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Boundary conditions
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The solution to the wave equation
| \displaystyle\frac{\partial^2 u}{\partial t^2}= c ^2\displaystyle\frac{\partial^2 u}{\partial x^2} |
is of the form
| z = z_0 e^{ i( k s - \omega t )} |
but it must satisfy the conditions of free or fixed edge. In the edge case
- free wave can move but has no support so the stress and thus the deformation must be zero.
- fixed the wave cannot move but it can generate tension and with it deformation
Graphically, we have
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Standing waves
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The equation
It means that there are two solutions.
\omega = \pm c k
so the solution is of the form
x_0 e^{ikx}(e^{i\omega t)}+e^{-i\omega t})
or with Euler's relation the real part is
2x_0 \cos(kx)\cos(\omega t)
In other words, a function of the position oscillates in the same place without moving:
This is called a standing wave.
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Solution modes
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Las condiciones de borde permiten soluciones que tienen mas nodos como se ve en el ejemplo fijo-libre
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Model
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Calculations
Variable 
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Target : Equation To be used 
Equations
z = z_0 e^{ i( k s - \omega t )}
z = z_0 * exp( %i *( k * s - omega * t ))
\displaystyle\frac{\partial^2 u}{\partial t^2}= c ^2\displaystyle\frac{\partial^2 u}{\partial x^2}
@DIF( u , t , 2) = c ^2*@DIF( u , x , 2)
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Wave equation
Equation
La ecuación de movimiento
| \displaystyle\frac{\partial^2 u_i}{\partial t^2}=\displaystyle\frac{ E }{ \rho }\displaystyle\frac{\partial^2 u_i}{\partial x^2} |
con la relación
| c ^2 = \displaystyle\frac{ E }{ \rho } |
representa la ecuación de onda del solido
| \displaystyle\frac{\partial^2 u}{\partial t^2}= c ^2\displaystyle\frac{\partial^2 u}{\partial x^2} |
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General solution of the wave equation
Equation
The general solution of the wave equation
| \displaystyle\frac{\partial^2 u}{\partial t^2}= c ^2\displaystyle\frac{\partial^2 u}{\partial x^2} |
can be written in the complex space as
| z = z_0 e^{ i( k s - \omega t )} |
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