Waves Study Guide

Level High SchoolReview time ~12 hours

Waves carry energy without carrying matter, and the same mathematics describes sound, light, and earthquakes. Master $v = f\lambda$ , the energy in oscillations, and the Doppler shift, and the whole subject clicks into place.

Key formulas

Hooke's Law — F = kx

The restoring force of a spring equals its stiffness times the displacement from equilibrium.

Kinetic Energy Formula — KE = ½mv²

Kinetic energy equals one-half times mass times velocity squared.

Power Formula — P = W/t

Power equals work done divided by the time taken.

Wave Equation — v = fλ

Wave speed equals frequency times wavelength.

Topic checklist

Wave basics

Use $v = f\lambda$ to connect speed, frequency, and wavelength.
Convert fluently between period and frequency ( $T = 1/f$ ).
Identify amplitude, wavelength, and nodes on a wave diagram.

Sound waves

Distinguish transverse from longitudinal waves with examples.
Know that sound needs a medium and travels about 343 m/s in air at 20 °C.
Explain why sound cannot travel through a vacuum.

Interference and standing waves

Describe superposition: waves add where they overlap.
Explain constructive versus destructive interference.
Relate standing waves to resonance and harmonics on strings.

Doppler effect

State the Doppler shift rule: approach raises pitch, recession lowers it.
Apply the Doppler idea to light (redshift) and radar.
Distinguish the moving-source case from the moving-observer case.

Oscillations and resonance

Connect simple harmonic motion to waves: the restoring force $F = -kx$ .
Know that oscillation energy splits between $\tfrac{1}{2}mv^2$ and $\tfrac{1}{2}kx^2$ .
Explain resonance: large amplitude at the natural frequency.

Wave energy and applications

Relate wave power to amplitude squared.
Explain diffraction: waves spread when passing openings near $\lambda$ in size.
Give one real use each for ultrasound, sonar, and noise cancellation.

Common mistakes

Mistake: Thinking frequency changes when a wave changes medium.

Fix:

Frequency is set by the source; speed is set by the medium. When a wave enters a new medium, $v$ and $\lambda$ change but $f$ stays fixed. Write $v = f\lambda$ and ask which variable the medium controls.

Mistake: Confusing the direction of particle motion with the direction of wave travel.

Fix:

In a longitudinal wave the medium oscillates parallel to travel (like a slinky pushed end-to-end); the wave pattern still moves forward. Sketch the particle motion arrows to keep the two directions separate.

Mistake: Mixing up nodes and antinodes in standing waves.

Fix:

Nodes are points of zero displacement that never move — the salt in the Chladni experiment collects there. Antinodes swing the most. Label both on every standing-wave diagram.

Mistake: Using the same Doppler formula for moving source and moving observer.

Fix:

Both change the observed pitch, but the formulas differ. Read the problem: is the source moving, the observer, or both? Then pick the matching form.

Mistake: Believing louder or bigger waves travel faster.

Fix:

Wave speed comes from the medium's properties (tension, density, temperature) — amplitude only sets the energy carried. Shouting louder does not make sound arrive sooner.

Exam tips

Draw the wave and label $\lambda$ , $A$ , and the direction of travel before calculating.
Keep frequency (Hz), period (s), and angular frequency ( $\omega = 2\pi f$ ) clearly separated.
For Doppler problems, write 'toward = higher, away = lower' first, then compute.
In interference questions, path difference of $n\lambda$ means constructive, $(n+\tfrac{1}{2})\lambda$ means destructive.