Q.Wave motion is doubly periodic. Explain.

[!TLDR] A wave repeats itself both in time (at one location, every period T) and in space (at one instant, every wavelength λ) -- these two independent repetitions together make it 'doubly periodic'. [!ANSWER] Periodic in time (period T) AND periodic in space (wavelength λ), simultaneously.

Consider a single fixed location in the medium as the wave passes through: the displacement recorded there rises and falls, repeating itself identically after every interval of time T (the period) -- this is periodicity IN TIME. Now instead consider a snapshot of the ENTIRE wave at one fixed instant: plotting displacement against distance from the source shows the same up-down pattern repeating after every distance λ (the wavelength) -- this is periodicity IN SPACE. A wave exhibits BOTH of these repetitions simultaneously and independently (one in the time domain, one in the space domain), which is exactly what 'doubly periodic' means -- Fig 8.1(a) illustrates the spatial repetition (many particles, one instant) and Fig 8.1(b) illustrates the temporal repetition (one particle, over time). [!ANSWER] A wave is doubly periodic because it repeats identically in TIME at any fixed point (period T) and, independently, repeats identically in SPACE at any fixed instant (wavelength λ).

Distinguish the two graphs used to describe a wave -- displacement vs distance (spatial period = wavelength) and displacement vs time (temporal period = T) -- and note both hold simultaneously.

Describing only ONE of the two periodicities (usually just the time-domain one, since that matches the everyday word 'periodic' from earlier SHM study) and forgetting the equally important spatial repetition unique to wave motion.