Solving 1D Wave Equation w/ Initial Values

In summary: This will allow you to find the solution u(x,t) for the given initial values and boundary conditions. In summary, for the first problem, the even extension of sin(πx) is sin(πx-π/2) and for the second problem, the even extension of sin(x) is cos(x-π/2).
  • #1
stunner5000pt
1,461
2
given the initial boundary value problem
[itex] u_{tt} - u_{xx} = 0 [/itex], 0<x<1, t>0,
[itex] u(x,0) =1 [/itex], [itex] 0\leq x \leq 1 [/itex],
[itex] u_{t}(x,0) = \sin(\pi x) [/itex], [itex] 0\leq x \leq 1 [/itex],
[itex] u_{x} (0,t) = 0 [/itex], [itex] t \geq 0 [/itex],
[itex] u_{x} (1,t) = 0 [/itex].

find u(0.5,1). Where u is d'Alembert's solution for the 1D wave equation
well f(x) = 0, and g(x) = sin (pi x)
but i need to even extend g(x) since the wave is not fixed at its endpoints. WOuld the even extension of sin (pi x) simply be a shift to the right by pi/2 that is [itex] \sin(\pi x - \frac{\pi}{2}) [/itex]


Another question i have is
Find u(x,t) corresponding to initial values f(x) = sin x, g(x) = 0, when u(0,t) =0 and [tex] u_{x} (\frac{\pi}{2} ,t) = [/itex] , c=1.
Nowi know i have to extend f as an evenfunction about x = pi/2 only (right?). But how would one go aboutenxtending sine as an evne function? do i make sine into the cosine function and then shift that by pi/2 to the right? that is cos (x - pi/2).

Is this correct? Please help!
 
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  • #2
Yes, your idea for the first problem is correct. The even extension of sin(πx) would be sin(πx-π/2). For the second problem, you are also correct that you need to extend f(x) as an even function about x=π/2. To do this, you can use the identity sin(x−π/2)=cos(x) and then extend the cosine function as an even function about x=π/2. So, the even extension of sin(x) should be cos(x-π/2).
 

1. What is the 1D wave equation and how is it solved?

The 1D wave equation is a mathematical model that describes the behavior of a wave propagating through a medium in one dimension. It is commonly used in physics and engineering to study various phenomena such as sound, light, and water waves. To solve the 1D wave equation with initial values, one must use a technique called separation of variables, where the equation is separated into two ordinary differential equations that can be solved independently.

2. Why are initial values important in solving the 1D wave equation?

Initial values, also known as boundary conditions, are crucial in solving the 1D wave equation as they provide the starting point for the wave to propagate through the medium. These values determine the amplitude, velocity, and direction of the wave at specific points in the medium, and without them, it is impossible to accurately solve the equation.

3. What are some common applications of the 1D wave equation with initial values?

The 1D wave equation with initial values has numerous practical applications, including predicting the behavior of seismic waves in earthquake studies, analyzing sound waves in musical instruments, and understanding the transmission of electromagnetic waves in communication systems. It also has applications in other fields, such as fluid dynamics and quantum mechanics.

4. What are some challenges in solving the 1D wave equation with initial values?

One of the main challenges in solving the 1D wave equation with initial values is finding an appropriate mathematical model that accurately represents the physical system being studied. This often requires simplifications and assumptions, which may not always reflect the real-world situation. Additionally, solving the equation can be computationally intensive and may require advanced mathematical techniques.

5. How does solving the 1D wave equation with initial values contribute to scientific research?

Solving the 1D wave equation with initial values allows scientists to understand and predict the behavior of waves in various physical systems. This information is essential for developing new technologies, improving existing ones, and advancing our understanding of the natural world. It also allows researchers to make predictions and test hypotheses, leading to new discoveries and advancements in various scientific fields.

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