Why are the filamentous body forms of slime molds (myxomycetes and acrasiomycetes), water molds (oomycetes) and the hyphae of fungi considered a case of convergent evolution?

Convergent evolution is the independent evolution of similar features in species of different lineages.

It creates analogous structures with similar form or function but these structures were not present in the last common ancestor of those groups.  

The filamentous body forms of slime molds, water molds and the hyphae of fungi  is considered a case of convergent evolution because the filaments shape is an adaptation for a nutritional mode as a decomposer.

1. the energy lost to disorder metabolism     -entropy

2. related to or possessing motion potential     -kinetic

3. all of the chemical reactions in an organism required for maintenance of the processes of life entropy     -metabolism

4. possible; ability to go into action or perform work kinetic     -potential

5. the study of energy and its transformations thermodynamics     -thermodynamics

when the displaced particles are perpendicular to the direction the wave travels, is called a transverse wave, whereas in longitudinal wave the displace practices are parallel to the direction of the wave.

in the given image:

first image shows - longitudinal wave

second image shows - transverse wave

third image shows - longitudinal wave

fourth image shows - longitudinal wave