Yes, for a wave to exist, it is necessary for there to be variation in time. Waves are characterized by the propagation of energy or disturbances through a medium or through spacetime. This propagation involves the oscillation or periodic variation of some physical quantity.
In most cases, waves exhibit variation in both space and time. The oscillatory behavior of a wave occurs over a specific period or cycle, which implies variation in time. As time progresses, the wave moves through space, and its shape or pattern evolves accordingly.
For example, in a common type of wave known as a sinusoidal wave, such as a water wave or a sound wave, the oscillation of the wave occurs at regular intervals, and the shape of the wave repeats itself over time. Each point along the wave undergoes periodic motion as time passes.
However, it is worth noting that some waves can exhibit variation only in time or only in space. In these cases, the wave is still defined by its periodic nature or oscillatory behavior, but the variation occurs exclusively in either time or space.
An example of a wave with variation only in time is a simple harmonic oscillator, such as a pendulum. The motion of the pendulum represents a wave-like behavior, but its variation occurs solely in time, with no spatial propagation.
Similarly, some waves can have variation solely in space, such as standing waves. These waves have fixed patterns or shapes that do not propagate through space but exhibit oscillatory behavior at different spatial locations.
In summary, while waves typically exhibit variation in both space and time, there are cases where waves can have variation solely in time or solely in space, depending on the specific characteristics of the wave and the system in which it occurs.