Wave Motion

Introduction to Wave Types and Wave Speed

Waves are disturbances that transfer energy through a medium (such as a string, water, or air) without transporting matter. They can be classified based on the direction of particle displacement relative to wave propagation.

• Transverse Waves: Displacement is perpendicular to the direction of wave motion (e.g., waves on a string).

• Longitudinal Waves: Displacement is parallel to the direction of wave motion (e.g., sound waves in air).

Key Terms:

• Wavelength (\(\lambda\)): The distance between two consecutive crests (transverse) or compressions (longitudinal).

• Amplitude (A): Half the vertical distance from crest to trough (transverse); not as important for longitudinal waves.

• Period (T): Time to complete one cycle.

• Frequency (f): Number of cycles per second, \(f = \frac{1}{T}\).

Wave Speed Relationship:

• All waves obey \(v = \lambda f\).

Example: The wavelength of a sound wave with frequency 260 Hz and speed 343 m/s is \(\lambda = \frac{v}{f} = \frac{343}{260} \approx 1.32\) m.

Velocity of Waves on a String

The speed of a wave on a string depends on the string's tension, mass, and length. The wave speed is given by:

• \(v = \sqrt{\frac{F_T}{\mu}}\), where \(F_T\) is the tension and \(\mu = \frac{m_{string}}{L_{string}}\) is the mass per unit length.

Example: For a string with tension 100 N, mass 0.5 kg, and length 1.2 m, and a wavelength of 0.15 m, the frequency is found using \(v = \lambda f\) and \(v = \sqrt{\frac{F_T}{\mu}}\).

Solving Waves on Strings Problems

• Wave speed (v) depends on tension, mass, and length.

• Frequency (f) is determined by the oscillator, not by the string's properties.

• Changing tension changes v; changing oscillator frequency changes f and \(\lambda\), not v.

Key Equations:

• \(v = \lambda f\)

• \(v = \sqrt{\frac{F_T}{\mu}}\)

Speed of Longitudinal Waves in Fluids and Solids

Longitudinal wave speed depends on the medium's properties:

• In fluids: \(v = \sqrt{\frac{\beta}{\rho}}\), where \(\beta\) is the bulk modulus and \(\rho\) is the density.

• In solids: \(v = \sqrt{\frac{Y}{\rho}}\), where Y is Young's modulus.

Example: For a liquid with density 1200 kg/m3 and frequency 400 Hz, wavelength 8 m, the bulk modulus can be calculated using the above formula.

Wave Intensity

Wave intensity measures the power transmitted per unit area:

• \(I = \frac{P}{A}\)

• For spherical waves: \(I = \frac{P}{4\pi r^2}\)

Units: W/m2

Example: A 500 W loudspeaker radiates sound in all directions. At 10 m, \(I = \frac{500}{4\pi (10)^2} = 0.398\) W/m2.

The Inverse-Square Law for Intensity

As waves spread out, intensity decreases with the square of the distance:

• \(I_1/I_2 = (r_2/r_1)^2\)

• Power remains constant; surface area increases; intensity decreases.

Example: If intensity is 0.25 W/m2 at 15 m, at what distance is intensity 0.01 W/m2? Use the inverse-square law to solve.

Wave Functions

A wave function describes the displacement of a wave as a function of position and time:

• \(y(x, t) = A \sin(kx \pm \omega t)\) or \(y(x, t) = A \cos(kx \pm \omega t)\)

• \(k = \frac{2\pi}{\lambda}\) (wavenumber)

• \(\omega = 2\pi f = \frac{2\pi}{T}\) (angular frequency)

The sign in \(kx \pm \omega t\) depends on the direction of wave travel.

Calculating Wave Speed Using the Wavefunction

Given a wave function, wave speed can be found by:

• \(v = \frac{\omega}{k}\)

Example: For \(y(x, t) = 3\cos(0.4x - 6t)\), \(v = \frac{6}{0.4} = 15\) m/s.

Transverse Velocity of Waves

The transverse velocity of a particle on a wave is the time derivative of the displacement:

• \(v_T(x, t) = \frac{\partial y}{\partial t}\)

• For \(y(x, t) = A \sin(kx \pm \omega t)\), \(v_T(x, t) = \pm A\omega \cos(kx \pm \omega t)\)

• Maximum transverse velocity: \(v_{T, max} = A\omega\)

Writing Wave Functions Using the Phase Constant

If a wave does not start at y = 0 or y = ±A, a phase constant \(\phi\) is included:

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

• \(\phi\) shifts the wave left or right.

Steps:

1. Write the equation with \(\phi\).

2. Determine the sign and value of \(\phi\) using initial conditions.

Superposition of Sinusoidal Waves

When two or more waves overlap, their displacements add algebraically (Principle of Superposition):

• \(y_{net} = y_1 + y_2\)

This applies to both sine and cosine functions.

Wave Interference & Superposition

When waves meet, they interfere:

• Constructive Interference: Displacements have the same sign; amplitudes add.

• Destructive Interference: Displacements have opposite signs; amplitudes subtract.

Example: Two pulses of amplitude 1 overlap: resultant amplitude is 2 (constructive); if one is inverted, resultant is 0 (destructive).

Introduction to Transverse Standing Waves

Standing waves form when two waves of the same frequency and amplitude travel in opposite directions and interfere. At certain frequencies (harmonics), the interference creates a stationary pattern.

• Fundamental Frequency (f1): The lowest frequency, n = 1.

• Harmonics: Higher frequencies, n > 1, are integer multiples of the fundamental.

• \(f_n = n f_1\)

Equations for Transverse Standing Waves

Standing waves on a string fixed at both ends have specific frequencies and wavelengths:

• \(f_1 = \frac{v}{2L}\) (fundamental)

• \(f_n = n f_1 = \frac{n v}{2L}\) (nth harmonic)

• \(\lambda_n = \frac{2L}{n}\) (wavelength of nth harmonic)

Overtones: The nth overtone is the (n+1)th harmonic.

Calculating Properties of Standing Waves Using Wavefunctions

The wavefunction for a standing wave is:

• \(y_n(x, t) = A_{SW} \sin(kx) \sin(\omega t)\)

• Nodes: Points where \(\sin(kx) = 0\); Antinodes: Points where \(\sin(kx) = \pm 1\).

Example: For \(y(x, t) = (5\text{ cm}) \sin[0.034x] \sin[50t]\), amplitude of original waves is twice the standing wave amplitude.

Summary Table: Key Wave Equations