Use of gerunds to mean "because"

From BBC.Co.UK World Service:

Participial clauses often express

• condition
• reason
• cause
• result
• time

in a similar way to full adverbial clauses, only more economically.

[And, as one would expect, with greater scope for ambiguity.]

The examples you give perhaps not too obviously express reason in the first instance (ie explaining why) rather than cause ('They awaited their deliverance with unwavering hope, the reason for this being that they knew that the public opinion was in their favour').

While the adjective 'causal' is certainly twinned with 'cause' and used to express the semantic relation, I wouldn't use 'rational' here, preferring 'expressing reason'.

They awaited their deliverance with unwavering hope, because they knew that the public opinion was in their favour.

I replaced because they knew with knowing:

They awaited their deliverance with unwavering hope, knowing that the public opinion was in their favour.

because they knew that the public opinion was in their favour. is a subordinate clause of reason, adverbially modifying They awaited their deliverance with unwavering hope (I would call it a “free modifier”)

knowing that the public opinion was in their favour. is a participle clause of state, adverbially modifying They awaited their deliverance with unwavering hope (I would call it a “free modifier”)

It is basically the same as in the knowledge that the public opinion was in their favour.

“Knowing” and “because” are quite different and do not mean the same thing at all. If you change one for the other, the meaning of the sentence changes.

“Knowing” can be replaced by a variety of participles “shouting”, believing, holding, repeating, etc. none of which give a reason.