Wavelength and Frequency — Question 2

Question 2

Angular and cyclic frequencies use different numerical scales

A wave is $y=.004\cos(8\pi x-40\pi t+\pi/5)$ in SI units.

1. Find amplitude, wavelength, period and frequency.

2. Find the phase change for $\Delta x=.050\,\mathrm{m}$ at fixed time, then express it as a fraction of a cycle.

Question 2 - Solution

Angular and cyclic frequencies use different numerical scales

Method and mathematical working

Read coefficients of position and time from the dimensionless phase, then convert radians to cycles. The constant phase affects position within a cycle but not its period.$$\begin{aligned}\Phi&=kx-\omega t+\phi_0,\\ k\lambda&=2\pi,\qquad \omega T=2\pi,\qquad \frac{\Delta\Phi}{2\pi}=\frac{\Delta x}{\lambda}.\\[4pt]\lambda&=\frac{2\pi}{8\pi}=0.25\,\mathrm m,\quad T=\frac{2\pi}{40\pi}=0.050\,\mathrm s,\\\Delta\Phi&=(8\pi)(0.050)=0.40\pi\quad(0.20\text{ cycle}).\end{aligned}$$

(a) Calculation and conclusion

$A=.004\,\mathrm{m}$, $k=8\pi\,\mathrm{rad/m}$ and $\omega=40\pi\,\mathrm{rad/s}$. Thus $\lambda=2\pi/k=.25\,\mathrm{m}$, $T=2\pi/\omega=.050\,\mathrm{s}$ and $f=20\,\mathrm{Hz}$.

(b) Interpretation and limits

$\Delta\Phi=k\Delta x=.4\pi\,\mathrm{rad}$, or $.20$ cycle. The constants $k$ and $\omega$ measure phase in radians; treating their numerical values as cycles per meter or hertz introduces a factor of $2\pi$.

Physical check

Both $kx$ and $\omega t$ are dimensionless. The frequency in hertz is $\omega/(2\pi)$, so units and cycle counts provide independent checks against a missing factor of $2\pi$.