In the section of the Torah detailing the laws of the trumpets (Bamidbar 10:1-10), a distinction is made between the tekiah תקיעה - "the long blast" and the teruah, "the short blast". The tekiah is used for gathering the camp together and happy occasions, whereas the teruah indicates the camp should move, and is used at times of war. However, the verse (10:5) uses a combination of the roots to describe the blowing of the teruah: וּתְקַעְתֶּם תְּרוּעָה (ut'kat'em teruah). Milgrom, in his JPS commentary there, explains as follows:
short blasts: Hebrew teru'ah, verbal form heri'a, in contrast to "blow long blasts," taka'. It should be noted that the term "blow long blasts" is expressed simply by the verb taka' (vv. 3-4), but "blow short blasts" requires the compound expression taka' teru'ah (vv. 5-6). The reason for these distinct forms is twofold.
(1) The term teru'ah and its corresponding verb heri'a refer elsewhere to a vocal shout by warriors (e.g. Josh. 6:5,10, 16, 20) and worshipers (e.g. Pss. 47:2; 95:2), whereas the sole verb signifying the blowing of a horn is taka' (e.g. Josh. 6:13). Hence when the text wishes to express the idea of blowing the teru'ah signal on the trumpet it must either use the verb taka', signifying blowing on an instrument, and the object teru'ah to indicate the appropriate signal, or, if it uses the verb heri'a, it must specify that the sound was produced by a trumpet (v. 9).
(2) Teru'ah can refer to a battle cry (cf. Amos 1:14, Jer. 14:19); and hence, its use in breaking camp implies signaling the Israelites to move from an encamped peaceful position to a mobile battle formation. Thus the trumpets taken into the Midianite war are actually called "the trumpets of teru'ah" (31:6, cf 2 Chron 13:12).
So we see from Milgrom that teruah (or the verb heria הריע) can be used to refer both to the noise of a trumpet/shofar, or the noise of people. Dr. Nissan Netzer discusses this in the latest issue of the parasha sheet Me'at Min Ha'or. He writes that the original meaning of the verb heria (from the root רו"ע) meant to blow on a shofar or trumpet, and later that was expanded to mean the shouts of a crowd of people (Klein seems to indicate the reverse development). Netzer then goes on to point out than in Rabbinic Hebrew, the verb hitria התריע was created by adding the letter tav from the word teruah (similar to the process we saw in the verb taram from teruma). Whereas originally the Biblical and Rabbinic forms of the verb had the same meaning, in Modern Hebrew they diverge: heria has a positive connotation - "to applaud", whereas hitria has a negative one - "to protest, to warn" (and neither meaning today refers to blowing a shofar; for that we only have the verb taka).
It is very easy to mix up hatra'ah התרעה - warning, with the similar sounding hatraah התראה - which also means warning. But the former means also "alert, alarm" (think of the shofar, and the original distinction of teruah), while the latter also has the sense of "give advance notice" (for example, as witnesses are required to do in capital cases). Or as the site Safa Ivrit has it, התראה means "warning someone not do something" and התרעה means "warning about something that is about to happen" (see also this explanation by the Hebrew Language Academy).