What are ways to develop web application for editing audio files in browser?

Your UI will obviously be pretty straightforward using HTML, CSS, and JavaScript. I imagine you will use the canvas tag a lot, perhaps for rendering waveforms, visualizer/eq, etc. As for the audio itself, the JS Web Audio API provides most of the tools you will need. I would code the mixing yourself in a Script Node, for example. You will need to develop a server side app to do authentication and basic file management. The server may occasionally need to process audio itself, especially for transcoding. Not all browsers support all audio codecs, and in fact WAV is the only one supported by Chrome, Safari, Firefox, and IE. You may need to use a service like Zencoder to transcode if you're on a platform like Google App Engine which doesn't allow calling out to command line tools like FFmpeg.