Waves, Sound, and Physical Optics

Notes

Wave basics

• Transverse waves vibrate perpendicular to the direction of travel (light, waves on a string). Longitudinal waves vibrate parallel (sound, with compressions and rarefactions).
• $v = f\lambda$. The medium sets the speed; the source sets the frequency.
• On a string, $\displaystyle v = \sqrt{\frac{F_T}{\mu}}$ (tension over mass per length).
• Mechanical waves need a medium; electromagnetic waves don't. Light travels at $c = 3.0 \times 10^8$ m/s in a vacuum.
• Sound intensity $\displaystyle I = \frac{P}{4\pi r^2}$ drops with distance squared.

Doppler effect

When a source moves toward you, you hear a higher frequency; moving away, lower. The same idea shifts light toward blue or red.

Interference and standing waves

• Superposition: overlapping waves add. Constructive interference (crests meet crests) makes bigger amplitude; destructive interference cancels.
• Standing waves form from reflected waves. On a string fixed at both ends: $\displaystyle \lambda_n = \frac{2L}{n}$, $\displaystyle f_n = \frac{nv}{2L}$.
• Pipes open at both ends: $\displaystyle \lambda_n = \frac{2L}{n}$. Pipes closed at one end: $\displaystyle \lambda_n = \frac{4L}{n}$, with only odd $n$.
• Beats: two close frequencies make $f_{beat} = |f_1 - f_2|$.

Physical optics

• Double-slit interference: bright fringes where the path difference is $m\lambda$: $d\sin\theta = m\lambda$. For small angles, fringe spacing $\displaystyle \Delta y = \frac{\lambda L}{d}$.
• Diffraction (spreading around edges) is greatest when the opening is about the size of the wavelength.
• Thin films (soap bubbles, oil) make colors from interference of light reflecting off the top and bottom surfaces.
• Diffraction gratings make sharp, widely spaced fringes.