ON UNIQUENESS IN LINEARIZED 2-DIMENSIONAL WATER-WAVE PROBLEMS FOR 2 SURFACE-PIERCING BODIES

In the study of linearized water waves interacting with obstacles, the question of the uniqueness of the solution is not yet fully answered. That is, are there non-radiating (and therefore persistent) oscillato ry modes at any frequency for some geometry? John (1) established uniq ueness for the case where the body is surface-piercing and has the pro perty that vertical lines from the free surface do not intersect the b ody. More recently, Simon and Ursell (2) generalized John's approach t o prove uniqueness for a wider class of problems. Each of these papers uses a bound on the potential energy of the non-radiating motion rela tive to its kinetic energy; as these quantities are equal a contradict ion is established. However, this approach cannot be employed directly in two dimensions when there are two surface-piercing bodies, essenti ally because the free surface between the bodies is separated (by the bodies) from both +/-(infinity). The purpose of the present work is to show how a conformal mapping can be used to help to derive a bound on this part of the potential energy, and thereby prove uniqueness; the end result is that the solution will be unique provided the frequency is no greater than a parameter which depends on the geometry.