Does d'Alembert's formula hold if the wave speed is not constant?
For example, if the speed c in the link is not a constant, but instead a function of the coordinate x. Is the formula still valid?
I think not, because the initial conditions are probably going to be in the form u(t=0,x)=f(x) and \frac{∂u}{∂t}(t=0,x)=g(x), because it is an infinite string (no boundary conditions). But if I we separate variables (i.e. u(x,t)=X(x)*T(t)), we have to put a single value and not a function for...
The wave equation:
(∂^2 u)/(∂t)^2 = c^2 (∂^2 u)/(∂x)^2,
where u is displacement of the string and c is the wave speed. c is not a constant, because c^2 = T/S, where T is the string tension and S is the cross-sectional area and is dependant on x. S[x]=S1+S2*Sin[x].
(Sorry about the...
Homework Statement
The cross-section of a long string (string along the x axis) is not constant, but it changes wit the coordinate x sinusoidally. Explore how a wave, caused with a short stroke, spreads through the string.
Homework Equations
Relevant is the one-dimensional wave...