Wave Phenomena — Formula Sheet

v = νλ

v = λ/T

y(x,t) = a sin(ωt − kx)

k = 2π/λ

ω = 2π/T = 2πν

v = √(F/m)

v = √(E/ρ)

v = √(P/ρ) [Newton's Formula]

v = √(γP/ρ) [Laplace's Formula]

v = √(γRT/M)

v = v₀ + 0.61t ≈ 333 + 0.61t

v ∝ √T (T in Kelvin)

A² = a₁² + a₂² + 2a₁a₂cosφ

A_max = a₁ + a₂ (constructive: φ = 2mπ or path diff = mλ)

A_min = |a₁ − a₂| (destructive: φ = (2m+1)π or path diff = (2m+1)λ/2)

I_max/I_min = (a₁ + a₂)² / (a₁ − a₂)²

I ∝ a²

Here, a = amplitude, ω = angular frequency, k = wave number.

The equation can also be written as

y = a sin 2π(t/T − x/λ) or y = a sin(2π/λ)(vt − x).

Newton assumed sound propagation in air is ISOTHERMAL (temperature constant). His formula: v = √(P/ρ).

At STP

P = 1.01 × 10⁵ N/m², ρ = 1.29 kg/m³. Newton's value: v ≈ 280 m/s. Experimental value: 333 m/s. Error ≈ 16%.

Laplace's correction

Sound propagation is ADIABATIC (rapid compressions and rarefactions; air is a poor heat conductor). Corrected formula: v = √(γP/ρ), where γ = Cₚ/Cᵥ.

Angular frequency ω = 2πν: rate of phase change with time

Using Laplace's formula v = √(γRT/M) where T = absolute temperature, M = molar mass, R = gas constant: (1) Effect of Temperature: v ∝ √T.