1. Reference problem

1.1. Geometry

The pipe is a hollow cylinder with a circular cross section, filled with fluid.

Characteristics of the pipe:

length:	\(L=\mathrm{1,0}m\)
outside diameter:	\(D=\mathrm{0,1}m\)
thickness:	\(\mathrm{ep}=\mathrm{0,01}m\)

For modeling C, the section is modified, which becomes variable: the diameter varies from 0.1 to 0.11 m while the thickness varies from 0.011 to 0.01 m.

1.2. Material properties

The physical characteristics of the material constituting the tube are as follows:

Young’s modulus: \(E=\mathrm{1,0}·{10}^{10}\mathrm{Pa}\)

Poisson’s ratio: \(\nu =\mathrm{0,3}\)

density: \({\rho }_{s}=\mathrm{1,0}·{10}^{4}\mathrm{kg}/{m}^{3}\)

Longitudinal wave speed: \({c}_{s}=\sqrt{\frac{E}{{\rho }_{s}}}=\mathrm{1,0}·{10}^{3}m/s\)

The physical characteristics of the fluid material in the tube are as follows:

density: \({\rho }_{f}=\mathrm{1,0}·{10}^{3}\mathrm{kg}/{m}^{3}\)

speed of sound: \({c}_{f}=\mathrm{1,0}·{10}^{3}m/s\)

1.3. Boundary conditions and loading

* Movement only along the :math:`x` axis.


* Embedding the pipe at the :math:`A` end.


* Free tubing at the :math:`B` end.


* For the fluid reservoir condition at the :math:`A` end.