Playing around with cosine-based formulas led to some images that seemed very close to Chadni's own diagrams, which can be found in an appendix to his book on Google books.
For a few of these I've uploaded the scripts that produced them.
Chladni image 40 b
Chladni image 41
Chladni image 53
Chladni image 58
Chladni image 63
And of course, many more are possible. Try a few yourself :)