Solution by Fourier Metod to Elastic Wave Equation
$begingroup$
I'm trying to find a solution using the Fourier transform and its inverse to initial value problems of elastic wave equations in two dimensional periodic media.
begin{equation}
begin{cases}
rho u_{tt}-text{div}Cepsilon(u)=f(x,t),quad&\
u(x,0)=g(x),quad u_t(x,0)=h(x),quad&\
text{$u(cdot,t)$ is periodic}quad&
end{cases}
end{equation}
Here $rho>0$ is a density (constant), $u=(u_1,u_2)$ is a displacement, $C$ is the elasticity, i.e., $C_{ijkl}=lambdadelta_{ij}delta_{kl}+mu(delta_{ik}delta_{jl}+delta_{il}delta_{jk})$, where $lambda,mu>0$ are Lame constants, and $epsilon$ is a strain that I assume $epsilon(u)=(nabla u+nabla u^intercal)/2$. I assume that all functions are well-behaved (ex. sufficiently smooth).
I often see how to derive a solution when $f$ is special (eg, Dirac delta) or when there is no initial condition, but I have not seen in more general cases.
First, I took Fourier transform wrt the space and reduced the problem to solving a certain second order ODE, but I couldn't come up with any ideas to solve it. I also attempted to take Fourier transform wrt time, not space, but I didn't handle the initial condition.
I don't make sure whether or not this problem can be solved, but I would be pleased if you could give us advice. Thank you.
calculus pde wave-equation
asked Dec 11 '18 at 2:28
useruser
18417
$endgroup$
add a comment |
$begingroup$
I'm trying to find a solution using the Fourier transform and its inverse to initial value problems of elastic wave equations in two dimensional periodic media.
begin{equation}
begin{cases}
rho u_{tt}-text{div}Cepsilon(u)=f(x,t),quad&\
u(x,0)=g(x),quad u_t(x,0)=h(x),quad&\
text{$u(cdot,t)$ is periodic}quad&
end{cases}
end{equation}
Here $rho>0$ is a density (constant), $u=(u_1,u_2)$ is a displacement, $C$ is the elasticity, i.e., $C_{ijkl}=lambdadelta_{ij}delta_{kl}+mu(delta_{ik}delta_{jl}+delta_{il}delta_{jk})$, where $lambda,mu>0$ are Lame constants, and $epsilon$ is a strain that I assume $epsilon(u)=(nabla u+nabla u^intercal)/2$. I assume that all functions are well-behaved (ex. sufficiently smooth).
I often see how to derive a solution when $f$ is special (eg, Dirac delta) or when there is no initial condition, but I have not seen in more general cases.
First, I took Fourier transform wrt the space and reduced the problem to solving a certain second order ODE, but I couldn't come up with any ideas to solve it. I also attempted to take Fourier transform wrt time, not space, but I didn't handle the initial condition.
I don't make sure whether or not this problem can be solved, but I would be pleased if you could give us advice. Thank you.
calculus pde wave-equation
asked Dec 11 '18 at 2:28
useruser
18417
$endgroup$
add a comment |
$begingroup$
I'm trying to find a solution using the Fourier transform and its inverse to initial value problems of elastic wave equations in two dimensional periodic media.
begin{equation}
begin{cases}
rho u_{tt}-text{div}Cepsilon(u)=f(x,t),quad&\
u(x,0)=g(x),quad u_t(x,0)=h(x),quad&\
text{$u(cdot,t)$ is periodic}quad&
end{cases}
end{equation}
Here $rho>0$ is a density (constant), $u=(u_1,u_2)$ is a displacement, $C$ is the elasticity, i.e., $C_{ijkl}=lambdadelta_{ij}delta_{kl}+mu(delta_{ik}delta_{jl}+delta_{il}delta_{jk})$, where $lambda,mu>0$ are Lame constants, and $epsilon$ is a strain that I assume $epsilon(u)=(nabla u+nabla u^intercal)/2$. I assume that all functions are well-behaved (ex. sufficiently smooth).
I often see how to derive a solution when $f$ is special (eg, Dirac delta) or when there is no initial condition, but I have not seen in more general cases.
First, I took Fourier transform wrt the space and reduced the problem to solving a certain second order ODE, but I couldn't come up with any ideas to solve it. I also attempted to take Fourier transform wrt time, not space, but I didn't handle the initial condition.
I don't make sure whether or not this problem can be solved, but I would be pleased if you could give us advice. Thank you.
calculus pde wave-equation
asked Dec 11 '18 at 2:28
useruser
18417
$endgroup$
I'm trying to find a solution using the Fourier transform and its inverse to initial value problems of elastic wave equations in two dimensional periodic media.
begin{equation}
begin{cases}
rho u_{tt}-text{div}Cepsilon(u)=f(x,t),quad&\
u(x,0)=g(x),quad u_t(x,0)=h(x),quad&\
text{$u(cdot,t)$ is periodic}quad&
end{cases}
end{equation}
Here $rho>0$ is a density (constant), $u=(u_1,u_2)$ is a displacement, $C$ is the elasticity, i.e., $C_{ijkl}=lambdadelta_{ij}delta_{kl}+mu(delta_{ik}delta_{jl}+delta_{il}delta_{jk})$, where $lambda,mu>0$ are Lame constants, and $epsilon$ is a strain that I assume $epsilon(u)=(nabla u+nabla u^intercal)/2$. I assume that all functions are well-behaved (ex. sufficiently smooth).
I often see how to derive a solution when $f$ is special (eg, Dirac delta) or when there is no initial condition, but I have not seen in more general cases.
First, I took Fourier transform wrt the space and reduced the problem to solving a certain second order ODE, but I couldn't come up with any ideas to solve it. I also attempted to take Fourier transform wrt time, not space, but I didn't handle the initial condition.
I don't make sure whether or not this problem can be solved, but I would be pleased if you could give us advice. Thank you.
calculus pde wave-equation
calculus pde wave-equation
asked Dec 11 '18 at 2:28
useruser
18417
asked Dec 11 '18 at 2:28
useruser
18417
asked Dec 11 '18 at 2:28
useruser
18417
asked Dec 11 '18 at 2:28
useruser
18417
asked Dec 11 '18 at 2:28
useruser
18417
18417
add a comment |
add a comment |
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