I know isotropic means equal properties in all directions , but the term "transverse" is not making it easy for me to understand.
You are correct. Isotropy means that the property at some specified point (X,Y,Z) is the same, whether the measurement is made along any direction.
If the property varies depending on which direction you make the measurement in, then that property is said to be "anisotropic". For example, the permeability of a material (i.e. its ability to conduct fluids) may be anisotropic if fluid can move preferentially in one direction and it would be difficult for it to move in a different direction (like the tubes below). This means, the permeability of this material at a given point (X,Y,Z) depends on which direction you make the measurement, i.e. high permeability parallel to the tubes, low permeability otherwise.
Transverse isotropy (also called hexagonal symmetry) is a special kind of anisotropy in which there exists a special "axis of symmetry" such that the property you are measuring appears to be the same as you make measurements perpendicular to this axis. Basically, rotating around this axis of symmetry, the property looks pretty much the same, i.e. it is isotropic in this transverse direction. However, once you go off axis, then the measured property will change.
In geology (especially in sedimentary geology), transverse isotropy is usually a good approximation to many phenomena (elasticity, permeability, heat conductivity, etc.) due to the fact rocks are deposited mostly horizontally. This means that things will look very similar laterally (in both N-S and E-W directions) but very different vertically.
Finally, transverse isotropy is just an approximation to the much more complex nature of real rocks. If you were to add multiple sets of fractures to a rock on top of its layered nature, you will need to use a "less symmetric" model to describe it: orthorhombic, monoclinic, or triclinic. But for most practical purposes, transverse isotropy does a pretty good job.
Hope this helps!
As I understand it, "tranverse isotropy" has to do with seismic properties of a rock, a rock unit, or even a specific mineral. It's an important term because even though a rock may appear "isotropic" visually, it may have an internal fabric (invisible to unassisted eyes) that treats seismic waves differently depending the axes through which they are propagated.
Internal fabrics can be the result of crystalline alignment from high temperature shearing and strongly aligned microcracking, to cite just two examples. There are fierce debates regarding the seismic properties of minerals in the mantle and there are few specialized labs that can seismically test these unstable, high-pressure minerals.
For example, a mylenite can be a mineralogically isotropic rock type but has an distinctly aligned mineral fabric that may do strange things to sound waves when they pass through: reflection, refraction, and polarization come to mind. Without knowing of the existence of this mylenite, in this example, a seismologist might mistake this highly reflective mylenitized unit for something very different.
This is not just a minor detail of a rock property but an important component behind understanding seismic transmission and reflection for exploration geologists as well as mantle geophysicists.