The D'Alembert solution for an $n+1$ dimensional wave equation

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I have been solving the wave equation in the upper half-space $mathbbR^n+1_+$ $$
D^2_ttu(x,t)=nabla^2u, quad u(x,0)=f(x), quad D_tu(x,0)=g(x), quad xin mathbbR^n, quad t>0
$$
using Fourier transform and obtained the following:



$$
hatu(xi,t)=hatf(xi)cosxi+
hatg(xi)fracsinxixi, quad xiin mathbbR^n, quad t>0.
$$
Hence, the inversion formula gives for $(x,t)in mathbbR^n+1_+$,
$$
u(x,t)=frac1(2pi)^nint_mathbbR^nhatf(xi)cos(e^ixicdot xdxi+
frac1(2pi)^nint_mathbbR^nhatg(xi)fracsin(xie^ixicdot xdxi=u_1+u_2.
$$
I consider first $$u_2=frac1(2pi)^nint_mathbbR^nhatg(xi)fracsin(xie^ixicdot xdxi=(gast T)(x).
$$
Where $$T(x)=(hatT(xi))^vee=left[left(fracsin(xiright)^veeright]^wedge.$$



For $n=1$ that is, in one dimension;



I consider $frac12chi_[-t,t]=left(fracsin(xiright)^vee$ from the fact that $hatchi_[-t,t]=int_-t^t1cdot e^-ixcdotxidx$.



  1. My first problem is that I am failing to get an expression of D'Alembert solution

$$u(x,t)=F(x+t)+ G(x-t).$$



  1. My second problem is that I want to see how the solution for $n=2$, that is, in $2$-dimension is obtained. Especially, on how to evaluate along the boundary of a circle.



fourier-analysis harmonic-analysis




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    I have been solving the wave equation in the upper half-space $mathbbR^n+1_+$ $$
    D^2_ttu(x,t)=nabla^2u, quad u(x,0)=f(x), quad D_tu(x,0)=g(x), quad xin mathbbR^n, quad t>0
    $$
    using Fourier transform and obtained the following:



    $$
    hatu(xi,t)=hatf(xi)cosxi+
    hatg(xi)fracsinxixi, quad xiin mathbbR^n, quad t>0.
    $$
    Hence, the inversion formula gives for $(x,t)in mathbbR^n+1_+$,
    $$
    u(x,t)=frac1(2pi)^nint_mathbbR^nhatf(xi)cos(e^ixicdot xdxi+
    frac1(2pi)^nint_mathbbR^nhatg(xi)fracsin(xie^ixicdot xdxi=u_1+u_2.
    $$
    I consider first $$u_2=frac1(2pi)^nint_mathbbR^nhatg(xi)fracsin(xie^ixicdot xdxi=(gast T)(x).
    $$
    Where $$T(x)=(hatT(xi))^vee=left[left(fracsin(xiright)^veeright]^wedge.$$



    For $n=1$ that is, in one dimension;



    I consider $frac12chi_[-t,t]=left(fracsin(xiright)^vee$ from the fact that $hatchi_[-t,t]=int_-t^t1cdot e^-ixcdotxidx$.



    1. My first problem is that I am failing to get an expression of D'Alembert solution

    $$u(x,t)=F(x+t)+ G(x-t).$$



    1. My second problem is that I want to see how the solution for $n=2$, that is, in $2$-dimension is obtained. Especially, on how to evaluate along the boundary of a circle.



    fourier-analysis harmonic-analysis




    share|cite|improve this question























      up vote
      0
      down vote

      favorite
      2









      up vote
      0
      down vote

      favorite
      2






      2





      I have been solving the wave equation in the upper half-space $mathbbR^n+1_+$ $$
      D^2_ttu(x,t)=nabla^2u, quad u(x,0)=f(x), quad D_tu(x,0)=g(x), quad xin mathbbR^n, quad t>0
      $$
      using Fourier transform and obtained the following:



      $$
      hatu(xi,t)=hatf(xi)cosxi+
      hatg(xi)fracsinxixi, quad xiin mathbbR^n, quad t>0.
      $$
      Hence, the inversion formula gives for $(x,t)in mathbbR^n+1_+$,
      $$
      u(x,t)=frac1(2pi)^nint_mathbbR^nhatf(xi)cos(e^ixicdot xdxi+
      frac1(2pi)^nint_mathbbR^nhatg(xi)fracsin(xie^ixicdot xdxi=u_1+u_2.
      $$
      I consider first $$u_2=frac1(2pi)^nint_mathbbR^nhatg(xi)fracsin(xie^ixicdot xdxi=(gast T)(x).
      $$
      Where $$T(x)=(hatT(xi))^vee=left[left(fracsin(xiright)^veeright]^wedge.$$



      For $n=1$ that is, in one dimension;



      I consider $frac12chi_[-t,t]=left(fracsin(xiright)^vee$ from the fact that $hatchi_[-t,t]=int_-t^t1cdot e^-ixcdotxidx$.



      1. My first problem is that I am failing to get an expression of D'Alembert solution

      $$u(x,t)=F(x+t)+ G(x-t).$$



      1. My second problem is that I want to see how the solution for $n=2$, that is, in $2$-dimension is obtained. Especially, on how to evaluate along the boundary of a circle.



      fourier-analysis harmonic-analysis




      share|cite|improve this question













      I have been solving the wave equation in the upper half-space $mathbbR^n+1_+$ $$
      D^2_ttu(x,t)=nabla^2u, quad u(x,0)=f(x), quad D_tu(x,0)=g(x), quad xin mathbbR^n, quad t>0
      $$
      using Fourier transform and obtained the following:



      $$
      hatu(xi,t)=hatf(xi)cosxi+
      hatg(xi)fracsinxixi, quad xiin mathbbR^n, quad t>0.
      $$
      Hence, the inversion formula gives for $(x,t)in mathbbR^n+1_+$,
      $$
      u(x,t)=frac1(2pi)^nint_mathbbR^nhatf(xi)cos(e^ixicdot xdxi+
      frac1(2pi)^nint_mathbbR^nhatg(xi)fracsin(xie^ixicdot xdxi=u_1+u_2.
      $$
      I consider first $$u_2=frac1(2pi)^nint_mathbbR^nhatg(xi)fracsin(xie^ixicdot xdxi=(gast T)(x).
      $$
      Where $$T(x)=(hatT(xi))^vee=left[left(fracsin(xiright)^veeright]^wedge.$$



      For $n=1$ that is, in one dimension;



      I consider $frac12chi_[-t,t]=left(fracsin(xiright)^vee$ from the fact that $hatchi_[-t,t]=int_-t^t1cdot e^-ixcdotxidx$.



      1. My first problem is that I am failing to get an expression of D'Alembert solution

      $$u(x,t)=F(x+t)+ G(x-t).$$



      1. My second problem is that I want to see how the solution for $n=2$, that is, in $2$-dimension is obtained. Especially, on how to evaluate along the boundary of a circle.




      fourier-analysis harmonic-analysis





      share|cite|improve this question












      share|cite|improve this question




      share|cite|improve this question








      edited Jul 31 at 16:41









      Davide Morgante

      1,654220




      1,654220









      asked Jul 31 at 16:20









      Sulayman

      1807




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