Vibroacoustical energy flow through straight pipes

At lower relative (i.e., non-dimensional) frequencies, four propagating waves exist in fluid-filled pipes. Each of these waves carries energy in the pipe wall, while three waves carry energy in the fluid as well. The otherwise fairly complex dispersion laws for waves in pipes simplify in the frequency region considered to simple rod- and beam-type laws. It is shown that these laws can be determined by approximate formulae fairly accurately, the accuracy decreasing with increase in frequency. Due to fluid-wall coupling, expressed again by simplifications, the energy flow in both the wall and the fluid can be evaluated in principle from knowledge of surface vibrations only. The portions of the flow in the solid and the fluid fluctuate along the pipe axis, and consequently spatial averaging has to be done in order to obtain useful results. In this way, the pipe becomes a homogeneous one-dimensional waveguide, suitable for measurements of energy flow by detection of surface vibrations only. Specific transducer patterns for this purpose are described. At higher frequencies however, where additional propagating waves take place, simplifications are no longer possible. The exact expression for the unit-length energy flow can be then employed in conjunction with averaging around the circumference to evaluate flow in the wall at a particular axial position.