Final answer:
The stage of sleep characterized by long, slow delta brain waves is stage 3 sleep, also known as deep sleep or slow-wave sleep.
Step-by-step explanation:
The stage of sleep marked by long, slow delta brain waves is stage 3 sleep. This stage is often referred to as deep sleep or slow-wave sleep and is characterized by delta waves, which are of low frequency (less than 3 Hz) and high amplitude. It is in this stage of sleep that the heart rate and respiration slow down significantly and it becomes much more difficult to wake the individual.
It should be noted that sleep spindles and K-complexes, which are patterns of brain activity associated with processing information and memory consolidation, are most often associated with stage 2 sleep. This stage is a period of deep relaxation with theta waves still present, but is differentiated from stage 3 by these bursts of brain activity.