# wave equation

• Jan 3rd 2009, 02:20 AM
James0502
wave equation
I have already shown T(x,t) = (A/sqrt(t))*exp((-x^2)/(4kt))

is a solution of the wave equation

I now need to show int (T(x,t)) dx between infinity and neg infinity is a constant function of T

obvioulsy I cant integrate the function.. I know integrating the wave equation would give me [(t^2)/2] is this enough to say?
• Jan 3rd 2009, 02:34 AM
mr fantastic
Quote:

Originally Posted by James0502
I have already shown T(x,t) = (A/sqrt(t))*exp((-x^2)/(4kt))

is a solution of the wave equation

I now need to show int (T(x,t)) dx between infinity and neg infinity is a constant function of T

obvioulsy I cant integrate the function.. Mr F says: Yes you can. See below.

I know integrating the wave equation would give me [(t^2)/2] is this enough to say?

To find $\displaystyle I = \frac{A}{\sqrt{t}} \int_{-\infty}^{+ \infty} e^{-\frac{x^2}{4 kt}} \, dx$ I suggest making the substitution $\displaystyle u = \frac{x}{2 \sqrt{kt}}$.

Note that $\displaystyle \int_{-\infty}^{+ \infty} e^{-u^2} \, du$ is a famous definite integral. See Gaussian Integral -- from Wolfram MathWorld
• Jan 3rd 2009, 04:22 AM
James0502
Brilliant, thankyou

I have another wave equation problem

http://people.maths.ox.ac.uk/~earl/sheet5b.pdf

question 3

I have verified it is a solution

now applying ICs I get

sum {sin(n.pi.x/l)[A_n] = alpha sin(pi.x/l)}

which I cant see how I would get A_n

many thanks
• Jan 3rd 2009, 06:21 AM
Chris L T521
Quote:

Originally Posted by James0502
Brilliant, thankyou

I have another wave equation problem

http://people.maths.ox.ac.uk/~earl/sheet5b.pdf

question 3

I have verified it is a solution

now applying ICs I get

sum {sin(n.pi.x/l)[A_n] = alpha sin(pi.x/l)}

which I cant see how I would get A_n

many thanks

• Jan 3rd 2009, 08:05 AM
Jester
Quote:

Originally Posted by James0502
I have already shown T(x,t) = (A/sqrt(t))*exp((-x^2)/(4kt))

is a solution of the wave equation

I now need to show int (T(x,t)) dx between infinity and neg infinity is a constant function of T

obvioulsy I cant integrate the function.. I know integrating the wave equation would give me [(t^2)/2] is this enough to say?

You say that the given solution is a solution of the wave equation. What equation do you refer to? When I think wave equation, I think

$\displaystyle T_{tt} = c^2 T_{xx}$

(although I've seen solutions like this of the heat equation $\displaystyle T_t = k T_{xx}$).
• Jan 3rd 2009, 08:12 AM
James0502
erm.. I have dT/dt = k.dT^2/Dx^2
• Jan 3rd 2009, 08:16 AM
Jester
Quote:

Originally Posted by James0502
erm.. I have dT/dt = k.dT^2/Dx^2

That's the heat equation. :)
• Jan 3rd 2009, 08:29 AM
James0502
oops.. sorry - I meant heat equation! (Worried)(Itwasntme)