  In this paper an approach for decreasing the computational effort required
for the spectral simulations of the water waves is introduced. Signals with
majority of the components zero, are known as the sparse signals. Like majority
of the signals in the nature it can be realized that water waves are sparse
either in time or in the frequency domain. Using the sparsity property of the
water waves in the time or in the frequency domain, the compressive sampling
algorithm can be used as a tool for improving the performance of the spectral
simulation of the water waves. The methodology offered in this paper depends on
the idea of using a smaller number of spectral components compared to the
classical spectral method with a high number of components. After performing
the time integration with a smaller number of spectral components and using the
compressive sampling technique, it is shown that the water wave field can be
reconstructed with a significantly better efficiency compared to the classical
spectral method with a high number of spectral components, especially for long
time evolutions.
  For the sparse water wave model in the time domain the well-known solitary
wave solutions of the Korteweg-deVries (KdV) equation is considered. For the
sparse water wave model in the frequency domain the well-known Airy (linear)
ocean waves with Jonswap spectrum is considered. Utilizing a spectral method,
it is shown that by using a smaller number of spectral components compared to
the classical spectral method with a high number of components, it is possible
to simulate the sparse water waves with negligible error in accuracy and a
great efficiency especially for large time evolutions.
