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Scalar Wave Equation

The scalar wave equation is given by 2ux2=1c22ut2\displaystyle \frac{\partial ^2 u}{\partial x^2} = \frac{1}{c^2}\displaystyle \frac{\partial ^2 u}{\partial t^2}


Show by substitution (NOT separation of variables) that
u = f(x ct) + g(x + ct)


is a general solution to the equation.


This is part of a maths module i'm taking, hence why it's in the maths section and not Physics.

I have started off defining u(x,t) = f(x-ct) and subbing it into the equation, but not sure where to go from there.
Reply 1
Original post by SwimGood
The scalar wave equation is given by 2ux2=1c22ut2\displaystyle \frac{\partial ^2 u}{\partial x^2} = \frac{1}{c^2}\displaystyle \frac{\partial ^2 u}{\partial t^2}


Show by substitution (NOT separation of variables) that
u = f(x ct) + g(x + ct)


is a general solution to the equation.


This is part of a maths module i'm taking, hence why it's in the maths section and not Physics.

I have started off defining u(x,t) = f(x-ct) and subbing it into the equation, but not sure where to go from there.


write the auxillary equation for the PDE
Original post by SwimGood
I have started off defining u(x,t) = f(x-ct) and subbing it into the equation, but not sure where to go from there.
The obvious thing to do is to find 2ux2\dfrac{\partial^2 u}{\partial x^2} and 2ut2\dfrac{\partial^2 u}{\partial t^2}, and it turns out that if you do so everything drops out nicely.
Reply 3
Original post by SwimGood
The scalar wave equation is given by 2ux2=1c22ut2\displaystyle \frac{\partial ^2 u}{\partial x^2} = \frac{1}{c^2}\displaystyle \frac{\partial ^2 u}{\partial t^2}


Show by substitution (NOT separation of variables) that
u = f(x ct) + g(x + ct)


is a general solution to the equation.


This is part of a maths module i'm taking, hence why it's in the maths section and not Physics.

I have started off defining u(x,t) = f(x-ct) and subbing it into the equation, but not sure where to go from there.


I missed that you only need to verify (rather than solve) so it is just a simple differentiation twice
Reply 4
Original post by TeeEm
I missed that you only need to verify (rather than solve) so it is just a simple differentiation twice


I'm not following sorry.


d/dt = -c d/dz is where I'm at I'm getting confused...its early.
Reply 5
Original post by SwimGood
I'm not following sorry.


d/dt = -c d/dz is where I'm at I'm getting confused...its early.


I will be back around 15.00
Reply 6
Original post by SwimGood
I'm not following sorry.


d/dt = -c d/dz is where I'm at I'm getting confused...its early.


I am assuming you just want to verify rather than prove or derive.

The general solution of this PDE for any kind of wave (stationary or progressive is

u(x,t) = F(x-ct) +G(x+ct)

where the type of F and G determine the shape of the wave.

it best to deal with each part of the solution separately.

e.g

u = F(x-ct)
ux = F'(x-ct)
uxx = F''(x-ct)

do the same with time
u = F(x-ct)
ut = -c ....
utt = -c(-c) ...

verification is complete for F

repeat with G(x+ct)

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