Sum of Equal and Opposite Harmonic Waves form Stationary Wave

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $\phi_1$ be a harmonic wave which is propagated along the $x$-axis in the positive direction with constant velocity $c$.


Let $\phi_2$ be a harmonic wave travelling with constant velocity $-c$, that is, at the same speed as $\phi_1$ but in the opposite direction.


Then their sum $\phi_1 + \phi_2$ describes a stationary wave.


Proof

From Equation of Harmonic Wave: Wave Number and Frequency:

$(1): \quad \map {\phi_1} {x, t} = a \map \cos {2 \pi \paren {k x - \nu t} }$

where:

$k$ denotes the wave number of $\phi_1$
$\nu$ denotes the frequency of $\phi_1$.


From Equation of Wave with Constant Velocity: Corollary, the equation of $\phi_2$ is:

$(2): \quad \map {\phi_2} {x, t} = a \map \cos {2 \pi \paren {k x + \nu t} }$


Then we have:

\(\ds \map {\phi_1} {x, t} + \map {\phi_2} {x, t}\) \(=\) \(\ds a \map \cos {2 \pi \paren {k x - \nu t} } + a \map \cos {2 \pi \paren {k x + \nu t} }\) from $(1)$ and $(2)$
\(\ds \) \(=\) \(\ds 2 a \map \cos {\dfrac {2 \pi \paren {k x - \nu t} + 2 \pi \paren {k x + \nu t} } 2} \map \cos {\dfrac {2 \pi \paren {k x - \nu t} - 2 \pi \paren {k x + \nu t} } 2}\) Cosine plus Cosine
\(\ds \) \(=\) \(\ds 2 a \map \cos {2 \pi k x} \map \cos {2 \pi \paren {-\nu t} }\) simplification
\(\ds \) \(=\) \(\ds 2 a \map \cos {2 \pi k x} \map \cos {2 \pi \nu t}\) Cosine Function is Even


This needs considerable tedious hard slog to complete it.
In particular: It remains to be shown that this is the equation of a stationary wave.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Finish}} from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Sum_of_Equal_and_Opposite_Harmonic_Waves_form_Stationary_Wave&oldid=621007"