How would you describe a sound "appearing"?

Stick with simplicity. I propose go._{(Oxford, sense 4.2)}

Make a sound of a specified kind:

the engine went bang

The elevator went ping and the doors opened.

It was dark. I couldn't see anything. Suddenly, I heard a gun go bang.

Speaking of guns, go off is related:

(Of a gun, bomb, or similar device) explode or fire:

(Of an alarm) begin to sound.

And then, I heard an alarm go off in the distance.

In case you're not still convinced, watch this.

Assuming you're asking for the sound equivalent of:

A light appeared in the forest

Emerge

to become manifest : become known

Along with "arise" or "emerge," you could use "ring out."

"Pistol shots ring out in the barroom night."

Or, along with those, and if you aren't talking about a sound, you could use sound as a verb. Like this:

"The trumpets sounded from the other side of the room."

But maybe don't say: "The sound sounded from the other side of the room." Unless you're after that.

A word that's sometimes used is arise. As in the line from A Visit From Saint Nicholas:

When out on the lawn there arose such a clatter,
I sprang from my bed to see what was the matter.