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4D · Physical principles of living systems
Light and sound
Wave behavior and its two MCAT showcases: sound (a longitudinal pressure wave — intensity, decibels, the Doppler effect) and light (an electromagnetic wave — reflection, refraction, total internal reflection, and image formation by lenses and mirrors).
Wave fundamentals
A wave carries energy without carrying matter. v = fλ ties speed, frequency, and wavelength; waves superpose (interfere constructively or destructively) and form standing waves at resonant frequencies.
Waves are transverse (oscillation perpendicular to travel, like light or a string) or longitudinal (oscillation along travel, like sound). Speed is set by the medium, while a source sets the frequency; wavelength then follows from λ = v/f. When waves overlap they add (superposition): aligned crests reinforce (constructive), crest-meets-trough cancels (destructive). Confined waves form standing waves only at specific resonant wavelengths — the basis of pipes, strings, and resonance.
Wavelength, frequency, and wave speed
v = fλ. Speed is fixed by the medium, so when a wave passes into a new medium its frequency stays constant while wavelength and speed change together.
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.
Interference, standing waves, and resonance
Overlapping waves superpose: in phase → constructive (louder/brighter), out of phase → destructive (cancellation). Confined systems resonate at discrete standing-wave frequencies (harmonics).
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|.
Sound
Sound is a longitudinal pressure wave that needs a medium and travels fastest in solids, slowest in gases. Loudness is the logarithmic decibel scale; relative motion of source and observer shifts pitch (the Doppler effect).
Because sound is a mechanical pressure wave, it cannot travel through vacuum, and its speed increases with the stiffness of the medium — counter to intuition, sound is fastest in solids and slowest in gases. The ear responds to a vast range of intensities, so loudness is compressed onto a logarithmic scale (decibels). A moving source or listener changes the perceived frequency without changing the source's actual output.
Intensity and the decibel scale
Sound level in decibels is β = 10·log(I/I₀). Because it's logarithmic, every +10 dB is a ten-fold increase in intensity (+20 dB = ×100), not a doubling.
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.
Don't confuse
Decibels are logarithmic: +10 dB means ×10 intensity, not "ten percent more" or "double." Adding decibels multiplies intensity.
The Doppler effect
Relative approach raises the perceived frequency; relative recession lowers it: f_obs = f_source·(v ± v_o)/(v ∓ v_s). Choose the signs so that approaching → higher f_obs.
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.
Don't confuse
It's the relative motion that matters; the perceived frequency shifts even though the source emits a constant frequency. The trap is plugging signs mechanically and getting a lower pitch on approach — sanity-check against "approaching = higher."
Light and geometric optics
Light is an electromagnetic wave (no medium needed). It reflects (angle in = angle out), refracts at boundaries (Snell's law, n₁sinθ₁ = n₂sinθ₂), can undergo total internal reflection, and forms images through lenses and mirrors via 1/f = 1/o + 1/i.
Geometric optics treats light as rays. At a surface, part reflects and part refracts; refraction bends the ray because light changes speed (n = c/v, higher index = slower light), bending toward the normal entering a denser medium. Beyond a critical angle, light going from dense to rare reflects entirely (TIR — the basis of fiber optics). Curved surfaces focus rays to form images whose size, orientation, and real/virtual nature follow from the thin-lens/mirror equation and a consistent sign convention.
The electromagnetic spectrum
Light is an EM wave traveling at c in vacuum; the spectrum runs (long λ, low energy → short λ, high energy) radio → microwave → IR → visible → UV → X-ray → gamma. Energy per photon is E = hf = hc/λ.
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.
Reflection, refraction, and total internal reflection
Reflection: angle of incidence = angle of reflection. Refraction: n₁sinθ₁ = n₂sinθ₂; light bends toward the normal entering a higher-index medium. Past the critical angle (dense→rare only), light is totally internally reflected.
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).
Don't confuse
TIR happens only when light travels from a higher to a lower index medium (e.g., glass→air), never the reverse. Refraction direction also trips people up: toward the normal when slowing down (entering denser medium).
Lenses, mirrors, and image formation
1/f = 1/o + 1/i, magnification m = −i/o. Converging lens/mirror: f > 0. A real image has i > 0 (inverted, projectable); a virtual image has i < 0 (upright, behind the lens/mirror).
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.
Don't confuse
Real images (i > 0, inverted, can be projected on a screen — the eye and camera) vs. virtual images (i < 0, upright, cannot be projected — mirrors, magnifiers). A flipped sign turns the answer upside down, literally.
The eye, lens power, and corrective lenses
Lens power is P = 1/f in diopters (1/m), and powers of stacked lenses add. The eye focuses an image on the retina; myopia (nearsighted) is corrected with a diverging lens, hyperopia (farsighted) with a converging lens.
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.
Diffraction, interference, and polarization
Light behaves as a wave: it diffracts (bends around edges and through slits) and interferes (Young's double-slit gives bright/dark fringes). Polarization filters light to a single oscillation plane — and is rotated by chiral molecules.
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.
Worked question
An ambulance with its siren sounding at a constant frequency drives past a stationary observer at steady speed. What does the observer hear as the ambulance approaches and then recedes?