The speed of sound in periodic ducts

The flow through corrugated pipes is known to lead to strong whistling tones which may be harmful in many industrial appliances. The mechanism is known to originate from a coupling between vortex shedding at the edges of the cavities forming the wall of the tube and the acoustical modes of the pipe. The latter depend upon the effective velocity of sound ceff within the corrugated pipe. The purpose of this paper is to compute accurately this effective velocity of sound through an asymptotic calculation valid in the long-wave limit. Results are given for a number of geometries used in previous works, and compared with a simple model in which the effective speed of sound is function of the geometry of the pipe. The latter is found to work best for short cavities but significant disagreement is found for longer cavities. The case of 2D channels with a corrugated wall is also considered.