Dead Sea MSS: 4QPsc & 4QPsu (42:5 ); 11QPsd (43:1-3)
Most commentators recognize that Psalms 42 and 43 comprise a single Psalm, containing three stanzas (42:2-6 [1-5], 7-12 [6-11], and 43:1-5), each of which ends with a common refrain. This is one of the clearest examples of a Psalm that, in its current form, would have been especially well-suited to being sung by a congregation in public worship. Metrically, it tends to follow a 3+2 bicolon (couplet) format (the so-called Qinah meter).
The superscription is distinctive, since it attributes the composition, not to David (as in most of Pss 1-41), but to the “sons of Qorah [Korah]”. The Korahites were priestly officials who served in the Temple, as attested in the books of Chronicles (1 Chron 9:19; 26:1, 19), and also as a company of singers (2 Chron 20:19). Elsewhere in the Old Testament, they are simply designated as Levite clan (Exod 6:21; 1 Chron 6:7, 23 [22, 38]), with no additional information provided. Clearly it is the group of Temple singers that is most relevant to the superscription here (as also in Pss. 44-49, 84-85, 87-88). It is possible that they were responsible for the editing of the ‘Elohist Psalter’ (cf. below).
The musical direction of the superscription also indicates that this composition is a lyK!c=m^, a term of uncertain meaning, but presumably derived from the root lk^c* which characteristically occurs in the Hiphil stem (= lyK!c=h!). The root fundamentally indicates the use of reason and intelligence—i.e., wisdom, understanding, prudence, etc. As a poetic or musical term, it could refer to a harmonious composition, a work of great skill and artistry (or requiring skill to perform), a poem/song used for instruction, or something else entirely. The term appears in the superscriptions of a number of Psalms in the ‘Elohist Psalter’ (44-45, 52-55, 74, 78, 88-89); cf. also the earlier study on Psalm 32.
Stanza 1: Verses 2-6 [1-5]
Verse 2 
“As a deer cries (out) upon channels of water,
so my soul cries (out) to you, Mightiest (One)!”
The meter of this initial couplet is 4+4, an expanded metrical form that creates a grand and solemn opening. The feminine form of the verb (gr)u&T^) in the first line does not match the noun-subject (lY`a^, “[male] deer”); we would rather expect hl*Ya^ (“[female] deer”), in agreement with the verb. Dahood (p. 255) suggests dividing the text grut lyak differently, as gr)u* tl#Y#a^K= (“like a [female] deer crying [out]”).
The verb gr^u* refers to a cry of longing; a crying out loud is indicated by the parallel between gr^u* and ar^q* in Joel 1:19-20 (the only other occurrence of gru in the OT). The idea may be of the deer’s longing to quench his (or her) thirst, but the parallel between the “channels of water” and God (“[the] Mightiest”) suggests rather a scene where the longing for thirst is fulfilled (upon finding water). The basic imagery is well-established in Semitic poetry, going back to the Canaanite poetic texts from Ugarit; most notably, the image of a deer/stag going to a spring to quench its thirst is compared to the ravenous appetite (hunger/thirst) of Death personified (Baal Epic, tablet V, column 1, lines 16-19, etc).
“Mightiest (One)” is my regular translation of the plural noun <yh!ýa$ (°§lœhîm), understood as an intensive plural when applied to El-Yahweh (in the context of ancient Israelite monotheism). It can be transliterated as a name/title (Elohim), though more often it is simply translated generically as “God”. Curiously, in Psalms 42–83 Elohim is used far more often than YHWH (more than 200 times, compared with only 45 for YHWH), in contrast to the rest of the Psalms, where YHWH dominates (more than 580 times, compared with little over 90 for Elohim). This has led to Pss 42-83 being referred to as the “Elohist Psalter”. The reasons for the difference are not entirely clear. It has been thought that the regular use of <yh!ýa$ reflects an intentional editing of compositions which originally used the divine name hwhy (YHWH) throughout.
Verse 3 
“My soul thirsts for (the) Mightiest,
for (the) Living Mighty (One)—
when will I come and be seen (by)
(the) face of (the) Mightiest?”
Here we have a pair of 3+2 couplets that builds upon the idea expressed in the opening verse. The motif of “drinking” has led Dahood (p. 256) to explain hara as a form of the root ary II (cognate with hwr), involving the idea of pouring out and watering (saturating) the ground, along with the related concept of a person (or animal) being filled (sated/satisfied) by drink. If his analysis turns out to be correct, then the second couplet above would be translated something like:
“when will I come and drink my fill
(of the) face of (the) Mightiest?” (cp. Ps 34:9)
In any case, the thing that will quench the Psalmist’s thirst is to experience the very presence of YHWH Himself (His “face”).
Verse 4 
“My tears have been food for me
(by) day and (by) night,
in (their) saying to me, all the day (long):
‘Where (is) your Mighty (One)?'”
The motif of thirst/drinking continues in this verse (again another pair of 3+2 couplets). While the Psalmist longs to drink from the very presence of YHWH, here on earth he has been been drinking only from his tears (toum*D=)—by which is meant his experience of sorrow and suffering. The idea of eating/drinking tears (as “food” [or bread, <j#l#]) reflects another ancient Canaanite poetic idiom. Again, an example is at hand in the Baal Epic from Ugarit (Tablet VI, column 1, lines 9-10), in which, following the death of Baal Haddu, the goddess Anat, in her mourning “she weeps her fill, drinks her tears like wine”.
The Psalmist’s sorrow/suffering is accompanied by mocking taunts from a group of wicked onlookers. This is a familiar motif in the Psalms (cf. the recent studies on Pss 40 and 41). The suffering of the Psalmist (often depicted as the result of an illness) brings into question his loyalty and trust in YHWH. The voice of the wicked and faithless ones, which can also serve to express his own doubts, asks “Where is your Mighty (One)?”. Here “Mighty One” = “Mightiest” (<yh!ýa$), the two titles El and Elohim being so close in meaning (and significance) as to be virtually identical. The theme of the suffering of the righteous—and with it, the apparent absence of God’s presence—was popular in ancient Near Eastern Wisdom literature, and is reflected in many of the Psalms as well.
Verse 5 
“These will I remember, and will pour out
my soul upon me (as I do)—
that I went over in(to the) cover of (the) <Majestic> (One),
unto (the) house of (the) Mightiest,
with a voice of shouting and casting (praise),
(amid the) noise of (those) circling.”
While these lines are difficult to interpret (and translate) in detail, the overall sense of them is clear enough. The protagonist of the Psalm, in his suffering, recalls his recent pilgrimage to the Jerusalem Temple, almost certainly on the occasion of a holy festival. This is indicated by the verb gg~j*, which, it seems, has the fundamental meaning of making a (circular) procession, but which early on took on the technical meaning of making a procession (to Jerusalem) for one of the three great pilgrimage feasts; a cognate root in Arabic was used later on for the pilgrimage to Mecca (the Hajj).
Here, the festival in question may be that of Booths/Tabernacles (Sukkot), for which an allusion seems to be present in the obscure phrase “I went over in(to the) cover [Es)] of Majesty [?]”. I follow Kraus (p. 437) in tentatively emending MT <D@D^a# as reflecting the root rda—possibly a plural substantive <yr!yD!a^, or the adjective ryD!a^ with an enclitic <-. According to this line of interpretation, ryD!a^ (“great, majestic”) is a title for YHWH, creating a clear parallel: “cover of (the) Majestic (One)” / “house of the Mightiest (One)”. The Es) (sœk, “covering”) is the Temple and its sanctuary, the “booth” (= house) of God.
At such a pilgrimage festival, the Psalmist would have come before “the face” of YHWH, to “be seen” by Him (cf. on verse 3  above), in a symbolic and ritual sense. Now, in the midst of his sorrow, the Psalmist longs for a real experience of God’s presence, one that he can “drink” to give him nourishment and to satisfy him in his time of need.
Refrain: Verse 6 
“(For) what are you bent down, my soul,
and make (such) a clamor upon me?
Wait for (the) Mightiest (One)—
for again will I throw Him (praise),
(the) Salvation of my face and my Mighty (One).”
This same refrain occurs in all three stanzas of the Psalm. The parallel occurrences in v. 12  and 43:5 make clear that the first word of v. 7  is misplaced and belongs at the end of the final line of v. 6 .
The Psalmist’s suffering and sorrow has led to his “soul” being “poured out”, and the same idea here is expressed by the verb jj^v* (“bend down [low]”) in the passive-reflexive (Hitpolel) stem. The sense of suffering/sorrow is reinforced by the image of the soul making a loud noise (‘clamor’), vb hm*h*. The Psalmist’s response to his own troubled soul is to wait (vb lj^y`) for God—that is, for Him to act, delivering the Psalmist from the source of his suffering. Indeed, the protagonist believes that he will once again, very soon, praise YHWH and worship Him just as he did at the pilgrimage festival (cf. above). The Psalmist remains firm in his belief that YHWH is his Savior (“the Salvation of my face”) and the true God (“Mighty/Mightiest [One]”) who will act on his behalf.
This reflects the theme of covenant loyalty that runs throughout many (indeed most) of the Psalms we have studied. Because the Psalmist has remained faithful and loyal, he is confident that he will receive help and protection from YHWH. Indeed, the promise of such protection is implicit in the very terms of the covenant (between YHWH and Israel). This extends to healing and deliverance from illness, as well as relief from the attacks (and taunts) by the wicked. Only a complete deliverance will confirm the trustworthiness of YHWH (as Sovereign), and vindicate the righteousness and loyalty of the Psalmist (His vassal).
References marked “Dahood” above are to Mitchell Dahood, S.J., Psalms I: 1-50, Anchor Bible [AB] vol 16 (1965).
References marked “Kraus” are to Hans-Joachim Kraus, Psalmen, 1. Teilband, Psalmen 1-59, 5th ed., Biblischer Kommentar series (Neukirchener Verlag: 1978); English translation in Psalms 1-59, A Continental Commentary (Fortress Press: 1993).