Question

Show that the wave equation a^2uxx = utt can be reduced to the form u xi×ita by the change of variable xi=x-at, ita = x+at.

Answer 1

Answer on Question #66343 – Math – Differential Equations

Question

Show that the wave equation

$$a ^ {2} u _ {x x} = u _ {t t}$$

can be reduced to the form

$$u _ {\xi \eta} = 0$$

by the change of variable

$$\xi = x - a t, \eta = x + a t.$$

Solution

We have the differential equation

$$a ^ {2} u _ {x x} = u _ {t t}$$

Change the variables $x,t$ by $\xi ,\eta$

$$\xi = x - a t, \qquad \eta = x + a t$$

Note that

$$\xi_ {x} = 1, \qquad \eta_ {x} = 1, \qquad \xi_ {t} = - a, \qquad \eta_ {t} = a$$

Find partial derivatives which are into the equation

$$u _ {x} = u _ {\xi} \xi_ {x} + u _ {\eta} \eta_ {x} = u _ {\xi} + u _ {\eta}$$ $$u _ {t} = u _ {\xi} \xi_ {t} + u _ {\eta} \eta_ {t} = - a u _ {\xi} + a u _ {\eta}$$ $$u _ {x x} = \left(u _ {\xi} + u _ {\eta}\right) _ {\xi} \xi_ {x} + \left(u _ {\xi} + u _ {\eta}\right) _ {\eta} \eta_ {x} = u _ {\xi \xi} + u _ {\eta \xi} + u _ {\xi \eta} + u _ {\eta \eta}$$ $$= u _ {\xi \xi} + 2 u _ {\xi \eta} + u _ {\eta \eta}$$ $$\begin{array}{l} u _ {t t} = \left(- a u _ {\xi} + a u _ {\eta}\right) _ {\xi} \xi_ {t} + \left(- a u _ {\xi} + a u _ {\eta}\right) _ {\xi} \eta_ {t} \\ = a ^ {2} u _ {\xi \xi} - a ^ {2} u _ {\eta \xi} - a ^ {2} u _ {\xi \eta} + a ^ {2} u _ {\eta \eta} = = a ^ {2} u _ {\xi \xi} - 2 a ^ {2} u _ {\xi \eta} + a ^ {2} u _ {\eta \eta} \\ \end{array}$$

Substitute the derivatives into the equation

$$a ^ {2} \left(u _ {\xi \xi} + 2 u _ {\xi \eta} + u _ {\eta \eta}\right) = a ^ {2} u _ {\xi \xi} - 2 a ^ {2} u _ {\xi \eta} + a ^ {2} u _ {\eta \eta}$$

Then we get

$$4 a ^ {2} u _ {\xi \eta} = 0$$

or

$$u _ {\xi \eta} = 0$$

Answer: the wave equation $a^2 u_{xx} = u_{tt}$ can be reduced to the form $u_{\xi \eta} = 0$ by the change of variables $\xi = x - at, \eta = x + at$ .

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