NEB Physics Crash Course: Sound & Wave Propagation (Day 4)

A wave is a mode of energy transfer in periodic manner. Key parameters: (1) Amplitude (A) - height of crest/depth of trough, measured in meters; (2) Wavelength (λ) - distance between consecutive crests/troughs, measured in meters; (3) Frequency (ν) - number of waves passing a point per second, measured in Hz; (4) Time period (T) - time for one complete oscillation, measured in seconds; (5) Velocity (v) - speed of wave propagation, measured in m/s; (6) Wave number (ν̄) - number of waves per unit length, measured in m⁻¹. The fundamental relationship is v = νλ, showing wave velocity equals frequency multiplied by wavelength.

Five fundamental parameters describe wave motion: (1) Amplitude is the maximum displacement from equilibrium position, indicating wave intensity, (2) Time period is the time for one complete oscillation, (3) Frequency is the number of oscillations per second (reciprocal of time period), (4) Wavelength is the distance traveled during one complete oscillation, (5) Wave speed is the distance traveled per unit time. These parameters are related by the fundamental equation v = fλ, where wave speed equals frequency multiplied by wavelength.

Waves are characterized by several fundamental parameters. Amplitude is the maximum displacement of a particle from its mean position, measured in meters, and directly relates to wave energy. Wavelength (λ) is the distance between consecutive points in phase (crest to crest or trough to trough), representing one complete wave cycle. Frequency (f or n) is the number of complete cycles passing a point per second, measured in Hertz. Time period (T) is the time for one complete cycle, being the reciprocal of frequency. These parameters are interconnected through the wave equation v = fλ, where v is wave velocity.

Key wave parameters include: (1) Wavelength (波長, λ) - the distance between two points in the same phase, such as crest to crest; (2) Amplitude (振幅) - the maximum displacement from equilibrium; (3) Period (周期, T) - the time for one complete wavelength to pass a point; (4) Frequency (振動数, f) - the number of oscillations per second, where f = 1/T; (5) Wave velocity (波速, v) - the speed at which the wave pattern propagates. The fundamental wave equation is v = λ/T = fλ, which relates these parameters. The unit of frequency is Hertz (Hz), equivalent to 1/second. This equation can be verified experimentally using a rope.

Amplitude is the maximum displacement from equilibrium, always taken as positive. Wavelength (λ) is the distance between consecutive points in the same phase (e.g., two crests). Frequency (f) is the number of complete cycles per second. Time period (T) is the time for one complete cycle. Frequency and time period are reciprocals: f = 1/T. These parameters define the fundamental characteristics of any wave.

Mediums have four main properties: elasticity, temperature, inertia, and density. Elasticity is the property that allows materials to return to their original shape after deformation, like a rubber ball returning to its shape after being pressed. Inertia is the property that resists changes in motion, directly related to mass—objects with greater mass have greater inertia and resist motion changes more strongly. Density is the amount of mass contained in a unit volume, calculated by dividing mass by volume. These properties determine how waves propagate through materials and how objects respond to forces, forming the foundation for understanding wave mechanics and material science.

This comprehensive section covers the foundational concepts of density and elasticity. Density (ρ) is mass per unit volume, calculated as ρ = m/V, with SI units of kg/m³ or g/cm³. Water has a density of approximately 1,000 kg/m³. Specific gravity is the ratio of a substance's density to a standard (water for liquids, air for gases). Elasticity is the property allowing a body to return to its original shape after deformation. Stress (σ) is force per unit area (σ = F/A), measured in Pascals. Strain (ε) is fractional deformation (ε = ΔL/L₀), a unitless quantity. The elastic limit is the maximum stress before permanent deformation occurs. Young's modulus (Y) measures tensile elasticity as stress divided by strain. Bulk modulus (B) describes volume elasticity under pressure: B = -P(V/ΔV). Compressibility (K) is the reciprocal of bulk modulus. Shear modulus (G) describes shape elasticity under tangential forces. Practical applications include calculating gasoline density (680 kg/m³, SG = 0.68) and oil density (745 kg/m³) using mass and volume measurements.

This segment explains material properties: Pressure formula P = F/A. Elasticity concepts: Steel is more elastic than rubber due to higher Young's modulus. Glass is brittle, not elastic. Newton's laws: Constant velocity means zero acceleration. Velocity can be positive, negative, or zero. These concepts form the basis of mechanics.

Elasticity is the property enabling materials to regain original shape after deforming force removal. Elastic materials (rubber) return to original shape; plastic materials (wax) do not. Stress is restoring force per unit area (Force/Area), measured in N/m² or Pascal. Strain is fractional change in dimension (ΔL/L), dimensionless. Three fundamental stress types exist: longitudinal stress (force along length causing stretching/compression), shearing stress (force parallel to surfaces causing layer sliding), and volume stress (fluid pressure causing compression without shape change). Corresponding strains are longitudinal strain (ΔL/L), shearing strain (ΔX/X in radians), and volume strain (-ΔV/V). Within the elastic limit, stress is directly proportional to strain (Hooke's Law). The modulus of elasticity is the constant of proportionality (Stress/Strain). Three main moduli describe material stiffness: Young's Modulus (longitudinal stress/strain), Shear Modulus (shearing stress/strain), and Bulk Modulus (volume stress/volume strain). Compressibility is the reciprocal of Bulk Modulus. Steel has higher Young's Modulus than rubber, making it more elastic and preferred for heavy-duty applications.

Elasticity is the property of solid materials that enables them to return to their original shape and size after deformation. Materials are classified based on their elastic behavior: perfectly elastic bodies recover 100%, plastic bodies show no recovery, and real materials fall between these extremes. No material is perfectly elastic or perfectly plastic in practice. Steel is more elastic than rubber because it resists deformation more effectively under the same force. Every elastic material has a limit of elasticity beyond which it cannot return to its original shape. Stress is the restoring force per unit area: Stress = Normal Force / Area, with units of N/m² (Pascal). Strain is the fractional change in dimension: Strain = Change in Dimension / Original Dimension, and is dimensionless. Three types of elastic moduli describe material behavior: Young's modulus (Y) measures resistance to length change: Y = (F/A) / (ΔL/L₀). Bulk modulus (B) measures resistance to volume change: B = (F/A) / (ΔV/V₀). Shear modulus (G) measures resistance to shape change: G = (F/A) / (Δx/h). Compressibility is the reciprocal of bulk modulus, with gases being most compressible and solids least compressible.

Sine and cosine waves are periodic functions that repeat indefinitely. A sine wave starts from zero, increases to maximum, decreases back to zero, then decreases to minimum, and returns to zero. The sine function is odd: sin(-θ) = -sin(θ). Sine and cosine waves are the same wave shifted by 90 degrees: cosine leads sine by 90 degrees, or sine lags cosine by 90 degrees. The mathematical relationship is sin(θ) = cos(θ - 90°) and cos(θ) = sin(θ + 90°).

This section covers the mathematical representation of waves using sine and cosine functions. The general form is y = A sin(ωt + φ) or y = A cos(ωt + φ), where A is amplitude, ω is angular frequency, and φ is the initial phase. The section explains how the choice between sine and cosine depends on initial conditions and demonstrates how to write wave equations with specific initial conditions.

Sine and cosine are trigonometric functions that describe the relationship between angles and side lengths in right triangles. For a right triangle with hypotenuse length 1, cosine of an angle θ equals the adjacent side length, while sine of θ equals the opposite side length. These functions can also be defined as ratios: cosine θ = adjacent/hypotenuse and sine θ = opposite/hypotenuse. The unit circle extends these definitions to all angles (0-360°) by defining cosine θ and sine θ as the x and y coordinates of a point on the unit circle at angle θ. Key values include: sin(0°)=0, cos(0°)=1; sin(90°)=1, cos(90°)=0; sin(180°)=0, cos(180°)=-1; sin(270°)=-1, cos(270°)=0.

Wave functions describe how amplitude varies with position or time. In mathematics, y represents vertical displacement (amplitude). Sine waves start at zero, reach maximum at π/2, return to zero at π, minimum at 3π/2, and complete a cycle at 2π. Cosine waves start at maximum, decrease to zero at π/2, minimum at π, and back to maximum at 2π. Both are sinusoidal waves. The general equations are y = a sin(kx) for sine and y = a cos(kx) for cosine, where 'a' is amplitude and k is the wave number.

Understanding trigonometric functions is essential for evaluating wave equations. Key values: sin(π/2) = 1, sin(π) = 0, sin(3π/2) = -1, sin(2π) = 0. The sine function is positive in the first and second quadrants (0° to 180°) and negative in the third and fourth quadrants (180° to 360°). For cosine: cos(0°) = 1, cos(90°) = 0, cos(180°) = -1, cos(270°) = 0, cos(360°) = 1. The sign of cosine depends on which side of the y-axis the angle falls: positive for angles between -90° and +90°, negative for angles between 90° and 270°.