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Shortest-time / next-instant questions
Convert distance to time through the wave's motion: the profile moves λ per T. For "shortest time until P has displacement X": slide the profile in the travel direction until the required feature reaches P; t = (distance moved/λ) × T.
What this actually means
The wave is a conveyor belt for shapes: whatever feature P needs (a crest, a particular displacement) already exists somewhere upstream, and it arrives after travelling its separation at speed v.
Recipe: find the nearest copy of the required feature on the side the wave comes FROM, measure its distance to P, then t = (distance/λ) × T.
Sliding the profile the wrong way gives the complementary, wrong time. Direction discipline is the entire question.
Prove it — watch it be true
- Pick particle P and note the nearest upstream crest
- STEP T/8 repeatedly, counting steps until the crest reaches P
- Check: steps × T/8 equals (separation/λ) × T