Light and sound

Frequency is set at the source and does not change when a wave crosses into a new medium — a crucial fact for refraction: as light slows entering glass, its wavelength shrinks proportionally so that v = fλ still holds with the same f. Period is T = 1/f. For light in vacuum, v = c ≈ 3×10⁸ m/s.

Standing waves have fixed nodes (no motion) and antinodes (maximum motion). A string or a pipe closed at both ends (or open at both ends) fits whole/half wavelengths in characteristic ways, producing a fundamental frequency and integer (or odd-integer) harmonics. Resonance — driving a system at a natural frequency — produces large-amplitude oscillation and is how instruments and the basilar membrane select tones. For a string (or a pipe open at both ends), the resonant frequencies are fₙ = nv/2L (all harmonics); a pipe closed at one end gives fₙ = nv/4L (odd harmonics only). Two slightly different frequencies sounded together produce beats at f_beat = |f₁ − f₂|.

Intensity is power per area (W/m²) and falls off with the square of distance from a point source (I ∝ 1/r²). The decibel scale references everything to the threshold of hearing I₀ = 10⁻¹² W/m². The logarithm is the trap engine: going from 40 to 60 dB is +20 dB, i.e. a 100-fold rise in intensity. Doubling intensity adds only about 3 dB.

When source and observer move closer, wavefronts bunch up and the observed frequency (pitch) rises; as they separate, wavefronts spread and pitch falls — the classic ambulance siren dropping as it passes. Rather than memorize the sign rules, anchor on the physics: closing the gap always raises pitch, so pick the top/bottom signs that make f_obs > f_source for approach. The effect also underlies Doppler ultrasound for blood-flow velocity and the redshift of receding galaxies.

All EM waves travel at c in vacuum regardless of frequency. Photon energy rises with frequency (E = hf) and falls with wavelength — so UV and X-rays are energetic (and ionizing) while IR and radio are not. Visible light spans roughly 400 nm (violet, higher energy) to 700 nm (red, lower energy). This E = hf relationship reappears in the photoelectric effect and atomic spectra in 4E.

Index of refraction n = c/v ≥ 1; a higher index means slower light and more bending. Entering a denser medium (low→high n), the ray bends toward the normal; leaving it (high→low n), away. Total internal reflection occurs only going from higher to lower index, when the angle exceeds the critical angle sinθ_c = n₂/n₁ — the principle behind fiber optics and the sparkle of a diamond (high n, small critical angle).

Converging (convex) lenses and concave mirrors have positive focal length and can form real, inverted images when the object is beyond the focal point — or a magnified virtual image when inside it (the magnifying glass). Diverging (concave) lenses and convex mirrors have negative focal length and always form upright, virtual, reduced images. Magnification m = −i/o: negative m means inverted, |m| > 1 means enlarged. Keeping the sign convention straight is the whole game.

Optometry uses diopters (P = 1/f, f in meters) because the powers of thin lenses in contact simply add. A myopic eye focuses images in front of the retina (eyeball too long) and needs a diverging (negative-power) lens to push the focus back; a hyperopic eye focuses behind the retina and needs a converging (positive-power) lens. These corrective-lens items are common passage applications of the thin-lens equation.

Wave optics is recognition-level on the MCAT. Diffraction and interference (the double-slit experiment, thin-film colors) are the evidence that light is a wave; constructive interference gives bright fringes where path differences are whole wavelengths. Polarization restricts the electric-field oscillation to one plane; optically active (chiral) molecules rotate the plane of polarized light — the link between this topic and stereochemistry.