Waves

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Waves transfer energy without transferring matter. They can be mechanical (require a medium) or electromagnetic (don’t require a medium).

Wave Equation: $$v = f\lambda$$

• $v$ = wave velocity (m/s)
• $f$ = frequency (Hz)
• $\lambda$ = wavelength (m)

Types of Waves:

Mechanical waves:

• Transverse: Displacement perpendicular to direction of propagation
  • Examples: Light, waves on strings, ripples on water
  • Have crests and troughs
• Longitudinal: Displacement parallel to direction of propagation
  • Examples: Sound, seismic P-waves
  • Have compressions and rarefactions

Electromagnetic waves:

• Don’t require a medium
• Travel at $c = 3 \times 10^8$ m/s in vacuum
• Include: radio, microwave, infrared, visible light, UV, X-ray, gamma ray

Doppler Effect: The observed frequency changes when source and observer move relative to each other: $$f’ = f \times \frac{v \pm v_o}{v \mp v_s}$$

• Source moving toward observer: denominator $-$ (lower frequency in denominator)
• Source moving away: denominator $+$
• Observer moving toward source: numerator $+$
• Observer moving away: numerator $-$

Stationary (Standing) Waves: Formed when two waves of the same frequency and amplitude travel in opposite directions. They have:

• Nodes: Points of zero amplitude (no displacement)
• Antinodes: Points of maximum amplitude

⚡ MDCAT Tip: For Doppler Effect, remember: when source and observer APPROACH each other, frequency INCREASES. When they RECEDE from each other, frequency DECREASES. Use the mnemonic “Approach = Up” for pitch.

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🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding.

Wave Properties:

Reflection:

• When waves hit a boundary, they reflect
• Angle of incidence = Angle of reflection
• For a fixed end: wave inverts on reflection
• For a free end: wave does not invert on reflection

Refraction: When waves enter a new medium: $$v_1 \sin\theta_1 = v_2 \sin\theta_2$$ or $$\frac{\sin\theta_1}{\sin\theta_2} = \frac{v_1}{v_2} = \frac{\lambda_1}{\lambda_2}$$

Note: Frequency $f$ does NOT change when waves enter a new medium.

Diffraction: Waves spread out when passing through an aperture or around an obstacle:

• Most noticeable when aperture/obstacle size ≈ wavelength
• Diffraction is more pronounced for longer wavelengths

Interference: When two or more waves meet, they superpose:

• Constructive interference: Path difference = $n\lambda$ (in phase)
• Destructive interference: Path difference = $(n + \frac{1}{2})\lambda$ (out of phase)

Beats: When two waves of slightly different frequencies interfere: $$f_{beat} = |f_1 - f_2|$$

Beats are used for tuning instruments.

Stationary Waves on Strings:

For a string fixed at both ends: $$\lambda_n = \frac{2L}{n}, \quad f_n = \frac{nv}{2L}$$

Mode	Wavelength	Frequency
Fundamental (1st harmonic)	$2L$	$f_1 = \frac{v}{2L}$
2nd harmonic	$L$	$2f_1$
3rd harmonic	$\frac{2L}{3}$	$3f_1$

Speed of wave on string: $$v = \sqrt{\frac{T}{\mu}}$$ Where $T$ = tension, $\mu$ = linear mass density (kg/m).

Sound Waves in Pipes:

Open pipe (open at both ends): $$\lambda_n = \frac{2L}{n}, \quad f_n = \frac{nv}{2L}$$ All harmonics present (n = 1, 2, 3…)

Closed pipe (closed at one end): $$\lambda_n = \frac{4L}{n}, \quad f_n = \frac{nv}{4L}$$ Only odd harmonics present (n = 1, 3, 5…)

⚡ Common Student Mistakes: Forgetting that frequency doesn’t change during refraction. Confusing which situations give higher/lower Doppler frequency. Mixing up node and antinode positions.

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🔴 Extended — Deep Study (3mo+)

Comprehensive theory for thorough preparation.

Wave Intensity: $$I = \frac{P}{A} = \frac{E}{At}$$ For a point source radiating uniformly: $$I \propto \frac{1}{r^2}$$

For a sinusoidal wave: $$I = \frac{1}{2}\rho v \omega^2 A^2$$ Where $A$ is amplitude (not to be confused with area).

Energy in Waves: For a progressive wave:

• Kinetic energy: $\frac{1}{2}m v^2$ (maximum at equilibrium)
• Potential energy: Maximum at maximum displacement
• Total energy is constant and proportional to $A^2$

Doppler Effect — Detailed:

Moving source: $$f’ = \frac{v}{v \mp v_s} f$$

Moving observer: $$f’ = \frac{v \pm v_o}{v} f$$

Moving source and observer: $$f’ = \frac{v \pm v_o}{v \mp v_s} f$$

Example: An ambulance moving at 30 m/s toward a stationary observer sounds its siren at 500 Hz. Speed of sound = 340 m/s. $$f’ = \frac{340}{340 - 30} \times 500 = \frac{340}{310} \times 500 \approx 548 \text{ Hz}$$

Doppler Effect for Light: $$f’ = f\sqrt{\frac{1+\beta}{1-\beta}}$$ Where $\beta = v/c$ (source velocity as fraction of light speed).

Redshift (receding source): observed wavelength increases. Blueshift (approaching source): observed wavelength decreases.

Huygens’ Principle: Every point on a wavefront acts as a source of secondary wavelets. The new wavefront is the envelope of all these wavelets.

Applications:

• Explaining reflection and refraction
• Diffraction patterns
• Single-slit diffraction

Young’s Double Slit Experiment: For two slits separated by distance $d$: $$\Delta x = \frac{\lambda D}{d}$$

Constructive: $\Delta x = n\lambda$ Destructive: $\Delta x = (n + \frac{1}{2})\lambda$

Where $D$ = distance to screen, $\Delta x$ = fringe separation.

Single Slit Diffraction: $$a\sin\theta = n\lambda$$ Where $a$ = slit width.

Central maximum is twice as wide as other maxima.

Intensity in Interference/Diffraction:

• Double slit: $I_{max} = 4I_0 \cos^2(\phi/2)$
• Single slit: $I = I_0 \left(\frac{\sin\alpha}{\alpha}\right)^2$ where $\alpha = \frac{\pi a\sin\theta}{\lambda}$

Shock Waves: When source speed $v_s >$ wave speed $v$: $$M = \frac{v_s}{v}$$

The shock wave front forms a cone (Mach cone): $$\sin\theta = \frac{v}{v_s} = \frac{1}{M}$$

Sonic boom occurs when the source passes the observer.

Seismic Waves:

• P-waves (primary): Longitudinal, faster (~6 km/s), travel through solids and liquids
• S-waves (secondary): Transverse, slower (~4 km/s), only travel through solids
• Surface waves: Travel along Earth’s surface, most destructive

Musical Instruments:

• String instruments: Standing waves on strings
• Wind instruments: Standing waves in air columns
• Percussion: Vibration of membranes, plates, or bars

Quality (Timbre): The same note sounds different on different instruments because of different harmonic content (overtones).

Electromagnetic Spectrum:

Type	Frequency	Wavelength	Production
Radio	< 3 GHz	> 0.1 m	Accelerating charges
Microwave	3-300 GHz	1 mm-0.1 m	Klystron, magnetron
Infrared	300 GHz-400 THz	0.7 μm-1 mm	Molecules, warm objects
Visible	400-750 THz	400-700 nm	Atoms
UV	750 THz-30 PHz	10-400 nm	Atoms
X-ray	30 PHz-30 EHz	0.01-10 nm	Inner electrons
Gamma	> 30 EHz	< 0.01 nm	Nuclear transitions

⚡ MDCAT Examination Patterns: Apply the wave equation $v = f\lambda$ in all contexts. Solve Doppler Effect problems for moving source and/or observer. Calculate standing wave frequencies for strings and pipes. Use Huygens’ Principle to explain wave phenomena. Solve double-slit and single-slit problems. Distinguish between transverse and longitudinal waves.