Wave effect on bottom text
Create your text layer. 
Create a 100% black solid layer
Create a 100% white solid layer, make sure it is above the black layer.
Precomp the two solid layers (cmd/control + shift + c) and jump into the comp.
Add a 'venetian blinds' effect to the white layer and delete the keyframes that automatically accompany it. 
Set the 'Direction' to -90deg, 'transition completion' to 50%, width to 100 and the feather to 50 (or half width). 
Set a keyframe to position of the white layer. Jump forward two seconds (or however long you want the loop to last) and add 100px to the y position. (Note: the difference in y pos needs to match the width of the venetian blinds effect. 100 = 100). 
Go to the previous frame by hitting the 'page up' button on your keyboard. 
Hit 'tab' then 'left arrow' then 'enter'. To go to the parent comp. 
Hit 'n' on the keyboard to set the animation duration. Duration should always be one frame less that the keyframes of the position on the white layer of the nested comp. 
Turn off the visibility of the nested comp and add a 'displacement map' effect to the text layer.
Set the 'Max Horizontal Displacement' to 50 (or your preference) and the 'Max Vertical Displacement' to 0. 
Set the 'Displacement Map' to the nested comp layer. 


Mine looks like this: 


You should be able to modify it to your liking. Good luck!

EDIT: I've just reread your post and realised that you wanted the top to remain static and have the bottom wave. 

In order to accomplish that, you'll want to add a couple of rectangular shape layers that will sit directly over your text layers (make them a fair bit wider than your text, but the same height). Add a 'gradient ramp' effect and make is so that it runs from black at the top of the layer to white at the bottom. Set the shape layer's blend mode to multiply. Then cut the layers out and paste them into the nested map layer. Your nested map layer should now look something like this: 

With text for reference: 


Final map, text turned off: 


Now the top of the text will remain static and the bottom will progressively wave more.