doop4,
I was afraid I was answering the wrong question, looks like I was right. You are talking about "acoustic coupling" and "impedance matching" for ultrasound drivers. (Hint: plug those terms in varying groups into Google or other search engines and see what you come up with).
Still, the basic idea here is the same (I think). Just to make sure, you are being told that the transfer material (aka impedance matching layers) are to be sized such that a down-and-back time for an acoustic wave traversing the material is equal to 1/4 of the piezo drive wavelength. You have two surfaces, the drive surface (piezo face) and the driven surface (pipe). You want to transfer the energy from driver to driven at best efficiency.
Think about what's happening in a stepwise fashion: 1. the piezo head moves and pushes on the transfer material, generating a pressure wave. 2.) the wave travels until it hits the pipe wall; some of the wave energy is deposited into the pipe, but some of the energy is reflected back through the material. 3.)a.) If the reflected wave coming back through your material hits the piezo head when it is still in the "compression stroke" (to borrow an analogy from IC engines)(i.e. the transfer material response is much less than 1/4 wave), then the wave will reflect again, and you have a lot of "ringing" or spurious echoes going on. Similarly, if the wave hits the piezo head at bottom dead center. But, if the wave hits the piezo head at 1/4 wavelength later, it hits at a point when the head is moving backwards at its maximum velocity, and thus the echo (re-reflection from the head) is minimized; also, the energy in the reflected wave is tranferred at best efficiency back into the piezo head.
It's like trying to pump a swing - you can kick your legs, wiggle or whatver you want at either end of the swing, and not much happens. But if you shift your weight near the center of the swing, when the velocity is at a maximum, you can keep the swing moving.
Another really good experiment is to go grab a "slinky" spring toy, and tie down or have a friend hold one end. You drive the other end back and forth along the axis of the spring. Watch what happens as you vary the timing (phase angle) of your "compression stroke". Hint: start really slowly relative to the frequency of the spring's wave motion.
As far as acoustic wave propagation thru solid materials, you may want to look at FEA solvers. Although, simple 1-D analysis should take you a long ways. Get a good freshman physics textbook, if for no other reason than to refresh your memory and load up your head with the right terms and buzzwords; then go do a Google search for both software and recent technical papers. I'd sit here and rattle off a list of codes (both acoustics and FEA), but since it's been >10 years since I did anything along those lines, you'd probably have a hell of a time finding anything I could mention.
Could anyone recommend a good book on this? I have tried the acoustic books I use and have had no luck 
