A siren gets higher as it approaches, lower as it recedes
Source moving toward you โ crests bunch up โ higher frequency โ higher pitch. Moving away โ crests spread out โ lower pitch. Used in radar, ultrasound, and astronomy.
Transverse waves (light, water surface): particles move perpendicular to wave travel. Longitudinal waves (sound): particles compress and rarefy parallel to travel direction. Sound cannot travel through a vacuum โ light can.
Resonance
Resonance: driving frequency matches natural frequency โ amplitude builds
Resonance
Push at the right frequency and things vibrate dramatically
Every object has a natural resonant frequency. Drive it at that frequency โ amplitude grows. Tacoma Narrows Bridge (1940) collapsed due to wind-driven resonance. Basis of MRI, musical instruments, and radio tuning.
Standing waves form when reflected waves interfere constructively
A standing wave appears stationary but is really two waves traveling in opposite directions. Nodes: points of zero amplitude. Antinodes: points of maximum amplitude. String instruments and organ pipes use this principle.
Two waves overlapping โ they can add up or cancel out
Constructive interference: waves in phase, amplitudes add. Destructive interference: waves out of phase (180ยฐ), amplitudes subtract. Superposition principle: resultant wave = sum of individual waves. Beats: two slightly different frequencies โ periodic loud/soft pattern = beat frequency = |fโ - fโ|.
Sound Intensity and Decibels
Sound intensity: decibels (dB). Every 10 dB increase = 10ร more intense. 20 dB = 100ร more intense.
Sound Intensity and Decibels
The logarithmic scale of sound intensity
0 dB: threshold of hearing. 60 dB: normal conversation. 120 dB: threshold of pain. Every 10 dB increase represents a 10-fold increase in intensity. 20 dB increase = 10 ร 10 = 100ร more intense. Perceived loudness doubles roughly every 10 dB.
Wave Properties Summary
Frequency determines pitch. Amplitude determines loudness. Speed depends on medium.
Wave Properties Summary
The relationship between frequency, amplitude, and speed
Frequency (Hz): number of complete cycles per second โ determines pitch for sound, color for light. Amplitude: maximum displacement from equilibrium โ determines loudness for sound, brightness for light. Speed: set by the medium โ sound travels faster in solids than liquids than gases.
Electromagnetic Waves
Electromagnetic spectrum: all EM waves travel at c = 3ร10โธ m/s in vacuum. Only speed, not frequency, changes in medium.
Electromagnetic Waves
What all EM waves have in common โ and what distinguishes them
All EM waves: travel at c in vacuum, are transverse, require no medium. Radio waves: lowest frequency, longest wavelength. Gamma rays: highest frequency, most energy. In a medium (glass, water): speed slows, wavelength shortens, but frequency stays the same โ this is why light bends at interfaces.
Diffraction
Diffraction: waves bend around obstacles or through openings. More diffraction when wavelength โ opening size.
Diffraction
Why waves spread out when they pass through gaps
Diffraction is most pronounced when the wavelength is comparable to the opening or obstacle size. Sound diffracts easily around corners (long wavelengths). Light diffracts less visibly (very short wavelength). Single slit and double slit diffraction patterns are classic exam problems.
Resonance in Pipes
Pitch of open pipe: f = v/2L. Closed pipe: f = v/4L. Closed has only odd harmonics.
Resonance in Pipes
Standing waves in air columns โ open vs closed pipes
Open pipe (open both ends): resonates at L = nฮป/2. Fundamental frequency fโ = v/2L. Has all harmonics. Closed pipe (one closed end): resonates at L = nฮป/4 (odd n only). Fundamental fโ = v/4L. Has only odd harmonics (1st, 3rd, 5th...). Flutes are open pipes, clarinets are closed.
Wave Polarization
Polarization: transverse waves can be polarized. Sound waves (longitudinal) cannot be polarized.
Wave Polarization
Why some waves can be polarized and others cannot
Transverse waves oscillate perpendicular to travel โ can be restricted to one plane (polarized). Light can be polarized by filters, reflection, or scattering. Polaroid sunglasses reduce glare by blocking horizontally polarized reflected light. Longitudinal waves (sound) oscillate parallel to travel โ cannot be polarized.
Wave Speed Equation
v = fฮป โ speed equals frequency times wavelength. Higher frequency = shorter wavelength (speed fixed in medium).
The fundamental wave equation linking speed, frequency, and wavelength
In a given medium, wave speed is fixed โ so higher frequency always means shorter wavelength
v = fฮป where v = wave speed (m/s), f = frequency (Hz), ฮป = wavelength (m). Speed of sound in air: ~343 m/s at 20ยฐC (increases with temperature). Speed of light in vacuum: c = 3ร10โธ m/s. In a medium: v = c/n where n is the refractive index. When a wave crosses into a new medium: frequency stays constant, speed and wavelength both change. Period T = 1/f. Angular frequency ฯ = 2ฯf.
v = fฮป
Speed = frequency ร wavelength โ the fundamental wave equation
New medium
Frequency unchanged, speed changes โ wavelength changes
T = 1/f
Period is reciprocal of frequency
Beat Frequency
f_beat = |fโ - fโ| โ two close frequencies interfere to create a pulsing beat at their difference
Why two slightly different frequencies create a periodic rise and fall in volume
Beats are constructive and destructive interference cycling at the difference frequency
When two waves of slightly different frequencies overlap, they alternately reinforce and cancel: f_beat = |fโ โ fโ|. At the beat: the waves are in phase (loud). Between beats: they are out of phase (quiet). Used for tuning instruments โ when beats disappear, the two sources are at the same frequency. Faster beats = greater frequency difference. Used in ultrasound imaging (beat frequency between transmitted and reflected waves gives velocity via Doppler).
f_beat = |fโ - fโ|
Absolute difference of the two frequencies
Tuning use
Beats disappear when frequencies are equal โ used to tune instruments
Fast beats
Large frequency difference. Slow beats = nearly in tune.
The full electromagnetic spectrum from lowest to highest energy
All EM waves travel at c in vacuum โ only frequency and wavelength differ, determining energy E = hf
Radio waves: longest wavelength, lowest frequency, lowest energy. Microwaves: used in communication and cooking (resonates water molecules). Infrared: heat radiation, TV remotes. Visible: 400โ700 nm (violet to red). Ultraviolet: causes sunburn, kills bacteria. X-rays: penetrates soft tissue, ionizing. Gamma rays: shortest wavelength, highest energy, most penetrating โ from nuclear decay. Energy E = hf = hc/ฮป. Ionizing radiation: UV, X-ray, gamma (enough energy to knock electrons off atoms).
RMIVUXG
Radio โ Micro โ IR โ Visible โ UV โ X-ray โ Gamma
Visible
400โ700 nm โ violet (400) to red (700)
Ionizing
UV and above โ enough energy to remove electrons from atoms
Shock Waves and Sonic Boom
Mach number = v_source / v_sound. Mach > 1 = supersonic โ shock wave โ sonic boom
What happens when a source moves faster than the waves it produces
When the source outruns its own wavefronts they pile up into a cone-shaped shock wave
Mach number M = v_source / v_sound. M < 1: subsonic. M = 1: transonic (sound barrier). M > 1: supersonic โ source outruns wavefronts โ Mach cone forms. Half-angle of cone: sin ฮธ = v_sound / v_source = 1/M. Sonic boom: the shock wave passing over an observer โ not a one-time event but a continuous cone. Cherenkov radiation: same concept for charged particles moving faster than light in a medium โ produces blue glow in nuclear reactors.
Mach 1
Source speed = sound speed โ wavefronts pile up at source
Mach > 1
Supersonic โ cone forms, sin ฮธ = 1/M
Sonic boom
Continuous cone passing observer โ not a single event
🎓 Common Exam Questions
Q: Explain the Doppler effect and derive the observed frequency equation.
A: The Doppler effect: when source and observer move relative to each other, the observed frequency differs from the emitted frequency. General formula: f_obs = f_source ร (v ยฑ v_observer)/(v โ v_source). Sign convention: upper signs when moving toward each other (higher frequency), lower signs when moving apart (lower frequency). Moving toward: f_obs > f_source (blue shift). Moving away: f_obs < f_source (red shift). Applications: police radar, medical ultrasound, astronomical redshift (Hubble's discovery that galaxies recede โ used to measure expansion of universe). Derivation intuition: if source moves toward observer, wavefronts are compressed โ shorter wavelength โ higher frequency.
Q: Describe standing waves โ how do they form and what are nodes and antinodes?
A: Standing waves form when two identical waves traveling in opposite directions superpose. Result: the pattern appears stationary โ nodes don't move. Nodes: points of zero displacement โ destructive interference always. Distance between adjacent nodes = ฮป/2. Antinodes: points of maximum displacement โ constructive interference always. Between each pair of adjacent nodes is one antinode. Open pipe (both ends open): both ends are antinodes. Harmonics: all integer multiples of fundamental. f_n = nv/2L. Closed pipe (one end closed): closed end is node, open end is antinode. Only odd harmonics: f_n = nv/4L (n = 1, 3, 5...). String fixed at both ends: same as open pipe formula. Applications: musical instruments, microwave ovens, laser cavities.
Q: What is resonance and why is it important in physics and engineering?
A: Resonance: when a system is driven at its natural frequency, amplitude grows dramatically (limited only by damping). Natural frequency: the frequency at which a system oscillates freely after being disturbed. Resonance condition: driving frequency = natural frequency. Q factor (quality factor): measure of sharpness of resonance peak. High Q = sharp resonance (low damping). Examples: Tacoma Narrows Bridge (1940) โ wind drove bridge at natural frequency โ catastrophic oscillations โ collapse. Wine glass shattered by singer hitting resonant frequency. MRI: nuclear magnetic resonance โ hydrogen nuclei resonate at specific radio frequency. Electrical LC circuits: resonant at f = 1/(2ฯโLC) โ used in radio tuning. Avoiding resonance: buildings, bridges, aircraft all must avoid natural frequencies of expected vibrations.
Q: Explain wave interference โ Young's double-slit experiment and its significance.
A: Young's double-slit (1801): light through two slits creates an interference pattern of bright and dark bands โ proved light is a wave. Bright fringes (constructive): path difference = mฮป (m = 0, ยฑ1, ยฑ2...). Dark fringes (destructive): path difference = (m + ยฝ)ฮป. Fringe spacing: y = mฮปL/d where L = screen distance, d = slit separation. Significance: (1) Proved light behaves as a wave โ contradicted Newton's corpuscle theory. (2) Later experiments with single photons showed same pattern โ quantum weirdness. (3) The experiment works with electrons, atoms, even molecules โ matter waves are real. Thin film interference: same principle โ path difference from front and back surface reflections. Soap bubble colors: different wavelengths constructively interfere at different thicknesses.
Q: Describe the electromagnetic spectrum โ properties and applications of each region.
A: All EM waves travel at c = 3ร10โธ m/s in vacuum. Energy E = hf = hc/ฮป. From lowest to highest frequency (longest to shortest wavelength): Radio waves (>1 mm): AM/FM broadcasting, radar, MRI. Microwaves (1 mmโ1 cm): WiFi, cell phones, cooking (resonates water at 2.45 GHz). Infrared (700 nmโ1 mm): heat radiation, TV remotes, thermal imaging, night vision. Visible (400โ700 nm): only region human eye detects. Ultraviolet (10โ400 nm): sunburn, vitamin D synthesis, sterilization (kills bacteria). X-rays (0.01โ10 nm): medical imaging, security scanning, ionizing radiation. Gamma rays (<0.01 nm): nuclear decay, cancer treatment (radiation therapy), highest energy, most penetrating. Ionizing radiation (UV and above): enough energy to remove electrons โ causes DNA damage. Non-ionizing (radio through IR): cannot ionize atoms directly.