Looks quite square wave to me. Set up some test app and a recording of
the real thing and dosbox for comparison.

I think all you can do is adjust the PC speaker rate from 11025Hz to 22050Hz or 44100Hz.

http://193.125.34.222:888/true-spk.wav what I need (recorded from real pc-speaker)
http://193.125.34.222:888/dosbox.wav DOSBox's emulation

How and where I can adjust bitrate of speaker emulation?

CD-Man 😀

pcrate in dosbox.conf

How are you recording the output from DOSBox? I'm only curious to know if something might be altering it in the process of recording.

One thing that stands out about the waveform from DOSBox is that the DC Bias in the right channel is shifted so much that some samples appear to get clipped in comparison to the left channel. I wonder why the two channels have such a difference in bias, and aren't simply identical.

Bitrate setting is helpful!
I replace all defaults 22050 to 88200 and now sound sharp like original! 😀
Maybe developers changes default conf values from 22050 to 44100 in future releases? I think it's a good idea.
Thanks!

P.S. About my wav files. I very novice in Linux and linux's sound subsystem. Many years I used 'cooledit' sound editor in windows, but now in Ubuntu, I can't find same 🙁 But old cooledit'96 runs under wine! Magic! But while recording, 1 or 2 most significant bit(s) is wrong, and it looks like DC offset in recordings (different each time, each channel). I do not know how to fix this 🙁
I record pc speaker using 2 resistor divider, and via capacitor to line-in (L+R tied together).
Common sound I record simply jack-jack cable line-out to line-in, because don't know how to redirect all playable sounds in linux direct to file. This give big noise, present in my dosbox.wav file (in speakers, all clear).
Onboard soundcard is used.

OFFTOPIC: If someone know native linux sound editor like Cooledit'96, please tell! 😀

You can record sounds generated by DosBox with Ctrl+F6.

Maybe developers changes default conf values from 22050 to 44100 in future releases?

No because you won't notice the difference most of the times,
and the current default is fine for even older pcs.

I normally run the PC speaker at 11025Hz and it sounds like the single speaker in my old systems. In fact, the menu in "DeathManager" for Doom/Doom2/Plutonia/TNT/Heretic/Hexen sounds identical to me. I don't play any games that really use the PC speaker beyond Mechwarrior, and it still sounds the same to me also.