User:
Boundary conditions
Description
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
ID:(14186, 0)
Standing waves
Description
The equation
| $$ |
It means that there are two solutions.
$\omega = \pm c k$
so the solution is of the form
$x_0 e^{ikx}(e^{i\omega t)}+e^{-i\omega t})$
or with Euler's relation the real part is
$2x_0 \cos(kx)\cos(\omega t)$
In other words, a function of the position oscillates in the same place without moving:
This is called a standing wave.
ID:(14205, 0)
Solution modes
Description
Las condiciones de borde permiten soluciones que tienen mas nodos como se ve en el ejemplo fijo-libre
ID:(14190, 0)
Standing waves
Description
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then, select the variable: 
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Equations
Examples
(ID 15572)
(ID 14182)
The solution to the wave equation
| $\displaystyle\frac{\partial^2 u}{\partial t^2}= c ^2\displaystyle\frac{\partial^2 u}{\partial x^2}$ |
is of the form
| $ z = z_0 e^{ i( k s - \omega t )}$ |
but it must satisfy the conditions of free or fixed edge. In the edge case
- free wave can move but has no support so the stress and thus the deformation must be zero.
- fixed the wave cannot move but it can generate tension and with it deformation
Graphically, we have
(ID 14186)
The equation
| $$ |
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.
(ID 14205)
Las condiciones de borde permiten soluciones que tienen mas nodos como se ve en el ejemplo fijo-libre
(ID 14190)
(ID 15583)
La ecuaci n 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 relaci n
| $ c ^2 = \displaystyle\frac{ E }{ \rho }$ |
representa la ecuaci n de onda del solido
| $\displaystyle\frac{\partial^2 u}{\partial t^2}= c ^2\displaystyle\frac{\partial^2 u}{\partial x^2}$ |
(ID 14180)
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 )}$ |
(ID 14187)
ID:(1888, 0)