Waves, Sound, and Physical Optics
Core concepts
Wave behavior is captured by $v=f\lambda$. Sound is a longitudinal pressure wave; its speed depends on the medium (faster in solids, ~343 m/s in air at room temperature). Standing waves on strings and in pipes produce harmonics: a string fixed at both ends has $f_n=nv/(2L)$; an open pipe matches that, while a closed pipe has only odd harmonics $f_n=nv/(4L)$. Interference and diffraction follow from path differences: constructive when $\Delta L=m\lambda$, destructive when $\Delta L=(m+\tfrac12)\lambda$. Double-slit maxima occur at $d\sin\theta=m\lambda$. The Doppler effect shifts observed frequency when source and observer move relative to each other.
Traps to watch
- Beat frequency equals the difference of two close frequencies, not the sum.
- Closed pipes support only odd harmonics-an easy discriminator in questions.
- For double slit, $m=0$ is the central maximum; adjacent maxima are separated by $\lambda L/d$ on a distant screen.
- Wave speed is set by the medium, not the source; changing frequency changes wavelength, not speed.
- In thin-film interference, a phase shift of $\pi$ can occur on reflection at a higher-index boundary.
Worked example
Light of $\lambda=600,\mathrm{nm}$ passes through slits $0.20,\mathrm{mm}$ apart onto a screen $2.0,\mathrm{m}$ away. Fringe spacing is $\Delta y=\lambda L/d=(600\times10^{-9})(2.0)/(2.0\times10^{-4})=6.0\times10^{-3},\mathrm{m}=6.0,\mathrm{mm}$.
Rule of thumb: for small angles, $\sin\theta\approx\tan\theta\approx y/L$, so fringe spacing simplifies to $\lambda L/d$.