Discussion
The speed of propagation, \(c\), of a capillary wave depends on the density of the fluid, \(\rho\), the wavelength of the wave, \(\lambda\), and the surface tension, \(\sigma\). If the density and wavelength remain constant, halving the surface tension would lead to a new velocity, \(c'\), given by
Answer: C
The new speed is \(c' = \frac{c}{\sqrt{2}}\).
By dimensional analysis, the only combination of \(\rho\) (kg/m3), \(\lambda\) (m) and \(\sigma\) (N/m = kg/s2) with units of speed is
\[ c \propto \sqrt{\frac{\sigma}{\rho\lambda}} \]
so \(c\) varies as \(\sqrt{\sigma}\). Halving \(\sigma\) with \(\rho\) and \(\lambda\) fixed multiplies c by \(1/\sqrt{2}\). \(\sqrt{2}c\) and \(2c\) would mean the speed rises when the restoring force weakens, and \(c' = c\) ignores the dependence on surface tension altogether.
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