non diffracting waves I came upon a proposed solution for a surface wave which was claimed to be a non diffracting wave. The wave at $z=0$ and $x=0$ (which is the propagating direction) is
 $E_z(0,y) = A\cos(ky)e^{-y^2/w^2}$.
After propagating along $x$ axis, the value of $E_z$ is shown to be
$E_z(x,y)= A'f(x)e^{-y^2/w^2}\cos(ky)$
The authors state that the field maintains the same transverse profile regardless of $x$. But the expression has got $f(x)$ which depends on $x$ so how can it maintain the same profile as we propagate along $x$ direction. what is exactly meant by transverse intensity profile?
 A: Maybe what they mean by same profile is the general shape of the function f(x). For example in a gaussian beam waist, the transverse profile is the gaussian function as the one you have mentioned, but the spread of the beam increases along the propagation axis . 
To sum up, the shape of the transverse profile is always a gaussian, but with a spread that changes along the propagation 
A: "Non diffracting waves" are indeed possible, but there is always a practical limitation.
The simplest example is the plane wave, which is an eigenmode of free space. It propagates by simply taking on the phase $\exp(i\,\vec{k}\cdot\vec{r})$. The practical limitation is of course that it has infinitely wide extent. Practically, diffraction is always present owing to the edge effects: you need a spread of Fourier components to make the wave have edges and thus have a finite extent.
OK, so that was probably a fairly uninteresting example. Another, less trivial, one is the Bessel Beam. This is a superposition of plane waves whose wavevectors would fill a cone if we collocated all their tails. A good approximation of the Bessel beam is made by passing a collimated wave through an axicon lens.
The practical Bessel beam is only nondiffractive over a small, but definitely nonzero region near the axicon's focus. Again, the truly nondiffracting beam would have infinite transverse extent, but the practical beam can also be surprisingly weakly diffractive. 
