I always wanted to do this, I think Julia is a good language to develop audio plugins and virtual instruments. But its currently missing real time tooling and binding to audio plugin frameworks such as CLAP JUCE VST3

There is a way to compile Julia to webassembly: https://tshort.github.io/WebAssemblyCompiler.jl/stable/examp... , it might be possible for you to run the julia script clientside. I don't know whether FFTW.jl can run though for the spectrogram plot

Another related demo is https://youtu.be/bujv4mxG7mE?si=LMGri56Gzs7mWTN_ , they solve differential equations of a saxophone