Proving Finite Speed of Propagation using Energy Methods
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So I'm working through a problem where I am trying to show that the wave equation in one dimension has a finite speed of propagation. Assuming that the initial data, $u(x,0)$ and $u_t(x,0)$ are supported in $(-R,R)$ form some $R>0$, by showing that the energy after some time $t$ outside of the domain $|x|<R+t$ is zero, i.e.
$$frac{1}{2}int_{|x|>R+t}u_x^2+u_t^2dx=0.$$
I am trying to show that it is non-increasing on this domain.
Now, attempting to differentiate this with respect to time I got
$$int_{|x|>R+t}u_xu_{xt}+u_tu_{tt}dx-[u_x^2(-R-t,t)+u_t^2(-R-t,t)]-[u_x^2(R+t,t)+u_t^2(R+t,t)].$$
Using the wave equation we can write that this is equal to
$$int_{|x|>R+t}u_xu_{xt}+u_tu_{xx}dx-[u_x^2(-R-t,t)+u_t^2(-R-t,t)]-[u_x^2(R+t,t)+u_t^2(R+t,t)].$$
Using integration by parts on the integral, I then get
$$u_t(-R-t,t)u_x(-R-t,t)-u_t(R+t,t)u_x(R+t,t)-[u_x^2(-R-t,t)+u_t^2(-R-t,t)]-[u_x^2(R+t,t)+u_t^2(R+t,t)].$$
A few questions:
I don't know if what I did with integration by parts is true as the terminals of the integrals are functions of t, not constants which I am used to. Do boundary terms come out of integration by parts?
How do I proceed in showing that this quantity is non-positive?
wave-equation hyperbolic-equations
asked Jan 10 at 4:33
Anthony SalibAnthony Salib
232
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add a comment |
$begingroup$
So I'm working through a problem where I am trying to show that the wave equation in one dimension has a finite speed of propagation. Assuming that the initial data, $u(x,0)$ and $u_t(x,0)$ are supported in $(-R,R)$ form some $R>0$, by showing that the energy after some time $t$ outside of the domain $|x|<R+t$ is zero, i.e.
$$frac{1}{2}int_{|x|>R+t}u_x^2+u_t^2dx=0.$$
I am trying to show that it is non-increasing on this domain.
Now, attempting to differentiate this with respect to time I got
$$int_{|x|>R+t}u_xu_{xt}+u_tu_{tt}dx-[u_x^2(-R-t,t)+u_t^2(-R-t,t)]-[u_x^2(R+t,t)+u_t^2(R+t,t)].$$
Using the wave equation we can write that this is equal to
$$int_{|x|>R+t}u_xu_{xt}+u_tu_{xx}dx-[u_x^2(-R-t,t)+u_t^2(-R-t,t)]-[u_x^2(R+t,t)+u_t^2(R+t,t)].$$
Using integration by parts on the integral, I then get
$$u_t(-R-t,t)u_x(-R-t,t)-u_t(R+t,t)u_x(R+t,t)-[u_x^2(-R-t,t)+u_t^2(-R-t,t)]-[u_x^2(R+t,t)+u_t^2(R+t,t)].$$
A few questions:
I don't know if what I did with integration by parts is true as the terminals of the integrals are functions of t, not constants which I am used to. Do boundary terms come out of integration by parts?
How do I proceed in showing that this quantity is non-positive?
wave-equation hyperbolic-equations
asked Jan 10 at 4:33
Anthony SalibAnthony Salib
232
$endgroup$
add a comment |
$begingroup$
So I'm working through a problem where I am trying to show that the wave equation in one dimension has a finite speed of propagation. Assuming that the initial data, $u(x,0)$ and $u_t(x,0)$ are supported in $(-R,R)$ form some $R>0$, by showing that the energy after some time $t$ outside of the domain $|x|<R+t$ is zero, i.e.
$$frac{1}{2}int_{|x|>R+t}u_x^2+u_t^2dx=0.$$
I am trying to show that it is non-increasing on this domain.
Now, attempting to differentiate this with respect to time I got
$$int_{|x|>R+t}u_xu_{xt}+u_tu_{tt}dx-[u_x^2(-R-t,t)+u_t^2(-R-t,t)]-[u_x^2(R+t,t)+u_t^2(R+t,t)].$$
Using the wave equation we can write that this is equal to
$$int_{|x|>R+t}u_xu_{xt}+u_tu_{xx}dx-[u_x^2(-R-t,t)+u_t^2(-R-t,t)]-[u_x^2(R+t,t)+u_t^2(R+t,t)].$$
Using integration by parts on the integral, I then get
$$u_t(-R-t,t)u_x(-R-t,t)-u_t(R+t,t)u_x(R+t,t)-[u_x^2(-R-t,t)+u_t^2(-R-t,t)]-[u_x^2(R+t,t)+u_t^2(R+t,t)].$$
A few questions:
I don't know if what I did with integration by parts is true as the terminals of the integrals are functions of t, not constants which I am used to. Do boundary terms come out of integration by parts?
How do I proceed in showing that this quantity is non-positive?
wave-equation hyperbolic-equations
asked Jan 10 at 4:33
Anthony SalibAnthony Salib
232
$endgroup$
So I'm working through a problem where I am trying to show that the wave equation in one dimension has a finite speed of propagation. Assuming that the initial data, $u(x,0)$ and $u_t(x,0)$ are supported in $(-R,R)$ form some $R>0$, by showing that the energy after some time $t$ outside of the domain $|x|<R+t$ is zero, i.e.
$$frac{1}{2}int_{|x|>R+t}u_x^2+u_t^2dx=0.$$
I am trying to show that it is non-increasing on this domain.
Now, attempting to differentiate this with respect to time I got
$$int_{|x|>R+t}u_xu_{xt}+u_tu_{tt}dx-[u_x^2(-R-t,t)+u_t^2(-R-t,t)]-[u_x^2(R+t,t)+u_t^2(R+t,t)].$$
Using the wave equation we can write that this is equal to
$$int_{|x|>R+t}u_xu_{xt}+u_tu_{xx}dx-[u_x^2(-R-t,t)+u_t^2(-R-t,t)]-[u_x^2(R+t,t)+u_t^2(R+t,t)].$$
Using integration by parts on the integral, I then get
$$u_t(-R-t,t)u_x(-R-t,t)-u_t(R+t,t)u_x(R+t,t)-[u_x^2(-R-t,t)+u_t^2(-R-t,t)]-[u_x^2(R+t,t)+u_t^2(R+t,t)].$$
A few questions:
I don't know if what I did with integration by parts is true as the terminals of the integrals are functions of t, not constants which I am used to. Do boundary terms come out of integration by parts?
How do I proceed in showing that this quantity is non-positive?
wave-equation hyperbolic-equations
wave-equation hyperbolic-equations
asked Jan 10 at 4:33
Anthony SalibAnthony Salib
232
asked Jan 10 at 4:33
Anthony SalibAnthony Salib
232
asked Jan 10 at 4:33
Anthony SalibAnthony Salib
232
asked Jan 10 at 4:33
Anthony SalibAnthony Salib
232
asked Jan 10 at 4:33
Anthony SalibAnthony Salib
232
232
add a comment |
add a comment |
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