angles>, which determine the ripple <wavelength> $\lambda$, independently of the mechanical <dispersion relation>. This optical geometry treats the surface as a reflection <diffraction grating> at the <laser> <wavelength>. https://web.mit.edu/8.13/www/JLExperiments/JLExp001.pdf[MIT Department of Physics, Introductory Optics, §III.1, equation (4), p. 3: grazing-angle reflection grating]{ref}

The inferred <surface tension> in the full deep-water model is
$$
\gamma=\rho\frac{\omega^2-gk}{k^3}.
$$
This formula makes useful experimental controls explicit. Measure <mass density> at the sample's <temperature>; check depth with $kh$; keep the driven <wave amplitude> small enough that $ka\ll1$; and distinguish $f$ from <angular frequency> $\omega$. For the stipulated parameters, the $100$–$290\,\mathrm{Hz}$ window corresponds to calculated <wavelengths> about $3.62$–$1.76\,\mathrm{mm}$ and $kh\simeq52$–$107$ at $h=30\,\mathrm{mm}$. Depth corrections are negligible here, but the <gravity> contribution is about $4.5\%$–$1.1\%$ of the capillary contribution. These are ideal predictions, not digitized results from the experiment.

Omitting <gravity> would infer $\gamma_{\mathrm{cap}}=\rho\omega^2/k^3=\gamma+\rho g/k^2$, biasing <surface tension> upward. In the capillary limit, the differential estimate $d\gamma/\gamma\simeq d\rho/\rho+2\,d\omega/\omega-3\,dk/k$ also shows why careful spot-spacing calibration matters. A clean surface and weak <viscosity> are modeling assumptions: appreciable <damping>, <surfactant> stresses or large <wave amplitudes> call for additional physics. The experiment tests the short-wave regime; its drive window does not establish the crossover or the long-wave <surface gravity wave> limit.


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 Incoming links: Laser-diffraction measurement of capillary waves