The verse as we have known it says, As the deer pants for streams of water. Yet there are some interesting Hebrew words behind this with meaning that don’t seem to get translated in our modern versions of the Bible.
Two particular Hebrew words of interest are ‘al (על), and aw-feek (אפיקי).
The proper meaning of the Hebrew word ‘al (על) according to Strong’s Hebrew is above, over, upon, or against yet it is translated as for or after in modern translations whereas the proper word for such translations is ‘el (אל) which is in fact the word used by the psalmist in reference to panting for God. The other Hebrew word, aw-feek (אפיקי) according to Strong’s Hebrew can, interestingly, be translated as strong or mighty river.
With these in mind, this would yield the translation, As the deer pants above the strong river, so my soul pants for you, O God.
Now you have a picture of a deer swimming through the current, struggling to keep his head above the water as he crosses. And is that not how our souls really pant after God?
This is seen in the counsel of Dhuoda, a 9th century mother, to her son (it is also referenced in Pliny’s Natural History):
Therefore, my son William, cherish and show respect to whatever one or many persons you wish to respect you. Love, revere, stand by, and honor all, so that you may be found worthy to receive appropriately honorable recompense in all the changeable situations of the world. Toward our edification in this regard a certain learned author offers a brief comparison—an important one, extraordinarily clear in its meaning—with dumb animals. He says in the forty-first Psalm, As the hart panteth. For this is what harts do when groups of them begin to cross seas or wide streams with churning waves—they lower their necks one after the other, each putting his head and horns on the back of the previous one, so that as they each rest a little they all may the more easily cross the swift current. The harts have such intelligence and such commensurate discretion that, when they perceive that the one in front is weakening, the leader becomes a follower and eventually the last in line so that the others may assist and support him; then they choose another to go first. Thus, as one individual takes the place of another, each feels the brotherly fellowship of love run through them all. Always careful that the head and antlers of any one of their kind not be plunged into the flood, the harts manage to hold up his head and to keep his antlers visible.
The point of this is not obscure to the learned, for everything is immediately clear in their sight. In the harts’ mutual support—in their changing places in line—they show that human beings too must have the brotherly fellowship of love for greater and lesser men alike, in all ways and in all circumstances. We read that this was fulfilled in the past by many men, especially among the holy apostles and those like them. It is written, For neither was there any one needy among them, but all things were common unto them. They had one heart and one soul in God, always feeling brotherly compassion for each other in Jesus Christ.
Just as the harts support and sustain each other’s heads and antlers, so those who have faith in Christ hold up their hearts and keep their minds always on him. He who was born king of David’s seed for the salvation of the human race and descended to the depth of this sea of battering waves has raised his horn to liberate his people. Acting of his grace, he has found those who were lying in darkness, and rising from that depth he has visited them and raised them to the heights. He offers his example lest we be lost in the turmoil of the deep sea or in the blinding mud of desire and cupidity, so that we may hold up our hearts in perseverance and say with the Apostle, But our conversation is in heaven.
 This is from Psalm 42 in our modern bibles.
 Acts 4:32-34
 Luke 1:68-69
 Philippians 3:20