The main and general meaning of Upāṁśu (Upamsu) indicates a kind of Mantrajapa (chanting of mantras or hymns). Upāṁśu (Upamsu) is also used as a conjunction to mean serenity-

Upāṁśujarpabhede (Upamsujarpabhede) syād (syad) upāṁśu (upamsu) vijane’vyayam|

In the Agni Purāṇa (Purana), it was said in the Mantraparibhāṣā (the text about the mantra or chants) that the mantra could be chanted in four ways- Uccsware Japa (Loud Chant), Upāṁśu- japa (Upamsu- japa or chant), Jihva Japa (Tongue Chant) and Mānasa Japa (Manas Japa or meditative chant). But amidst these chants, in the Agni Purāṇa (Purana) it was said that it was ten times better to perform Upāṁśu- japa (Upamsu- japa or chant) silently than to perform the Loud Chant

Uccairjapād (Ucchairjapad) viśiṣtaḥ (visistah) syād (syad) Upāṁśurdaśābhirguṇaiḥ (Upamsurdasabhirgunaih)|

In the epics and the Purāṇas (Puranas), it was not clearly said how to perform the Upāṁśu- japa (Upamsu- japa or chant). This mantra chanting was a kind of indistinct murmuring to yourself. You have to concentrate more on this indistinct chanting to assimilate the mantra or hymns than what you generally concentrate while loudly chanting the mantras or hymns. Perhaps for this reason, it was said in the Manusaṁhitā (Manusamhita) that the yajña (yajna) by chanting was ten times more effective than the Brahminic yajña (yajna). Then, Upāṁśu- japa (Upamsu- japa or chant) was hundredth times more effective than the yajña (yajna) by chanting-

Vidhiyajñājjapayjño (Vidhiyajnajjapayjno) viśiṣto (visishto) daśabhirguṇaiḥ (dasabhirgunaih)|
Upāṁśuḥ (Upamsuh) syācchataguṇaḥ (syacchatagunah) sahasre mānasaḥ (manasah) smṛtaḥ (smritah)||

While analysing this word ‘Upāṁśu’ (Upamsu) in this Manuśloka (Manusloka), the commentator Kullūkabhatta (Kullukabhatta) wrote- Upāṁśu (Upamsu) was a kind of chanting which the paraloka or the afterlife could not hear even if you were chanting in front of it-

Yat samīpastho’pi (samipastho’pi) paro na śṛṇoti (srinoti) tadupāṁśuḥ (tadupamsuh) |

In the Śavdakalpadruma (Savdakalpadruma), without naming the āgama (agama), it was proved that Upāṁśu- japa (Upamsu-japa) was performed by chanting the mantras in a way that the mantras could only be heard to himself by the least movement of tongue and lips, while concentrating on the God-
Jihvaṣthou (Jihvashthou) cālayet (chalayet) kiṁcid (kimchid) devatāgatomānasaḥ (devatagatomanasah)|
Nijaśravaṇayojyoḥ (Nijasravanayojyoh) syād upāṁśu (upamsu) sa japaḥ (japah) smṛtaḥ (smritah)||

In the Mahābhārata (Mahabharata), there was no specific mention of Upāṁśu- japa. But it was analysed in the Anugītā (Anugita) upaparva or sub-division of the Mahābhārata (Mahabharata) that, for the mantrajapa or chanting of the hymns, nirghoṣa (nirghosha or not loud) and almost silent chanting was better than the loud and powerful chanting of the mantras or hymns. Here, two epitomes of the Vāgdevī (Vagdevi) were represented. She was set between Prāṇa (Prana or life) which was epitomized in the manifestation of Saraswatī (Saraswati) as mantra and Apāna Vāyu (Apana Vayu). Bāgdevī was sometimes ‘Ghoṣiṇī’ (‘Ghoshini’ or someone who loudly spoke) and sometimes she was Nirghoṣā (Nirghosha or someone who remained silent). But Nirghoṣā (Nirghosha or someone who remained silent) Vāk (Vak) was better than the representation of Vāk (Vak) as ‘Ghoṣiṇī’ (‘Ghoshini’ or someone who loudly spoke)-

Ghoṣiṇī (Ghoshini) Jātanirghoṣā (Jatanirghosha) nityameva pravartate|

Tayorapi c ghoṣiṇya (ghoshinya) nirghoiṣava (nirghoishava) garīyasī (gariyasi)||

Here Nirghoṣā (Nirghosha) indicates Upāṁśu japa (Upamsu) or Mānasa japa (Manasa japa or meditative silent chant). Nīlkaṇtha wrote in his commentary- ‘Nirghoṣā  Haṁsamantrarūpā’ (‘Nirghosha Hamsamantrarupa’) which meant that there was no space of the loud sound in the Nirghoṣā (Nirghosha) mantra.Only connection could be established between the subtle life soul (Aham) and the Paramātma  [‘Saḥ’ (Sah)] or the ultimate soul by the chanting of the hymns through silent breathing.

Here Nirghoṣā (Nirghosha) indicates Upāṁśu japa (Upamsu) or Mānasa japa (Manasa japa or meditative silent chant). Nīlkaṇtha wrote in his commentary- ‘Nirghoṣā  Haṁsamantrarūpā’ (‘Nirghosha Hamsamantrarupa’) which meant that there was no space of the loud sound in the Nirghoṣā (Nirghosha) mantra.Only connection could be established between the subtle life soul (Aham) and the Paramātma [ ‘Saḥ’ (Sah)] or the ultimate soul by the chanting of the hymns through silent breathing – and this connection should be made always and in all conditions-
Nirghoṣā (Nirghosha) tu haṁsamantrarūpā (hamsamantrarupa) sarvaswavasthāsu (sarvaswavasthasu) pravartate iti garīyasī (gariyasi)- for this reason it is better than the Ghoṣiṇī (Ghoshini) mantrajapa or the chanting of the hymns.

David Frawley writes-
Yet at a higher level beyond duality, Ha and Sa are the natural sounds of the Self, which is the inner breath of awareness, the unitary Prana that is Self-existent and immortal. Ha is the Self as I (aham) and Sa is the Self as that or the inner Being. Hamsa also refers to the supreme or Paramahamsa, which is the liberated soul that dwells in the state of the Supreme Shiva. In this regard, Hamsa teachings are an integral part of Shiva Yoga and Shiva is also Hamsa. Hamsa as sound and prana vibrations also Om or Pranava, of which Lord Shiva is the indicator.
Hamsa represents the union of Shiva and Shakti, which are Ha and Sa, Sun and Moon, Prana and Apana, the incoming and outgoing vital energies. All dualities, starting with the breath, are a reflection of the greater two-in-one power of Shiva and Shakti, which gets divided in the lower worlds.

There are different types of mantras or hymns. One of the type is ‘likhita’ or written. May be this type of mantra is written to get the effect of repeated chanting. But generally, there are three types of chanting the mantras or hymns and this is described by Chaitanya Parikara Gopālbhatta (Gopalbhatta) Goswāmī (Goswami) who quoted from the Narasiṁha (Narasimha) or Nṛsiṁha (Nrisimha) Purāṇa (Purana) and wrote in the book, Haribhakti Vilāsa (Vilasa). He commented that Yajña (Yajna or pious fire) is of three types- Vācika (Vachika), Upāṁśu (Upamsu) and Mānasa (Manasa). Amomg these Japayajñas (Japayajnas), Upāṁśu (Upamsu) and Mānasa (Manasa) are more effective. The Narasiṁha (Narasimha) or Nṛsiṁha (Nrisimha) Purāṇa (Purana) indicates the Upāṁśu (Upamsu) mantrajapa by commenting that when someone slowly chants the mantras or hymns by moving the lips slightly and only he can hear the mantras himself or he can know the mantras, he is performing the Upāṁśu (Upamsu) mantrajapa-
Śanairuccārayenmantramīṣadoṣthou pracālayet (Sanairuchcharayenmantramishadoshthou prachalayet).
Kiṁcinmātraṃ (Kimchinmatram) swayaṃ (swayam) vidyād (vidyad) upāṁśuḥ (upamsuh) sa japa smṛtaḥ (smritah).

[Narasimha Purana (Maharshi) 58.78-82; Haribhakti Vilasa, 17.73-76, p.1099]