stringwaves
My question is short and simple. If a damped, travelling wave (say on a string) could be described as
$y\left( x,t\right) =Ae^{-\gamma x}\sin \left( kx-\omega t\right)$
how could/would one determine the decay/dampening constant $\gamma$ mathematically?
Propagation of waves is described by the wave equation $$\frac{d^2u}{dx^2}=\frac{1}{c^2}\frac{d^2u}{dt^2}.$$ But on its own this is not enough: we also need boundary conditions. You are describing what's called fixed boundary conditions or Dirichlet boundary conditions. This amounts to setting $u(x,t)$ to a fixed value on the boundary. It turns out in 1D this is quite easy; in 1D you can always decompose a solution to the wave equation in a left- and right travelling part $$u(x,t)=u_R(x-vt)+u_L(x+vt).$$ Now let's say we want to impose $u=0$ at $x=0$. In that case we get \begin{align} u(0,t)=0&\implies u_R(-vt)=-u_L(+vt)\\ &\implies u_R(-z)=-u_L(z) \end{align} where I used $z$ to remind you that $u_R,u_L$ are just functions of one variable. So we can conclude $u_R,u_L$ have the same shape, but they are reflected in the y-axis (one is upside down) and reflected in the x-axis (they are mirrored around $x=0$). So as a recap because we set $u$ to some fixed value at a particular $x$ the left- and right travelling solutions are mirror images of eachother: the reflected wave is minus the incoming wave. To see this in action look at the following animation.
Here the green line shows the actual solution $u$: it is a sum of the blue and red lines which are the right- and left travelling solutions. The blue and red lines are extended past the domain to make it more clear what is happening. Notice how blue and red work together to keep the boundary point of green at a fixed height.
Best Answer
If you have a string moving in a fluid in which the damping force $F$ is proportional to the transverse velocity of the wave $v$, then you can compute the damping per wavelength that will occur - and from this you can derive the factor $\gamma$ that you are asking about.
A very deep treatment can be found at http://www.people.fas.harvard.edu/~djmorin/waves/transverse.pdf