11.1 Vibrations, Waves, and Optics

11.1 Vibrations, Waves, and Optics on NMAT Physics

The Center for Educational Measurement (CEM) NMAT Physics subtest is a 30-item block with a recommended ~30 minutes. Official content spans mechanics, thermodynamics, vibrations, waves, and optics, electricity and magnetism, and modern physics. This section trains the wave–optics family at introductory college depth: conceptual SHM, traveling-wave relations, sound, geometric optics, and a light touch of interference.

Quick frame: Oscillations repeat in time; waves transport energy through space. Optics asks where light goes (rays) and when phases matter (interference). Always state units and which medium or interface you are on.

Simple harmonic motion (SHM) — conceptual core

Simple harmonic motion is periodic motion in which the restoring force (or acceleration) is proportional to displacement from equilibrium and directed toward equilibrium:

$F = -kx \quad \text{or} \quad a = -\omega^2 x$

Displacement, velocity, and acceleration are sinusoidal in time (phase-shifted). High-yield properties:

Energy view: Total mechanical energy is constant if undamped; it converts between kinetic and potential. At extreme positions velocity is zero and potential is max; at equilibrium speed is max.

Worked conceptual scenario A
Doubling amplitude of an ideal mass–spring oscillator doubles the extreme displacement but does not change T. It quadruples stored energy (E ∝ A²). If an item claims period rises with amplitude, reject it for ideal SHM.

Worked conceptual scenario B
A pendulum’s small-angle period depends on length and g, not mass. Replacing the bob with a denser mass of equal size leaves T essentially unchanged (air resistance aside).

Waves: properties and the master relation

A wave is a disturbance that transfers energy without net transport of the medium’s bulk (for many mechanical waves). Key descriptors:

• Wavelength λ — distance between successive crests (or equivalent phase points)
• Frequency f — oscillations per second of a point on the medium
• Period T — 1/f
• Amplitude — maximum displacement of the medium element
• Wave speed v — how fast the disturbance advances

Master equation:

$v = f\lambda$

Also v = λ/T. For a string under tension, speed depends on tension and linear density; for sound in air, speed depends mainly on temperature (and medium), not on loudness or frequency in the ideal linear model.

Superposition: Overlapping waves add displacements. Constructive interference reinforces; destructive cancels when crests meet troughs of equal amplitude.

Worked example C — wave speed
A sound wave has f = 340 Hz and λ = 1.0 m in air. Speed:

$v = f\lambda = 340 \times 1.0 = 340\ \text{m/s}$

If frequency doubles while the medium (hence v) stays the same, λ halves to 0.5 m.

Sound and the Doppler effect (intro)

Sound is a longitudinal mechanical wave. Loudness relates to intensity (energy per time per area); pitch relates to frequency. The Doppler effect changes the observed frequency when source and/or observer move relative to the medium:

• Source approaching observer → higher observed f (shorter λ ahead of moving source)
• Source receding → lower observed f
• Observer motion also shifts f, with the classic formula combining source and observer speeds relative to the wave speed

NMAT-level items often ask direction of the shift and qualitative comparison, not always the full algebraic plug-in.

Worked conceptual scenario D
An ambulance siren sounds higher-pitched as it approaches and lower as it recedes. The source frequency is fixed; relative motion changes the observed frequency. A stationary wall reflection can produce beats or a second Doppler path, but the core idea is motion-relative frequency shift.

Light: reflection and refraction

Law of reflection: angle of incidence equals angle of reflection, both measured from the normal to the surface.

Refraction is the change in direction when light crosses into a medium with different index of refraction n = c/v (c = speed of light in vacuum, v = speed in medium). Larger n means slower light in that medium.

Snell’s law:

$n_1 \sin\theta_1 = n_2 \sin\theta_2$

Light bends toward the normal when entering a higher-n medium and away from the normal when entering a lower-n medium.

Worked example E — Snell
Light in air (n₁ ≈ 1.00) hits water (n₂ ≈ 1.33) at θ₁ = 30°. Find θ₂:

$\sin\theta_2 = \frac{n_1}{n_2}\sin\theta_1 = \frac{1.00}{1.33}\sin 30° ≈ 0.376 \Rightarrow \theta_2 ≈ 22°$

The ray bends toward the normal, as expected.

Total internal reflection (TIR)

When light travels from higher n to lower n, θ₂ > θ₁. There is a critical angle θ_c where θ₂ = 90°:

$\sin\theta_c = \frac{n_2}{n_1} \quad (n_1 > n_2)$

For incidence θ₁ > θ_c, no refracted ray enters the second medium — the interface totally reflects. Fiber optics and some prisms use TIR.

Worked example F — critical angle
Glass to air: n₁ = 1.50, n₂ = 1.00.

$\sin\theta_c = 1.00/1.50 = 0.667 \Rightarrow \theta_c ≈ 42°$

A ray inside glass at 50° to the normal undergoes TIR at a glass–air face.

Mirrors and thin lenses

Spherical mirrors and thin lenses form images whose location follows the lens/mirror equation (paraxial form):

$\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$

Lateral magnification:

$m = -\frac{d_i}{d_o} = \frac{h_i}{h_o}$

Converging lens (f > 0): object beyond 2f → real, inverted, reduced image between f and 2f; object between f and 2f → real, inverted, enlarged beyond 2f; object inside f → virtual, upright, enlarged (same side as object).

Diverging lens (f < 0): always virtual, upright, reduced for real objects.

Worked example G — thin lens
A converging lens has f = 20 cm. Object at d_o = 30 cm. Find d_i and m.

$\frac{1}{d_i} = \frac{1}{f} - \frac{1}{d_o} = \frac{1}{20} - \frac{1}{30} = \frac{1}{60} \Rightarrow d_i = 60\ \text{cm}$

$m = -\frac{60}{30} = -2$

Real image (d_i > 0 in this convention), inverted, twice as tall as the object. Place a screen 60 cm on the opposite side of the lens.

Worked example H — virtual image
Same lens, object at d_o = 10 cm (inside f):

$\frac{1}{d_i} = \frac{1}{20} - \frac{1}{10} = -\frac{1}{20} \Rightarrow d_i = -20\ \text{cm}$

$m = -(-20)/10 = +2$

Virtual, upright, magnified — classic magnifying-glass geometry.

Interference and diffraction (conceptual)

Interference requires coherent sources (stable phase relation). Young’s double-slit: path difference of mλ → bright (constructive); (m+½)λ → dark (destructive), for the usual far-field small-angle pattern in air.

Diffraction is the spreading of waves when they pass apertures or edges comparable to λ. Narrower aperture → wider diffraction pattern. Single-slit minima relate to a sinθ = mλ (m = ±1, ±2, …) in the standard model.

Exam distinction: Interference is about superposition of multiple coherent paths; diffraction is wave bending/spreading from limited wavefronts. Both rest on wave phase.

Worked conceptual scenario I
Why do radio waves diffract around buildings more readily than visible light? Radio wavelengths are much longer, so everyday obstacles are not huge compared to λ. Visible light’s tiny λ makes geometric-shadow ray optics a good approximation for large objects.

High-yield error traps

Study protocol

1. Derive T for mass–spring and pendulum from memory twice weekly.
2. Drill ten Snell problems and one TIR critical-angle calculation.
3. For every lens item, sketch rays (parallel → focal; through center undeviated) before algebra.
4. State Doppler shift direction in one sentence before any formula.

Section checkpoint

You are ready when you can: (1) state SHM restoring-force and period independence of amplitude, (2) use v = fλ fluently, (3) apply Snell and identify TIR conditions, (4) solve 1/f = 1/do + 1/di with magnification and interpret real vs virtual, and (5) explain interference vs diffraction in one clear contrast.