One-dimensional wave equation with non-constant speed

[j5rp]

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 equation, where the wave speed c is not a constant (i.e. c=√T/ρS, where T is the string tension, ρ is the density of the string, and S is the cross-section).

The cross-section:
S=S1+S2*Sin[x]

The Attempt at a Solution

I thought about using Laplace transformation so that I get an ordinary differential equation. I also have trouble with the initial conditions, I don't know what is meant by short stroke, or if it really matters.

 

[haruspex]

I interpret 'short stroke' as simply meaning it is a small perturbation.
Can you write down the wave equation?

 

[j5rp]

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 dependent on x. S[x]=S1+S2*Sin[x].

(Sorry about the formatting. The ∂ stands for derivative.)

 

[haruspex]

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 dependent on x. S[x]=S1+S2*Sin[x].

(Sorry about the formatting. The ∂ stands for derivative.)

In LaTeX: $\frac{\partial^2 u}{\partial t^2} = c^2\frac{\partial^2 u}{\partial x^2}$.
So, plugging in the expression for c(x), can you apply e.g. separation of variables?

 

[j5rp]

So, plugging in the expression for c(x), can you apply e.g. separation of variables?

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 initial conditions (example: X(0)=value, instead of u(t=0,x)= function).

I was wandering if d'Alembert's formula applies if speed of propagation (also c in the link) is not constant?