  When a mechanical wavemaker at one end of a water-wave tank oscillates with a
frequency, $\omega_0$, time series of downstream surface waves typically
include the dominant frequency (or first harmonic), $\omega_0$, along with the
second, $2\omega_0$; third, $3\omega_0$; and higher harmonics. This behavior is
common for the propagation of weakly nonlinear waves with a narrow band of
frequencies centered around the dominant frequency such as in the evolution of
ocean swell, pulse propagation in optical fibers, and Langmuir waves in
plasmas. Presented herein are measurements of the amplitudes of the first and
second harmonic bands from four surface water wave laboratory experiments.
  The Stokes expansion for small-amplitude surface water waves provides
predictions for the amplitudes of the second and higher harmonics given the
amplitude of the first harmonic. Similarly, the derivations of the NLS equation
and its generalizations (models for the evolution of weakly nonlinear,
narrow-banded waves) provide predictions for the second and third harmonic
bands given measurements of the first harmonic band. We test the accuracy of
these predictions by making two types of comparisons with the experimental
measurements. First, we consider the evolution of the second harmonic band
while neglecting all other harmonic bands. Second, we use explicit Stokes and
generalized NLS formulas to predict the evolution of the second harmonic band
using the first harmonic data as input. Comparisons of both types show
reasonable agreement, though predictions obtained from dissipative
generalizations of NLS consistently outperform the conservative ones. Finally,
we show that the predictions obtained from these two methods are qualitatively
different.
