preposition + which [closed]

I know that normally this structure, preposition+which, is used because the preposition is bound with a verb phrase or noun phrase in the following relative clause, but I came across a sentence in the New York Times which I find does not follow the rule:

We looked carefully at the physical phenomena by which exposure to such fields might perturb biological systems and lead to cancers

Why is "by which" used here? I cannot find any phrases in the relative clause including "by".

Solution 1:

We looked carefully at the physical phenomena [by which exposure to such fields ____ might perturb biological systems and lead to cancers].

The sentence is fine and perfectly grammatical.

The relative clause is "by which exposure to such fields might perturb biological systems and lead to cancers".

Within the relative clause is the relative phrase, the PP "by which", where the complement of "by" is the relative pronoun "which", whose antecedent is "physical phenomena".

The PP "by which" then functions as complement of "exposure" in the relative clause, as indicated by the notation '___' (called 'gap').

We understand that exposure to such fields by physical phenomena might perturb biological systems and lead to cancers.

Solution 2:

I believe 'by which' here to be a shortened version of "by means of which". By means of is defined by Lexico as

With the help of; by using.

with the example

supplying water to cities by means of aqueducts

'By means of' is usually used to refer to deliberate human action but it can also be used for natural processes as well.

'By means of' is normally followed by the enabling object, system or whatever as in the Lexico example above but the addition of 'which' to 'by means of' refers 'by means of' back to something described previously as in "There was a ring set into the top of the block by means of which it could be raised". This type of sentence is often shortened by missing out the 'means of' to give a sentence like "There was a ring set into the top of the block by which it could be raised"

I believe that this is what has happened to the sentence in the question so that the full version would have been:

We looked carefully at the physical phenomena by means of which exposure to such fields might perturb biological systems and lead to cancers.

In this case 'by means of which' and, therefore, its shortened form refer back to 'the physical phenomena' and the sentence could have been written (less well) as "Such fields might perturb biological systems and lead to cancers by means of some physical phenomena. We looked carefully at these physical phenomena"

In this case 'which' is not required as the physical phenomena had not been mentioned previously.