Hi everybody,
We all know that in mathematics, any wave can be thought of as the plot of a circle in the 2D coordinate plane (considering 2D waves only). A wave [math]W[/math] may be represented as:
[math]W(x,t)=Acos(kx-\omega t)[/math]
Where [math]W(x,t)[/math] is the function of the wave's position [math]x[/math] and time [math]t[/math], which gives the displacement from the x-axis, [math]k[/math] is the wavenumber, [math]\omega[/math] is the angular frequency of the circle; [math]\omega=2\pi f[/math], where [math]f[/math] is the frequency of the wave.
Thus the wave [math]W[/math] is a curved line, consisting of points of the form [math](x,Acos(kx-\omega t))[/math]
But, I think (and that's my question too) that the position [math]x[/math] and time [math]t[/math] can be represented as functions of the angle of the circle [math]\theta[/math].
Here's how...
Let the wave velocity be [math]v[/math]. Then,
[math] v=\frac{x}{t}[/math]
Or, [math]x=vt[/math]
But, [math]v=f\lambda[/math], [math]f[/math] is frequency and [math]\lambda[/math] is the wavelength.
So, [math]x=f\lambda t[/math]
But, [math]f=\frac{\omega}{2\pi}[/math]
So, [math]x=\frac{\omega}{2\pi}\lambda t[/math]
But, [math]\omega=\frac{\theta}{t}[/math]
So, [math]x=\frac{\theta \lambda}{2\pi}[/math]
Similarly,
[math] t=\frac{\theta}{\omega}[/math]
Am I correct ?