# Acoustics: Construction of the pressure field with transmitted and reflected plane waves

So I have been reading one of the papers about wave manipulation, which takes advantage of the phase shifted reflection by tailoring the design of the system. The design has two domains: air and foam (which is rigidly backed). For the sound pressure expression in the foam domain, as shown in figure, the paper gives:

$$P_{II}=A_{i2}e^{ik_2(y-y_2)}+A_{r2}e^{-ik_2(y-y_2)}$$

What I don't understand about this equation is the $$y-y_2$$ part. It seems to me that this operation is equivalent to shifting the x-axis to $$y=y_2$$, which would be the new origin. But since it's the phase difference between $$p_{i1}$$ and $$p_{r1}$$ is at interest here and I think the part of the phase information will be lost if one do this operation, more specifically, the phase difference between the old and the new coordinate.

In my opinion, in order to keep all the phase information, the pressure field in domain II would be given by:

$$P_{II}=A_{i2}e^{ik_2(y-y_2)+ik_1y_2}+A_{r2}e^{-ik_2(y-y_2)-ik_1y_2}$$

But if one use this definition, the reflection coefficient will have a different form, which will lead to different results. And curious enough, the experiment results in the paper match really well with his analytical solution. So I think maybe there might be some faults in my chain of thought.

Can anyone help to clarify this?

By the way, the paper that I referenced is: Acoustic porous metasurface for excellent sound absorption based on wave manipulation. This equation appears at Appendix B.

• It might be a good idea to provide a link to the referenced paper. Commented Oct 11, 2018 at 14:56
• Thanks for the advice. I have added the reference to that paper. Commented Oct 11, 2018 at 15:01

For a longer one, it's the phase information of a plane wave can either be store in amplitude or $$e^{jkx}$$ depending on the form one devised to describe the field. No matter which form is used, the final result will be the same because of the same boundary condition at each zone. This means, although the calculated amplitude and $$e^{jkx}$$ will be different respectively if different forms are used, but the combined result which contains the whole phase information will be the same. And if the field is defined as: $$A_{i2}e^{ik_2(y-y_2)}+A_{r2}e^{-ik_2(y-y_2)}$$ instead of the other form, the missing phase information will be accounted for by the value of $$A_{i2}$$ and $$A_{r2}$$ respectively. I have validated it in MATLAB the result supports my conclusion.