# Fourier series solution of wave equation

## Homework Statement

Suppose a horizontally stretched string is heavy enough for the effects of gravity to be significant, so that the wave equation must be replaced by ##u_{tt} = c^2u_{xx} - g## where ##g## is the acceleration due to gravity. The boundary conditions are ##u(0,t) = u(l,t) = 0##.

Find the steady state solution ##\phi(x)##

Suppose that initially ##u(x,0) = u_{t}(x,0) = 0##. Find the solution ##u(x,t)## as a Fourier series.

## The Attempt at a Solution

The steady state differential equation is ##u_{xx} = g/c^2## which has solutions ##\frac{gx^2}{2c^2} -\frac{gxl}{2c^2} ##. Here is where I have problems if I try to find solutions to the homogeneous differential equation with the same boundary conditions and initial conditions replaced with ##-\frac{gx^2}{2c^2} +\frac{gxl}{2c^2}##. It would be a sum of ##\sin(\pi n x / l)## terms due to the boundary conditions, but multiplied by a sum of ##\sin(\pi n c t / l) ## and ## \cos ( \pi n c t / l)## terms. The book outlines a way to find the coefficients by expanding the initial conditions to the Foerier series and equating, but the example given was with the heat equation which only had one exponential term that become 1 at ##t = 0##.

Edit: writing this out has made me realise I can that the ##\sin ( \pi n c t / l)## coefficients are zero.

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