Types of Waves and Wave Equations: Understanding Wave Motion
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Chapters 16, 17 Waves
Types of waves • Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) • Electromagnetic – governed by electricity and magnetism equations, may exist without any medium • Matter – governed by quantum mechanical equations
Types of waves • Depending on the direction of the displacement relative to the direction of propagation, we can define wave motion as: • Transverse – if the direction of displacement is perpendicular to the direction of propagation • Longitudinal – if the direction of displacement is parallel to the direction of propagation
Types of waves • Depending on the direction of the displacement relative to the direction of propagation, we can define wave motion as: • Transverse – if the direction of displacement is perpendicular to the direction of propagation • Longitudinal – if the direction of displacement is parallel to the direction of propagation
The wave equation • Let us consider transverse waves propagating without change in shape and with a constant wave velocityv • We will describe waves via vertical displacementy(x,t) • For an observer moving with the wave • the wave shape doesn’t depend on time y(x’) = f(x’)
The wave equation • For an observer at rest: • the wave shape depends on time y(x,t) • the reference frame linked to the wave is moving with the velocity of the wave v
The wave equation • We considered a wave propagating with velocity v • For a medium with isotropic (symmetric) properties, the wave equation should have a symmetric solution for a wave propagating with velocity –v
The wave equation • Therefore, solutions of the wave equation should have a form • Considering partial derivatives
The wave equation • Therefore, solutions of the wave equation should have a form • Considering partial derivatives
The wave equation • Therefore, solutions of the wave equation should have a form • Considering partial derivatives
The wave equation • The wave equation (not the only one having solutions of the form y(x,t) = f(x ± vt)): • It works for longitudinal waves as well • v is a constant and is determined by the properties of the medium. E.g., for a stretched string with linear density μ = m/l under tension τ
Superposition of waves • Let us consider two different solutions of the wave equation • Superposition principle – a sum of two solutions to the wave equation is a solution to the wave equation +
Superposition of waves • Overlapping solutions of the wave equation algebraically add to produce a resultant (net) wave • Overlapping solutions of the wave equation do not in any way alter the travel of each other
Chapter 16 Problem 27
Reflection of waves at boundaries • Within media with boundaries, solutions to the wave equation should satisfy boundary conditions. As a results, waves may be reflected from boundaries • Hard reflection – a fixed zero value of deformation at the boundary – a reflected wave is inverted • Soft reflection – a free value of deformation at the boundary – a reflected wave is not inverted
Sinusoidal waves • One of the most characteristic solutions of the wave equation is a sinusoidal wave: • ym - amplitude, φ - phase constant
Wavelength • “Freezing” the solution at t = 0 we obtain a sinusoidal function of x: • Wavelengthλ – smallest distance (parallel to the direction of wave’s travel) between repetitions of the wave shape
Wave number • On the other hand: • Angular wave number: k = 2π / λ
Angular frequency • Considering motion of the point at x = 0 • we observe a simple harmonic motion (oscillation) : • For simple harmonic motion (Chapter 15): • Angular frequencyω
Frequency, period • Definitions of frequency and period are the same as for the case of rotational motion or simple harmonic motion: • Therefore, for the wave velocity
Chapter 16 Problem 7
Interference of waves • Interference – a phenomenon of combining waves, which follows from the superposition principle • Considering two sinusoidal waves of the same amplitude, wavelength, and direction of propagation • The resultant wave:
Interference of waves • If φ = 0 (Fully constructive) • If φ = π (Fully destructive) • If φ = 2π/3 (Intermediate)
Interference of waves • Considering two sinusoidal waves of the same amplitude, wavelength, but running in opposite directions • The resultant wave:
Nodes Antinodes • Interference of waves • If two sinusoidal waves of the same amplitude and wavelength travel in opposite directions, their interference with each other produces a standing wave
Chapter 16 Problem 54
Standing waves and resonance • For a medium with fixed boundaries (hard reflection) standing waves can be generated because of the reflection from both boundaries: resonance • Depending on the number of antinodes, different resonances can occur
Standing waves and resonance • Resonance wavelengths • Resonance frequencies
Harmonic series • Harmonic series – collection of all possible modes - resonant oscillations (n – harmonic number) • First harmonic (fundamental mode):
More about standing waves • Longitudinal standing waves can also be produced • Standing waves can be produced in 2 and 3 dimensions as well
Phasors • For superposition of waves it is convenient to use phasors – vectors that have magnitude equal to the amplitude of the wave and rotating around the origin • Two phase-shifted waves with the same frequency can be represented by phasors separated by a fixed angle
Phasors • To obtain a resultant wave (add waves) one has to add phasors as vectors • Using phasors one can add waves of different amplitudes
Rate of energy transmission • As the wave travels it transports energy, even though the particles of the medium don’t propagate with the wave • The average power of energy transmission for the sinusoidal solution of the wave equation • Exact expression depends on the medium or the system through which the wave is propagating
Sound waves • Sound – longitudinal waves in a substance (air, water, metal, etc.) with frequencies detectable by human ears (between ~ 20 Hz and ~ 20 KHz) • Ultrasound – longitudinal waves in a substance (air, water, metal, etc.) with frequencies higher than detectable by human ears (> 20 KHz) • Infrasound – longitudinal waves in a substance (air, water, metal, etc.) with frequencies lower than detectable by human ears (< 20 Hz)
Speed of sound • Speed of sound: • ρ – density of a medium, B – bulk modulus of a medium • Traveling sound waves
Chapter 17 Problem 12
Intensity of sound • Intensity of sound – average rate of sound energy transmission per unit area • For a sinusoidal traveling wave: • Decibel scale • β – sound level; I0 = 10-12 W/m2 – lower limit of human hearing
Chapter 17 Problem 18
Sources of musical sound • Music produced by musical instruments is a combination of sound waves with frequencies corresponding to a superposition of harmonics (resonances) of those musical instruments • In a musical instrument, energy of resonant oscillations is transferred to a resonator of a fixed or adjustable geometry
Open pipe resonance • In an open pipe soft reflection of the waves at the ends of the pipe (less effective than form the closed ends) produces standing waves • Fundamental mode (first harmonic): n = 1 • Higher harmonics:
Organ pipes • Organ pipes are open on one end and closed on the other • For such pipes the resonance condition is modified:
Musical instruments • The size of the musical instrument reflects the range of frequencies over which the instrument is designed to function • Smaller size implies higher frequencies, larger size implies lower frequencies
Musical instruments • Resonances in musical instruments are not necessarily 1D, and often involve different parts of the instrument • Guitar resonances (exaggerated) at low frequencies:
Musical instruments • Resonances in musical instruments are not necessarily 1D, and often involve different parts of the instrument • Guitar resonances at medium frequencies:
Musical instruments • Resonances in musical instruments are not necessarily 1D, and often involve different parts of the instrument • Guitar resonances at high frequencies:
Beats • Beats – interference of two waves with close frequencies +
Sound from a point source • Point source – source with size negligible compared to the wavelength • Point sources produce spherical waves • Wavefronts – surfaces over which oscillations have the same value • Rays – lines perpendicular to wavefronts indicating direction of travel of wavefronts
Interference of sound waves • Far from the point source wavefronts can be approximated as planes – planar waves • Phase difference and path length difference are related: • Fully constructive interference • Fully destructive interference
Variation of intensity with distance • A single point emits sound isotropically – with equal intensity in all directions (mechanical energy of the sound wave is conserved) • All the energy emitted by the source must pass through the surface of imaginary sphere of radius r • Sound intensity • (inverse square law)
