Hi Seongjun,
The shape of the initial pressure distribution will indeed determine which frequencies appear in the resulting ultrasonic pulse. It can be modelled accurately with k-Wave for frequencies below the cut-off. If making a comparison to measurements you need to be sure that your grid can support the maximum frequency you are measuring.
If you try to model an initial pressure distribution with sharp edges you may see what look like oscillations in the output. This is related to the Gibbs phenomenon, explained below, and well worth being aware of when using spectral methods.
With any numerical model it is necessary to replace the continuous initial pressure distribution that you are interested in with a discretized version; in other words, one defined only on the grid points. The numerical model will use some assumptions about what is going on in between the grid points: if it is a simple finite difference code then it might assume that the values at the grid points are linearly interpolated, for example. k-Wave is a spectral method, so it interpolates between the grid points using a Fourier series fitted to all the values across the whole grid. If there is a sharp step up from one grid point to the next, then the interpolated function will oscillate either side of that step (between the grid points) in order to achieve that rapid change. The oscillations occur because the model is necessarily constrained to use a finite number of Fourier components and a very large number are required to model a step function. (The Fourier transform of a step function decays very slowly in the Fourier domain.) It is this interpolated function (sometimes called the 'band limited interpolant') that is propagated across the grid. This makes it looks as though spurious oscillations have appeared, but really they were in the Fourier version of your initial pressure distribution all along. They are not due to an instability. If you want to remove this effect (as it is rather unsightly) smoothing the initial pressure distribution will do it.
This post is relevant.
Hope that helps,
Ben