One-Dimensional Propagation

Abstract

Having already invested in understanding both the equation of state and the hydrodynamic equations, only straightforward algebraic manipulations will be required to derive the wave equation, justify its solutions, calculate the speed of sound in fluids, and derive the expressions for acoustic intensity and the acoustic kinetic and potential energy densities of sound waves. The “machinery” developed to describe waves on strings will be sufficient to describe one-dimensional sound propagation in fluids, even though the waves on the string were transverse and the one-dimensional waves in fluids are longitudinal. These results are combined with the thermal and viscous penetration depths to calculate the frequencies and quality factors in standing wave resonators. The coupling of those resonators to loudspeakers will be examined. The introduction of reciprocal transducers that are linear, passive, and reversible will allow absolute calibration of transducers using only electrical measurements (i.e., currents and voltages) by the reciprocity method, if the acoustic impedance that couples the source and receiver is calculable. Reflection and transmission at junctions between multiple ducts and other networks will be calculated and applied to the design of filters. The behavior of waves propagating through horns will provide useful impedance matching but introduce a low-frequency cut-off.