hey I was just wondering what would happen if you had a length of string with varying densitities with one end fixed to a wall and you sending a pulse down it. Would the waves travel faster along some bits or change amplitude at all or do something i havent mentioned? Can anyone enlighten me :) thanks in advance
As far as I remember the elastic wave speed in a solid is a function of its density: [tex]c=\sqrt{E/\rho}[/tex] being [tex]E[/tex] the Young Modulus.
The wave would do a lot of funky things. As Clausius pointed out, wavespeed depends on the density of the solid, so the wave would change speed. However, the impedance (Z) of the wave is [itex] \rho \cdot c [/itex], so if [itex] \rho [/itex] changes, so does Z. This is important, because anytime the medium the wave is travelling along changes impedance, there is a reflected wave. So if you sent a pulse down the line, it's velocity would alter predictably with density, according to above eqn, and it's amplitude would decrease, because some of the energy would be reflected. Did you mean that the density is a continuous function of position, or that there are a bunch of different kinds of string with different p's tied together? The latter isn't too hard, you can find the necessary equations for reflection/transmission at a boundary in string in any intro waves text, and just apply them to each boundary separately. To solve the former I think you'd have to model the string as an infinite amount of bits of string with length dl, density [itex]\rho[/itex], and take the limit as dl goes to 0. You'd wind up with an infinite series, but I think it might convert to an integral due to the above limit. Interesting problem.
Because the wave equation remains linear, then you will have another modes, which will not be simple harmonic functions. These new waves will have different, but stable shapes, they can form standing waves as well. But they can not be characterized by a single velocity. Now, your velocity depends on the subject of propagation. For usual harmonic waves everything (energy, phase, impulse...) propagates with the same speed. Inis case this will not be so.