Transient Sounds in Rooms

The propagation of transient sound waves in bounded one- and three-dimensional regions is studied by means of Laplace transform methods. In the one-dimensional case, a plane wave is considered propagated down a rigid-walled tube from a source at one end toward an acoustic termination at the other end. The velocity potential for an arbitrary particle-displacement input is found as a series, each term of which represents the effect of a reflection from the ends of the tube. In the three-dimensional case spherical wave from an arbitrary input source is considered, first in an unbounded region, and then in regions containing one wall, three perpendicular walls, and two parallel walls; finally, the case of a point source in a rectangular room is solved. An image method is used, the results being found in the form of a series whose terms represent reflections from individual walls, as well as cross reflections between walls. The terms of the series are in the form of plane-wave expansions around the image points; these integrals are approximated by the method of steepest descents. In the last section some sample calculations are made for both the one- and three-dimensional systems.