
dringen
to penetrate
To force one's way through an obstacle or into something; to enter or pass through.
Das Wasser drang durch die undichte Stelle im Dach.
The water penetrated through the leaky spot in the roof.
to penetrate / to insist
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to penetrate
To force one's way through an obstacle or into something; to enter or pass through.
Das Wasser drang durch die undichte Stelle im Dach.
The water penetrated through the leaky spot in the roof.

to insist on
To insist on something emphatically or demand something forcefully.
Die Gewerkschaft drang auf eine Lohnerhöhung für alle Angestellten.
The union insisted on a pay raise for all employees.

to reach
Of a sound, light, or information: to become audible or perceptible; to reach someone.
Aus der Ferne drang leise Musik zu uns herüber.
Soft music reached us from the distance.
The auxiliary verb for the perfect tense of 'dringen' (to penetrate / to insist) depends on the meaning. If it describes movement or entering into something, 'sein' is used. If it means 'auf etwas bestehen' (to insist on something), 'haben' is used.
Rauch ist aus dem Fenster gedrungen.
Smoke came out of the window.
In the meaning 'energisch fordern' (to demand forcefully), 'dringen' (to insist) always takes the preposition 'auf' followed by the accusative. One insists on a thing.
Die Anwohner dringen auf eine schnelle Lösung des Verkehrsproblems.
The residents are insisting on a quick solution to the traffic problem.
When 'dringen' (to penetrate) describes movement, it is often combined with prepositions such as 'in (+Akk)' (into + accusative), 'durch (+Akk)' (through + accusative), or 'bis zu (+Dat)' (up to + dative) to describe the path or destination.
Das Licht der Taschenlampe drang in die dunkle Höhle.
The flashlight's beam penetrated into the dark cave.
to come
For an audible sound, kommen states its origin neutrally. Dringen emphasizes more strongly how the sound reaches the listener or spreads through a spatial boundary. When a door or wall is important, dringen is particularly appropriate.

Leise Musik drang durch die geschlossene Tür aus dem Nachbarzimmer.
Soft music came through the closed door from the next room.
Leise Musik kam aus dem Nachbarzimmer.
Soft music was coming from the next room.