Occasionally players may find themselves facing off against someone whose knowledge of magic is greater than theirs. In such cases a question that comes up often is how to handle the description of spells cast by the more powerful wizard. There is no question with the big, flashy spells. However, some of the subtler spells would be harder to determine until the effect takes place (for example, if a wizard casts haste on someone and that person doesn't move).
Piecing Together the Spell
An optional rule for this kind of situation could be that a character knowledgeable in magic can try to figure out the effect of a spell by making a check while it is being cast. The harder part to decide is when such a check would be required. At the very least, a wizard should be able to recognize the spells they know. However, at there could be many spells of a given level that a character doesn't know. In those cases, we can have two variants.
Variant 1: When a spell is being cast that a magic class character does not know, they can make a DC 10 + level of the spell arcana check to determine the effect the spell is about to have.
Variant 2: When a spell of a higher level than a magic class character knows is cast, the character can make a DC 10 + level of the spell arcana check to determine the effect the spell is about to have.
Variant B: The above variants but use a DC of 10 + proficiency + modifier.
I tend to use variant 2 + B.
Frequency of Use
The variants above generally assume that the characters won't be facing characters much higher level than them very often (if this isn't true, these kinds of rolls will happen more often). It also assumes that a lower level wizard would be able to determine what spell is being cast through their arcana knowledge, even when they cannot cast it themselves. If either of these assumptions proves problematic, it may be a good idea for the Dungeon Master to use a more restricted version of the rules.
There examples of spells that allow for multiple effects (right now this is mostly for house rules). The rule can be adapted to only apply to these kinds of spells instead to reduce the number of these kinds of rolls. This kind of rule needs to be carefully applied in order to prevent the game slowing down. As an alternative, the Dungeon Master can keep more in their arsenal as a more random occurrence instead of a rule to still allow for it to be used when needed while not being used all the time (weird spells, as below, is an example of this, though more rigid in the way it is applied).
I've seen situations where certain wizards have special spells that they came up with. In these cases, particularly when the spell is more subtle, it can be hard to determine and react to the spell when it is cast. Instead, the effect of the spell needs to be revealed with time. In such cases, this kind of rule could apply to just the weird spells of this spell caster. I've also seen examples where a necromancer, for example, will have modified spells in their disposal (a fireball ends up being a ball of necromatic energy). These situations can be hard for the characters because these spells may not even be available for them. In these kinds of cases, this kind of rule could apply (knowing it's a modified fireball can help players make inferences about the number of spell slots left for the wizard, etc.).
It is important to realize that ritual magic is also magic. In these cases, this kind of rule gives a straight forward and consistent way to determine the intention of a ritual or the remnants of a ritual site (circles, components, etc.). Even there is hesitation to apply this rule in combat it can still be used as a Dungeon Master tool during dungeon construction to decide DCs.