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Is the wave equation a continuous analytical PDE?
The wave equation as shown by (eq. 1) is a continuous analytical PDE, in which x can take infinite values between 0 and 1, similarly t can take infinite values greater than zero.
Is the wave equation a second order partial differential equation?
The wave equation describes the propagation of waves and it is a second order partial differential equation (PDE) that involves second derivatives in space and time. In the one dimensional wave equation, there is only one independent variable in space.
How to find a solution to the wave equation?
To find a unique solution, we need to impose initial conditions (IC) and boundary conditions (BC). To impose Initial conditions, we define the solution u at the initial time t=0 for every position x. Since the wave equation has 2 partial derivatives in time, we need to define not only the displacement but also its derivative respect to time.
How is Taylor series used to approximate the wave equation?
To approximate the wave equation (eq. 1), we will use Taylor series expansion. Taylor series is a way to approximate the value of a function at a given point by using the value it takes at a nearby point. In this case we use Taylor series to approximate the displacement u at x +/- Δx knowing its value at x.
How to solve the wave equation by numerical methods?
To solve the wave equation by numerical methods, in this case finite difference, we need to take discrete values of x and t : For instance we can take nx points for x and nt points for t, where nx and nt are positive integer numbers. Then the values that x and t take are: where Δx = L/ (nx-1) and Δt = T/ (nt-1).
Is the wave equation a partial differential equation?
The wave equation describes the propagation of waves and it is a second order partial differential equation (PDE) that involves second derivatives in space and time.
Which is the independent variable in the wave equation?
In the one dimensional wave equation, there is only one independent variable in space. In this case we assume that x is the independent variable in space in the horizontal direction. Additionally, the wave equation also depends on time t.