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Numerical Analysis of PDEs: A Finite Difference Method for the Wave Equation, Damped Wave Equation and Free Vibration of Beam

Exact solution

Exact solution of Wave Equation

It is convenient to rescale equations to reduce to a problem with dimensionless parameters.

Method of separation of variables is used to find the solution of the equation. Solution can be written as the product of two functions, one a function of x alone, the other a function of t alone.

Initial vertical velocity of the string defined as g(x), equal to zero to reduce complexity of the problem.

g(x)=0 => bn=0

General solution:

Now it is possible to take frequencies in dimensionless form and relate them back to the natural frequencies of the system with familiar units of inverse time.

The solution is a superposition of many modes, where each mode oscillates at different frequency. The first factor , shows how mode vibrates with time. The time that it takes for the n mode to go through one cycle of vibration is just the period of the cosine function.

Natural frequencies of the vibrating string are

Notice that the difference between any two consecutive harmonics is the fundamental frequency.

Specific solution with Initial conditions

The coefficients in the series can be computed by integration (Exact solution) or approximate coefficients can be obtained using FFT. The latter approach is chosen for comparison to other methods. This Fourier Approximation will be used for solution of Wave Equation and Beam vibration. Implementation of Fourier approximation is included according to H.B. Wilson and L.H. Turcotte "Advanced Mathematics and Mechanics Applications Using MATLAB" book.

Exact solution of Damped wave equation

This equation can be solved as a Fourier series

Free Vibration of a Beam

Method of separation of variables is used:

u(x,t)=X(x)T(t)

So the modal solution is:

Boundary Conditions

1. Pinned Beam

Zero deflection and zero moment at x=0 and x=1.

Now it is possible to take frequencies in dimensionless form and relate them back to the natural frequencies of the system with familiar units of inverse time.

It have been already seen that for an ideal string the frequencies of the standing waves are spaced by fundamental frequency, independent of the frequency. However, for the free vibration beam this separation increases as we move to higher frequencies.
The general solution can be used to solve an initial value problem in the same way in which the initial value problem for ideal string was solved.