Chapter 10 Wave Motion • §1. Several Concepts of Mechanical Wave • §2. Wave Function of Simple Harmonic Wave • §3. Energy in Wave Motion, Energy Flux Density • §4. Huygens Principle, Diffraction and Interference of Waves • §5. Standing Waves • §6. Doppler Effect • §7. Plane Electromagnetic Waves
§1. Several Concepts of Mechanical Wave • 1. The formation of mechanical wave • 2. Transverse wave and longitudinal wave • 3. Wavelength, wave period and frequency, wave speed • 4. Wave line, wave surface, wave front
Notice 1. The formation of mechanical wave 1 Wave source An object which is oscillating mechanically 2 Medium Elastic medium which can propagate mechanical oscillation What are propagated is the oscillation states, while the mass points do not flow away.
2. Transverse wave and longitudinal wave 1 Transverse wave Characteristics: The oscillation directions of mass points are perpendicular to the direction of travel of the wave.
2 Longitudinal wave Characteristics: The oscillation directions of mass points are parallel to the direction of travel of the wave.
3 Complex wave Characteristics: Any complex wave motions can be viewed as a superposition of transverse waves or longitudinal waves.
1 Wavelength - A 3. Wavelength, wave period and frequency, wave speed y A O
3 Frequency 2 Period T The period (or frequency) of wave is equal to the oscillation period (or frequency) of the wave source.
4 Wave velocity The magnitude of the wave velocity depends on the nature of media.
In solid (transverse wave) (longitudinal wave) In liquid and gas (longitudinal wave)
4. Wave line, wave surface, wave front 1 Wave line The lines drawn with arrows along the direction of wave propagation 2 Wave surface The curved surface by connecting the points with the same phase on the different wave lines Wave front At one instant, the curved surface connected by every point with the original state of wave source In isotropic medium, wave line is perpendicular to wave surface.
Classification (1) Plane wave (2) Spherical wave
§2. Wave Function of Simple Harmonic Wave • 1. Wave function of simple harmonic wave • 2. The physical meaning of wave function
1. Wave function of simple harmonic wave In the homogeneous and non-absorbing medium as the wave source is in simple harmonic motion, the wave formed is called plane simple harmonic wave.
P x O A mass point is in simple harmonic motion at origin O. Its motion equation is:
At time t, the displacement of point P is: This is the function of plane simple harmonic wave spreading along the positive direction of Ox axis, and it is also called the wave equation of plane simple harmonic wave.
The equation can be written in the following three commonly used forms:
if 2. The physical meaning of wave function 1 When x is fixed then The equation gives the displacement of the mass point, which is x away from origin O, at different time.
The curve of displacement versus time for a simple harmonic motion of every mass point on wave line
if y ox x1 x2 2 When t is fixed then The equation represents the distribution of displacement of every mass point at the given time.
O x 3 When both x and t are in variation The equation expresses the overall situation of displacement varying with time of all mass points. waveform at time t waveform at time t+
P x O 4 If the plane simple harmonic wave travels along the negative x-direction:
§3. Energy in Wave Motion, Energy Flux Density • 1. The propagation of wave energy • 2. Energy flux and energy flux density
x O x O 1. The propagation of wave energy 1 Wave energy Take the longitudinal wave in a rod as an example: Kinetic energy of oscillation:
Elastic potential energy: Total energy of this volume element:
1) have the same phase. Discussion They all reach the maximum at the equilibrium position, whereas they are all zero at the maximum displacement. 2) The mechanical energy in each volume element is not constant. 3) Wave motion is a mode of dissemination of energy.
x O x O Energy density: Average density of energy :
udt S 2. Energy flux and energy flux density Energy flux: Average energy flux:
udt S Energy flux density：
§4. Huygens Principle, Diffraction and Interference of Waves • 1. Huygens principle • 2. The diffraction of waves • 3. The interference of waves
plane wave spherical wave O 1. Huygens principle The every point of a wave front in the medium may be considered the sources of emitting secondary wavelets that spread out in all directions, and at any later time the envelope of these secondary wavelets is the new wave front.
2. The diffraction of waves When wave strikes a barrier in the process of spreading, it can round the edge of the barrier and go on spreading in the shade area of the barrier. diffraction phenomena formed by water wave diffraction
3. The interference of waves 1 Superposition principle of waves The waves will keep their own properties without any change after they meet, and keep traveling in their original directions as if they had never met each other. The oscillation at any point in the area where the waves meet is the vector sum of their separate oscillation displacements produced by every wave existing at the same point independently.
2 The wave’s interference If there are two waves with the same frequency, the parallel oscillation direction, the same phase or the invariant phase difference, when they meet the oscillations of some areas are always strengthened and the oscillations of some other areas are always weakened.
* The conditions for constructive and destructive interference Oscillations of wave sources: Oscillations at point P:
“Phase Difference” conditions for interference when when Discussion
if then The difference of wave paths Phase difference constructive destructive
when when “Wave Path Difference”conditions for interference
§5. Standing Waves • 1. Formation of standing waves • 2. Equation of standing waves • 3. Phase jump • 4. Energy in standing waves • 5. Normal modes of oscillation
1. Formation of standing waves 1 Phenomena 2Conditions
3 Formation of standing waves The standing wave is a particular interference phenomenon that produced by two coherent waves with the same amplitude, frequency and wave speed traveling in the opposite direction along the same straight line.
2. Equation of standing waves the positive x-direction the negative x-direction
1 0 Discussion Equation of standing waves (1) Amplitude, , only depends on x
a when nodes antinodes b when
Conclusions some points remain still all the time; while the amplitudes of some other points are the maximum. Distance between two adjacent nodes Distance between two adjacent node and antinode antinode y node x
y x (2)Phase Conclusion 1 Between two adjacent nodes, the phase of every point is the same.
y x Conclusion 2 The phases of both sides of one node are opposite.
y x At any time, the standing wave has a certain waveform, but it does not appear to be moving in either direction along string. Every point oscillates in the vicinity of its own equilibrium position with the certain amplitude.