PX275 - G5 - method of separation of variables

assuming that solutions can be found of the form $u (x, t) = X (x) Y (t)$
so, assuming that the dependence of individual variables can be separated

\begin{aligned} \frac{\partial^{2} X Y}{\partial t^{2}} & = c^{2} \frac{\partial^{2} X Y}{\partial x^{2}} \\ X \frac{d^{2} Y}{d t^{2}} & = c^{2} Y \frac{d^{2} X}{d x^{2}} \\ \underset{only dependent on t}{\underset{⏟}{\frac{1}{c^{2}} \frac{1}{Y} \frac{d^{2} Y}{d t^{2}}}} & = \underset{only dependent on x}{\underset{⏟}{\frac{1}{X} \frac{d^{2} X}{d x^{2}}}} = α_{s} \end{aligned}

where, $α_{s}$ is a constant of separation

three cases to consider:
- $α_{s} = 0$
- $α_{s} > 0$
- $α_{s} < 0$

case 1:

α = 0

looking at the $x$ -dependent term:

\begin{matrix} \frac{1}{X} \frac{d^{2} X}{d x^{2}} = 0 \forall x \\ ⟹ \frac{d^{2} X}{d x^{2}} = 0 \\ ⟹ X (x) = A_{0} x + B_{0} \end{matrix}

where, $A_{0}$ and $B_{0}$ are constants of integration

now, the $y$ -dependent term:

\frac{1}{c^{2} Y} \frac{d^{2} Y}{d t^{2}} = 0 ⟹ Y = C_{0} t + D_{0}

where, $C_{0}$ and $D_{0}$ are constants of integration

therefore, for $α_{s} = 0 :$

u (x, t) = (A_{0} x + B_{0}) (C_{0} t + D_{0})

this is a 'steady state'

case 2

α > 0

trying $α_{s} = k^{2}$

\frac{1}{c^{2} Y} \frac{d^{2} Y}{d t^{2}} = k^{2} ⟹ Y = k^{2} c^{2} Y

the solutions to this are exponentials:

Y (t) = C e^{- c k t} + D e^{i k t}

but these are divergent, and solutions are unphysical

case 3

α < 0

trying $α_{s} = - k^{2}$
the $x$ -dependent term:

\frac{1}{X} \frac{d^{2} X}{d x^{2}} = - k^{2} ⟹ \frac{d^{2} X}{d x^{2}} = - k^{2} X

the solutions to this are sinusoidal:

X (x) = A \cos k x + B \sin k x

now, the $y$ -dependent term:

\frac{1}{c^{2} Y} \frac{d^{2} Y}{d t^{2}} = - k^{2} ⟹ \frac{d^{2} Y}{d t^{2}} = - (k c)^{2} Y

Y (t) = C \cos k c t + D \sin k c t

the wavenumber:

k = \frac{2 π}{λ}

conclusion

combining the first and the third cases:

u (x, t) = (A_{0} x + B_{0}) (C_{0} t + D_{0}) + (A \cos k x + B \sin k x) (C \cos k c t + D \sin k c t)