Method
Similar to the way that breath sounds are amplified through the tuba, singing through the instrument creates a unique and relatively effortless audible effect. This is achieved by singing through the instrument, using the mouthpiece as the entry point for the voice. Provided the performer is comfortable with singing, the types of sounds created by singing through the instrument are as numerous as those created solely with the voice.
Necessary information
The limiting factors with this technique involve the physics of the instrument itself. When singing into the instrument, the lips must necessarily be restricted by the rim of the mouthpiece. This cuts down on the number of discernible consonants and vowels, affecting the range of words that are capable of being sung clearly. Furthermore, the instrument itself provides a fair amount of feedback; when singing into the instrument, any pitch not contained within the harmonic series available at that moment may experience a variable amount of interference. To put it simply, the tuba and euphonium both act as reverse resonators, affecting the vocal cords in ways that are tangible.
The author has found through their own practice that it is possible to avoid some of these pitfalls by allowing a small portion of the mouth to slip outside of the mouthpiece rim while singing through the instrument. This tends to mitigate the negative feedback of the instrument, while also allowing for a wider range of vocal sounds. This does necessitate a louder sung dynamic, which might not be comfortable for inexperienced vocalists.
One particularly effective way to vocalize through the tuba and euphonium is by singing sustained vowel sounds. The instruments tend to amplify these sounds at a higher level, although this is a purely subjective opinion.
Notation
There are a few different options when notating a sung portion of a tuba/euphonium solo. If it is clear to do so in the score, the sung part may be notated on the staff above the tuba or euphonium component (although this is more valid for tuba rather than the higher-pitched euphonium). Barring that, the sung portion may be notated on a separate staff. When notating speech by itself (i.e., not combined with ordinario playing), it is common practice to use x-shape noteheads to notate the relative pitch content, a la Sprechstimme. If combined with other techniques (as in the provided musical example), the sung portion may be notated with a simple text direction, or on a separate stave.
Relative Difficulty
Intermediate to Professional
Works to consider (bolded titles are particularly representative examples of this technique)
Three Essays – William Penn
The First Dream of Light – Morton Subotnick
Alter Ego – George Heussenstamm
Post-prae-ludium ‘per Donau’ – Luigi Nono
The Composer's Guide to the Tuba 