PX154 - F1b - the wave function

• considering sinusoidal behaviour
• the wavelength is $\lambda$, so, at $x=\lambda$, we need $1$ complete cycle
• as $x$ goes from $0\to \lambda$, consider $\cos ax$ to go from $0\to 2\pi$
• take $\cos (\frac{2\pi }{\lambda }x)$: $\cos \left(\frac{2\pi }{\lambda }x\right)=1$ at $2n\pi$, so, when $x=\lambda ,2\lambda ,3\lambda ,etc$
• take

where $k=$ wave number with units $rad{m}^{-1}$ or $m^{-1}$
- we have, $\cos (kx)$

• for time dependence at one x position
  
• we can use $\cos (\frac{2\pi }{T}t)$:
  • $f=\frac{1}{T}$
  • we have $\cos (2\pi ft)$
  • and, we can write $\cos (\omega t)$
    • where, $\omega =$ angular frequency
• to put this together:
• the waveform translates, but doesn't change shape: in the figure above, it translates from left $\to$ right
• we write:

• for right $\to$ left translation: $u(x,t)=A\cos (kx+\omega t)$

where, $A=$ amplitude, $k=$ wave number, $\omega =$ angular frequency

• where, $-$ for right; $+$ for left travelling