Wave Properties

Overview

A wave is a disturbance that transfers energy through a medium (or through space) without transferring matter. Understanding wave properties is fundamental to physics.

Types of Waves

Mechanical Waves

• Require a medium to travel
• Examples: sound, water waves, waves on a string

Electromagnetic Waves

• Do not require a medium
• Examples: light, radio waves, X-rays

By Direction of Oscillation

Transverse waves: Oscillation perpendicular to wave direction

• Example: light, waves on a string

Longitudinal waves: Oscillation parallel to wave direction

• Example: sound waves

Wave Parameters

Wavelength ($\lambda$)

Distance between two consecutive points in phase (e.g., crest to crest)

• Unit: meters (m)

Frequency ($f$)

Number of complete waves passing a point per second

• Unit: Hertz (Hz) = 1/s

Period ($T$)

Time for one complete wave to pass

• Unit: seconds (s)

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

Amplitude ($A$)

Maximum displacement from equilibrium

• Unit: meters (m)

Wave Speed ($v$)

Speed at which the wave propagates:

$$v = f\lambda = \frac{\lambda}{T}$$

Wave Equation

For a wave traveling in the +x direction:

$$y(x,t) = A \sin(kx - \omega t + \phi)$$

For a wave traveling in the -x direction:

$$y(x,t) = A \sin(kx + \omega t + \phi)$$

Wave Number

$$k = \frac{2\pi}{\lambda}$$

Angular Frequency

$$\omega = 2\pi f = \frac{2\pi}{T}$$

Relationship

$$v = \frac{\omega}{k} = f\lambda$$

Wave Speed in Different Media

Wave on a String

$$v = \sqrt{\frac{T}{\mu}}$$

Where $T$ = tension, $\mu$ = linear mass density (kg/m)

Sound in Air

$$v = 331 \text{ m/s} \times \sqrt{\frac{T}{273 \text{ K}}}$$

At 20°C: $v \approx 343$ m/s

Sound in General

$$v = \sqrt{\frac{B}{\rho}}$$

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

Energy in Waves

Intensity

Power per unit area:

$$I = \frac{P}{A}$$

Unit: W/m²

Intensity and Amplitude

$$I \propto A^2$$

Intensity and Distance (Point Source)

$$I = \frac{P}{4\pi r^2}$$

Intensity decreases as $1/r^2$ from a point source

Superposition Principle

When two waves overlap, the net displacement is the sum of individual displacements:

$$y_{\text{total}} = y_1 + y_2$$

Constructive Interference

Waves in phase add up:

• Path difference = $n\lambda$ ($n = 0, 1, 2, ...$)
• Maximum amplitude: $A_1 + A_2$

Destructive Interference

Waves out of phase cancel:

• Path difference = $(n + \frac{1}{2})\lambda$
• Minimum amplitude: $\lvert A_1 - A_2 \rvert$

Standing Waves

Formed by superposition of two waves traveling in opposite directions.

Nodes

Points of zero displacement

• Occur at $x = \frac{n\lambda}{2}$

Antinodes

Points of maximum displacement

• Occur at $x = \frac{(n + \frac{1}{2})\lambda}{2}$

Standing Wave Equation

$$y(x,t) = 2A \sin(kx) \cos(\omega t)$$

Standing Waves on a String

For a string fixed at both ends, length $L$:

Wavelengths:

$$\lambda_n = \frac{2L}{n} \quad (n = 1, 2, 3, ...)$$

Frequencies:

$$f_n = \frac{nv}{2L} = nf_1$$

Where $f_1$ = fundamental frequency (first harmonic)

Examples

Example 1: Wave Parameters

A wave has $\lambda = 2$ m and $f = 5$ Hz. Find wave speed and period.

$$v = f\lambda = 5 \times 2 = 10 \text{ m/s}$$ $$T = \frac{1}{f} = 0.2 \text{ s}$$

Example 2: Wave on String

A string ($\mu = 0.01$ kg/m) under 100 N tension. Find wave speed.

$$v = \sqrt{\frac{T}{\mu}} = \sqrt{\frac{100}{0.01}} = 100 \text{ m/s}$$

Example 3: Standing Waves

A 1.2 m guitar string produces a fundamental frequency of 220 Hz. Find wave speed.

$$\lambda_1 = 2L = 2.4 \text{ m}$$ $$v = f_1\lambda_1 = 220 \times 2.4 = 528 \text{ m/s}$$

Example 4: Harmonics

Find the first three harmonic frequencies for Example 3.

$$f_1 = 220 \text{ Hz}$$ $$f_2 = 2f_1 = 440 \text{ Hz}$$ $$f_3 = 3f_1 = 660 \text{ Hz}$$

Example 5: Intensity

Sound intensity at 5 m from a source is 0.4 W/m². Find intensity at 10 m.

$$I_1 r_1^2 = I_2 r_2^2$$ $$I_2 = I_1 \left(\frac{r_1}{r_2}\right)^2 = 0.4 \times \left(\frac{5}{10}\right)^2 = 0.1 \text{ W/m}^2$$