Shofar awakens the soul's root

בענין קול שופר י"ל ע"פ מ"ש אא"ז מו"ר ז"ל בפ' בשלח ע"ד המד' יעקב מתגאה בירושתו כו' שע"י הקול מעורר כח שורשו שלמעלה מהקול ושואלין אח"כ מה תצעק כו' ועכ"ז בא הכל ע"י הקול כו' ע"ש דפח"ח.

Regarding the sound of the shofar, it may be explained according to what my saintly grandfather, my teacher, of blessed memory, said on Parshas Beshalach concerning the Midrash, "Yaakov takes pride in his inheritance," etc. — that through the kol (sound/cry) one awakens the power of one's root that is above the kol, and afterward they are asked, "Why do you cry out?" etc., and yet it all comes about through the kol — see there, for it is words of the living G-d.

The Sefas Emes builds on the Chiddushei HaRim: a person's cry awakens the inner power of his soul-root, which lies above speech and sound. At the Yam Suf, Hashem said to Moshe Rabbeinu, "Why do you cry out to Me?" — implying the moment was beyond the cry — yet the salvation still came through that very cry. The kol reaches upward to a place higher than itself.

וכן איתא ג"כ שופרי' דיעקב מעין שופרי' דאדם.

And so too it is brought that the beauty (shufrei) of Yaakov is a semblance of the beauty (shufrei) of Adam HaRishon.

Chazal teach that Yaakov Avinu's radiance was a reflection of the original beauty of Adam HaRishon. The Sefas Emes hears in "shufrei" an echo of "shofar" — the shofar's sound reaches back toward that primordial radiance and root.

ולכן הקול בלי דיבור כדי לרמוז שאין הסמיכה על הקול רק לעורר השורש שלמעלה מהקול ע"י שהקול יש בו מעין שופרי' דלמעלה.

Therefore the kol (of the shofar) is without speech, in order to hint that one does not rely on the kol itself, but only to awaken the root that is above the kol — since the kol contains a semblance of the "shofar" of Above.

The shofar produces sound without words specifically to teach that the cry is not an end in itself. The wordless blast reaches past articulated speech to awaken the soul's root above all sound, because the shofar's sound carries within it a trace of the higher, supernal "shofar."

וזה קול שופר כי כל מיני תפארת הכל מעולם העליון כי בעוה"ז אין נמצא תפארת כלום ולכן עוה"ב נק' נועם ה'.

And this is "the sound of the shofar," for all forms of tiferes (beauty/glory) are entirely from the upper world — for in this world there is no true beauty at all, and therefore Olam HaBa is called "the pleasantness (no'am) of Hashem."

All genuine beauty has its source in the higher world; whatever radiance appears in this world is only a borrowed reflection. That is why the World to Come is called "no'am Hashem" (Tehillim 27:4) — true beauty belongs there, and the shofar draws from that supernal source.

והנה עיקר השופר הוא לעורר ולתקן המדרגה הראשונה שהי' קודם חטא הראשון ושהי' עומד להיתקן במתן תורה אך ע"י החטא בעגל נאמר כאדם תמותון ולכן צריכין לבקש לחזור למעלה הראשונה:

And behold, the essence of the shofar is to awaken and to rectify the first level (madreigah) that existed before the first sin, and which was poised to be fully restored at Mattan Torah — but through the sin of the Eigel (Golden Calf) it was said, "you shall die like Adam." Therefore we must beseech to return to that first, original level.

The deepest purpose of the shofar is to reawaken the exalted madreigah Adam HaRishon held before the cheit. That level was about to be restored at Sinai, but the cheit ha'eigel brought mortality back — "ke'adam temusun" (Tehillim 82:7). So on Rosh Hashanah the wordless cry of the shofar is our plea to climb back to that original height.

Summary: The shofar is sounded without words because its cry is not an end in itself but a means to awaken the soul's root, which lies above all sound — a trace of the supernal "shofar" and the primordial radiance of Adam HaRishon. Since all true beauty belongs to the upper world, the shofar's essential avodah is to reawaken and restore the lofty level held before the first sin, which the cheit ha'eigel undid, and to plead to return there.