Wave Equation is Linear

Theorem

Let $\phi_1$ and $\phi_2$ be particular solutions to the wave equation.

Then:

$a_1 \phi_1 + a_2 \phi_2$ is also a particular solution to the wave equation.

Proof

\(\ds \map {\dfrac {\partial^2} {\partial t^2} } {a_1 \phi_1 + a_2 \phi_2}\)

\(=\)

\(\ds a_1 \map {\dfrac {\partial^2} {\partial t^2} } {\phi_1} + a_2 \map {\dfrac {\partial^2} {\partial t^2} } {\phi_2}\)

Linear Combination of Derivatives

\(\ds \map {\dfrac {\partial^2} {\partial x^2} } {a_1 \phi_1 + a_2 \phi_2}\)

\(=\)

\(\ds a_1 \map {\dfrac {\partial^2} {\partial x^2} } {\phi_1} + a_2 \map {\dfrac {\partial^2} {\partial x^2} } {\phi_2}\)

Linear Combination of Derivatives

\(\ds \map {\dfrac {\partial^2} {\partial y^2} } {a_1 \phi_1 + a_2 \phi_2}\)

\(=\)

\(\ds a_1 \map {\dfrac {\partial^2} {\partial y^2} } {\phi_1} + a_2 \map {\dfrac {\partial^2} {\partial x^2} } {\phi_2}\)

Linear Combination of Derivatives

\(\ds \map {\dfrac {\partial^2} {\partial z^2} } {a_1 \phi_1 + a_2 \phi_2}\)

\(=\)

\(\ds a_1 \map {\dfrac {\partial^2} {\partial z^2} } {\phi_1} + a_2 \map {\dfrac {\partial^2} {\partial x^2} } {\phi_2}\)

Linear Combination of Derivatives

Hence:

\(\ds \map {\dfrac {\partial^2} {\partial t^2} } {a_1 \phi_1 + a_2 \phi_2}\)

\(=\)

\(\ds a_1 \map {\dfrac {\partial^2} {\partial t^2} } {\phi_1} + a_2 \map {\dfrac {\partial^2} {\partial t^2} } {\phi_2}\)

from above

\(\ds \)

\(=\)

\(\ds a_1 c^2 \paren {\map {\dfrac {\partial^2} {\partial x^2} } {\phi_1} + \map {\dfrac {\partial^2} {\partial y^2} } {\phi_1} + \map {\dfrac {\partial^2} {\partial z^2} } {\phi_1} } + a_2 c^2 \paren {\map {\dfrac {\partial^2} {\partial x^2} } {\phi_2} + \map {\dfrac {\partial^2} {\partial y^2} } {\phi_2} + \map {\dfrac {\partial^2} {\partial z^2} } {\phi_2} }\)

as $\phi_1$ and $\phi_2$ satisfy the wave equation

\(\ds \)

\(=\)

\(\ds c^2 \paren {\map {\dfrac {\partial^2} {\partial x^2} } {a_1 \phi_1 + a_2 \phi_2} + \map {\dfrac {\partial^2} {\partial y^2} } {a_1 \phi_1 + a_2 \phi_2} + \map {\dfrac {\partial^2} {\partial z^2} } {a_1 \phi_1 + a_2 \phi_2} }\)

from above

and it is seen that $a_1 \phi_1 + a_2 \phi_2$ also satisfies the wave equation.

$\blacksquare$

Sources

• 1955: C.A. Coulson: Waves (7th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Equation of Wave Motion: $\S 6$