Equation of Harmonic Wave/Wavelength and Velocity
Jump to navigation
Jump to search
Theorem
Let $\phi$ be a harmonic wave which is propagated along the $x$-axis in the positive direction with constant velocity $c$.
Then the disturbance of $\phi$ at point $x$ and time $t$ can be expressed using the equation:
- $\map \phi {x, t} = a \map \cos {\dfrac {2 \pi} \lambda \paren {x - c t} }$
where $\lambda$ is the wavelength of $\phi$
Proof
\(\ds \map \phi {x, t}\) | \(=\) | \(\ds a \map \cos {\omega \paren {x - c t} }\) | Equation of Harmonic Wave | |||||||||||
\(\ds \) | \(=\) | \(\ds a \map \cos {\dfrac {2 \pi} \lambda \paren {x - c t} }\) | Wavelength of Harmonic Wave: $\lambda = \dfrac {2 \pi} \omega$ |
$\blacksquare$
Sources
- 1955: C.A. Coulson: Waves (7th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Equation of Wave Motion: $\S 3$: $(4)$