Transverse waves:

• Particle motion perpendicular to wave propagation
• Examples: waves on string, electromagnetic waves
• Can be polarized

Longitudinal waves:

• Particle motion parallel to wave propagation
• Examples: sound waves, compression waves
• Cannot be polarized

Both have same wave equations but different particle motion.

• Amplitude (A): maximum displacement from equilibrium
• Wavelength (λ): distance between successive identical points (crest to crest)
• Frequency (f): number of complete cycles per second (Hz)
• Period (T): time for one complete cycle

Relationships: $T = \frac{1}{f}$ $f = \frac{1}{T}$

$v = \lambda f$

Where:

• v = wave speed
• $\lambda$ = wavelength
• f = frequency

For waves on string: $v = \sqrt{\frac{T}{\mu}}$

Where T = tension, $\mu$ = mass per unit length

For sound: $v = \sqrt{\frac{B}{\rho}}$

Where B = bulk modulus, $\rho$ = density

Sound is a longitudinal mechanical wave.

Properties:

• Requires medium (cannot travel through vacuum)
• Longitudinal: particle motion parallel to propagation
• Speed depends on medium properties

Speed of sound:

• Air (20 degrees C): 343 m/s
• Water: ~1480 m/s
• Steel: ~5960 m/s

Frequency range of human hearing: 20 Hz - 20 kHz

Standing waves: superposition of two waves traveling in opposite directions.

String fixed at both ends:

• Wavelengths: $\lambda_n = \frac{2L}{n}$ (n = 1, 2, 3, ...)
• Frequencies: $f_n = \frac{nv}{2L} = nf_1$
• Fundamental: $f_1 = \frac{v}{2L}$

Pipe open at both ends:

• Similar to string: $\lambda_n = \frac{2L}{n}$, $f_n = \frac{nv}{2L}$

Pipe open at one end, closed at other:

• Wavelengths: $\lambda_n = \frac{4L}{n}$ (n = 1, 3, 5, ... odd only)
• Frequencies: $f_n = \frac{nv}{4L}$ (odd harmonics only)

Nodes and antinodes:

• Node: point of zero displacement
• Antinode: point of maximum displacement

Beat frequency when two waves of slightly different frequencies interfere:

$f_{beat} = |f_1 - f_2|$

Physical mechanism:

• Constructive and destructive interference alternates
• Audible as periodic variation in loudness

Applications:

• Tuning musical instruments
• Detecting frequency differences

Apparent frequency change when source and/or observer move.

Moving source: $f' = \frac{v}{v \pm v_s} f$

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

General case (both moving): $f' = \frac{v \pm v_o}{v \mp v_s} f$

Sign conventions:

• • for approach (higher frequency)
• • for recession (lower frequency)

Applications: radar, medical ultrasound, astronomy

Electromagnetic waves organized by frequency/wavelength:

All travel at speed c in vacuum.

$c = 3.00 \times 10^8$ m/s (in vacuum)

In medium: $v = \frac{c}{n}$

Where n = refractive index

Max possible speed according to special relativity.

Restriction of wave oscillation to specific direction.

Linear polarization:

• Transverse waves only
• E-field oscillates in single plane

Achieving polarization:

• Polarizing filters (only allow specific direction)
• Reflection (Brewster's angle)
• Birefringent materials

Malus's law: $I = I_0\cos^2\theta$

Where $\theta$ = angle between polarization directions

Superposition of waves can produce constructive or destructive interference.

Constructive interference:

• Waves in phase (peaks align with peaks)
• Amplitude increases
• Path difference: $\Delta d = n\lambda$ (n = 0, 1, 2, ...)

Destructive interference:

• Waves out of phase (peaks align with troughs)
• Amplitude decreases
• Path difference: $\Delta d = (n + \frac{1}{2})\lambda$

Superposition principle: resultant wave = sum of individual waves.

Bending of waves around obstacles or through openings.

Single-slit diffraction:

• First minimum at: $a\sin\theta = \lambda$
• a = slit width
• Broader diffraction for smaller slit width
• Stronger diffraction for longer wavelength

Diffraction limit on resolution: $\theta = 1.22\frac{\lambda}{D}$

Classic demonstration of wave nature of light.

Bright fringes (constructive): $d\sin\theta = m\lambda$ (m = 0, ±1, ±2, ...)

Dark fringes (destructive): $d\sin\theta = (m + \frac{1}{2})\lambda$

Fringe spacing: $\Delta y = \frac{\lambda L}{d}$

Where:

• d = slit separation
• L = distance to screen
• $\lambda$ = wavelength

Multiple slits produce sharper interference pattern.

Principal maxima: $d\sin\theta = m\lambda$

Where d = grating spacing (distance between adjacent slits)

Used in spectroscopy to separate wavelengths.

Advantages over double slit:

• Sharper, brighter maxima
• Better wavelength resolution

Interference from reflections at top and bottom of thin film.

Phase shifts:

• Reflection off denser medium: $\pi$ (180 degrees ) phase shift
• Reflection off less dense medium: no phase shift

Constructive interference (maxima):

• With one phase shift: $2t = (m + \frac{1}{2})\frac{\lambda}{n}$
• With two phase shifts (or none): $2t = m\frac{\lambda}{n}$

Destructive interference (minima):

• With one phase shift: $2t = m\frac{\lambda}{n}$
• With two phase shifts (or none): $2t = (m + \frac{1}{2})\frac{\lambda}{n}$

Where t = film thickness, n = film index

Applications:

• Anti-reflective coatings
• Colorful soap bubbles, oil films
• Optical filters