0

Suppose there is a dog, in an infinitely large garden, tied with a leash to a fixed pole positioned at a point $P$. Let us suppose there is a circular obstacle (for simplicity's sake) that the dog enjoys crossing while playing and is the leash long enough (of finite length $L$) so that it can completely pass the obstacle.

How much ground can the dog walk on?

A little modification due to diffraction would be asking:

Suppose at a point P there is a source of a two-dimensional wave and suppose there is a circular obstacle in front of it. We fix a time $T_0$ and consider the wave crest generated at $T_0$. After a long enough time $T$, how much space will be covered by that wave?

Is there a simple answer to such an elementary question$^1$?

Is it possible to add the shape of the object in the general solution on how it affects the covered space?

$^1$note that elementary is not the same as simple, it just means that anyone without any specific background can understand it.

Gabrielek
  • 1,946
  • Are you asking about the maximum path length of the dog? What constrains it? Because at first glance the dog can just walk around indefinitely – Lourenco Entrudo Aug 18 '23 at 14:10
  • No, the space the dog can cover. Trying to go around the obstacle, since the rope is of a fixed length, the dog will be able to walk only on a minor fraction of the space as if the obstacle was not there. Call $Q$ the point of tangency between the obstacle and P. As the dog goes around the obstacle it is like if he has a new constraint in $Q$ which prevents it from walking over the entire coverable space the obstacle was not there. – Gabrielek Aug 18 '23 at 14:16
  • So the obstacle is not closed, it's like a circular arc? And the rope interacts with the obstacle? – Lourenco Entrudo Aug 18 '23 at 14:22
  • The obstacle is a closed circle and of course the rope interacts with it. – Gabrielek Aug 18 '23 at 14:23
  • 1
    These questions are usually about a cow or goat. Also here. – Jaap Scherphuis Aug 18 '23 at 15:04

0 Answers0