Wave Dynamics Chapter 1. Waves
Recommended Article : 【Physics】 Physics Table of Contents
1. Definition of Waves
⑴ Wave : Energy transmission without the movement of a medium (substance)
① Standing waves can be seen as exceptions.
② Thermal conduction fits the definition, but it is not a wave.
⑵ (Distinct Concept) Vibration : Periodic occurrence at a specific location. Waves propagate to different places.
⑶ (Distinct Concept) Pulse : A disturbance that passes through only once.
⑷ Wave Source : The point where a wave originates.
⑸ Medium : Substance that transmits waves.
2. Types of Waves
⑴ Classification based on the presence of a medium
① Elastic waves : Mechanical waves. Require a medium.
○ External force causes deformation → Deformation induces stress → Stress causes deformation of adjacent molecules.
○ Examples: Water waves, sound waves, seismic waves.
② Particle waves : Do not require a medium.
○ Electromagnetic waves (light) : Gauge invariance, involvement of scalar, vector, tensor.
○ Gravitational waves : Gauge invariance, involvement of scalar, vector, tensor.
○ Matter waves : Probability amplitude density, Fermions → Spinors (approximately in the context of relativity)
⑵ Classification based on the direction of vibration
① Longitudinal waves (e.g., P-waves in earthquakes, sound waves, longitudinal waves in slinky)
○ Definition: Waves in which the direction of propagation aligns with the direction of vibration of the medium.
○ Vibration direction and energy transfer direction are parallel.
○ Involvement of compressional forces.
② Transverse waves (e.g., S-waves in earthquakes, electromagnetic waves, transverse waves in slinky)
○ Definition: Waves in which the direction of propagation is perpendicular to the direction of vibration of the medium.
○ Vibration direction and energy transfer direction are perpendicular.
○ Involvement of shear forces.
③ While water waves are classified as transverse waves, they also have a longitudinal component due to the circular motion of water particles.
⑶ Classification based on waveform
① Wavefront : A line or surface connecting points of equal phase, like a crest or trough.
② Plane waves : Wavefront forms a straight line.
③ Spherical waves : Wavefront forms a circle.
3. Representation of Waves
⑴ Crest : The point of maximum displacement in a wave.
⑵ Trough : The point of minimum displacement in a wave.
⑶ Amplitude : Maximum displacement of the medium during vibration.
⑷ Wavelength : Distance between two neighboring points in phase, such as crest to crest or trough to trough.
⑸ Period : Time T for a repeating pattern of a wave.
⑹ Frequency : Number of pattern repetitions per unit time, reciprocal of period (f).
⑺ Wave Velocity
⑻ Antinode : Point of maximum amplitude.
⑼ Node : Point of minimum amplitude.
⑽ Example
Figure 1. Determination of wavelength, frequency, and velocity of a wave example
① (a) represents the situation at t = 0, and (b) represents the situation at t = 1 s.
② From (a), we can determine that λ = 4 m.
③ From (a) and (b), we can determine that v = 1 m/s.
④ Frequency can be determined using the formula.
4. Wave Functions
⑴ Definition : Language used to describe progressive waves and periodic waves.
⑵ General Expressions
① Wavenumber :
② Angular frequency :
⑶ Wave Velocity : Speed at which a wave propagates.
① Phase velocity : Tracing points with the same phase.
○ ω/k ＞ 0 : vp ＞ 0, indicating forward propagation.
○ ω/k ＜ 0 : vp ＜ 0, indicating backward propagation.
② Example 1. String Waves
○ Proof 1. Dimensional analysis
○ [μ] = ML-1
○ [T] = MLT-2
○ [v] = LT-1
○ [（T / μ）0.5 ] = [v] ⇔ v ∝ （T / μ）0.5
○ Proof 2. Derived from the wave equation
Figure 2. Derivation of velocity of string waves through the wave equation Wave Equation
○ Proof 3. Derived through thought experiment : Utilizing the centripetal force formula
Figure 3. Derivation of velocity using the centripetal force formula
③ Example 2. Slinky Waves : Faster with higher spring constant
④ Example 3. Speed of Water Waves : Detailed explanation in Oceanography
○ General formula for wave speed
○ Speed of deep-water waves
○ Speed of shallow-water waves
⑤ Example 4. Speed of Sound
○ Solid > Liquid > Gas, in order of increasing speed
○ Faster at higher air temperatures
⑥ Example 5. Speed of Electromagnetic Waves : Same in vacuum for all electromagnetic waves
⑷ Displacement-Related Physical Quantities
① Displacement velocity
② Displacement acceleration
③ Curvature
④ Others
⑸ Energy
① Tension and linear density remain constant when amplitude is small.
② Kinetic energy : Utilized by
③ Potential energy : Utilized by
④ Total energy
○ Elastic waves’ total energy is proportional to the square of frequency.
○ Quantum particles’ total energy is proportional to frequency, i.e., E = hf.
○ Photons, as quantum particles, have total energy proportional to frequency.
⑤ Energy transmission rate
Input : 2019.05.03 17:14