Math expression for a developing waveform?

[Sturk200]

Hi all! I have a question about representing waves mathematically. I hope it will make sense.

The waves I've been learning about are of the form F(x-vt), which represents a wave-form, every point of which is translated spatially with velocity v. I am wondering if there is a way to mathematically represent something that might be called a "developing wave." (I don't know if there is a name for this -- probably there is but I'm not aware of it). I am imagining something like a sine function which at time t=0 is defined on the interval, say, x in [0,1], but as time goes on the interval widens. So the function would be like sin[x], x in [0,vt]. This would be like a wave whose individual points do not themselves propagate, but which sort of grows in one direction and perhaps fades from the other end with passing time. Would it even make sense to call this a wave? Is there a mathematical apparatus that might help me to think about this kind of object? Are there physical phenomena that display this kind of form?

Thanks for your help.

 

[Sturk200]

So I just ran a search on "time dependent domains" and found a few interesting studies of wave equations in time dependent domains.

http://www.sciencedirect.com/science/article/pii/0022039679900275

http://www.sciencedirect.com/science/article/pii/0022247X9090230D

However, these are slightly above my level. I have never been exposed to time-dependent domains in my coursework. Does anybody have a textbook-style reference for someone interested in learning about equations in time-dependent domains? Thanks again.

(Or maybe I'm thinking about this the wrong way and there is a way to describe the "developing wave" behavior using some kind of parametric equations? Maybe x=sin(t), y=vt)