I first came across this exercise on an Estill course and it is now widely used in many a pedagogy. It’s an exercise which aids a super swift warm up and can be practised pretty much anywhere with no embarrassment.
Please don’t interpret the descriptive name as a direct imitation of an emergency vehicle, however. The sound should be a gentle “ng” (as in the end of the word “sing”) sliding throughout the whole range. You can siren over a specific range of a 5th or an 8ve or simply take the siren for a gentle stroll through the whole range. It’s not always easy to, but endeavour to keep the jaw soft and the middle of the tongue high.
I suggest to my students to imagine that the sound is effortlessly gliding up and down a very wet pain of glass that’s in front of them. The aim is to keep the note gently attached to the glass. This will help
to make the sound even, with no jarring or loss of sound. Don’t rush this slide; in fact where it seems more tricky to keep the sound even, slow down more.
The siren is also useful when learning melodies: instead of using the words, use the siren to get familiar with the geography of the phrase. Because you are sliding, you will have automatically a sense of linking the notes together, so that when you sing the phrase with a vowel or with the words, the portamento (“carrying the sound from one pitch to another”) should easily be achieved.