Wave Equation is Linear
Jump to navigation
Jump to search
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: $\S 6$