Mechanical Waves and Wave Equation A wave is a nonlocal perturbation traveling in media or vacuum. A wave carries energy from place to place without a bulk flow of matter. A mechanical wave is a wave disturbance in the positions of particles in medium. Types of waves Electromagnetic waves (light), plasma waves, gravitational waves, …

Periodic and solitary waves compression rarefaction Parameters of periodic waves: (i)period T, cyclic frequency f, and angular frequency ω : T = 1/ f = 2 π / ω ; (ii) wavelength λ and wave number k : λ = 2π / k ; (iii) phase velocity (wave speed) v = λ/T=ω/k (iv) group velocity v group = dω/dk. Sinusoidal (harmonic) wave traveling in +x: Solitons

Longitudinal Sound Waves

Wave Equation Longitudinal waves in a 1-D lattice of identical particles: y n = x n – nL is a displacement of the n-th particle from its equilibrium position x n0 = nL. Restoring forces exerted on the n-th particle: from left spring F nx (l) = - k (x n -x n-1 -L), from right spring F nx (r) = k (x n+1 -x n -L). Newton’s 2 nd law: ma nx = F nx (l) + F nx (r) = k [x n+1 -x n -(x n -x n-1 )], a nx = d 2 y n /dt 2. Limit of a continuous medium: x n+1 -x n = L∂y/∂x, x n+1 -x n -(x n -x n-1 )= L 2 ∂ 2 y/∂x 2 Transverse waves on a stretched string: y(x,t) is a transverse displacement. Restoring force exerted on the segment Δx of spring: (n-1)L nL(n+1)L X n-1 XnXn X n+1 X y n-1 ynyn y n+1 F is a tension force. μ = Δm/Δx is a linear mass density (mass per unit length). Newton’s 2 nd law: μΔx a y = F y, a y = ∂ 2 y/∂t 2 Slope= F 2y /F=∂ y/∂x Slope = -F 1y /F=∂y/∂x

Wave Intensity and Inverse-Square Law Power of 1D transverse wave on stretched string = Instantaneous rate of energy transfer along the string For a traveling wave y(x,t) = A cos (kx – ωt), F y does work on the right part of string and transfers energy. X y 0 3-D waves since v y = - v ∂y/∂x = = ωA sin (kx - ωt).

Exam Example 33: Sound Intensity and Delay A rocket travels straight up with a y =const to a height r 1 and produces a pulse of sound. A ground-based monitoring station measures a sound intensity I 1. Later, at a height r 2, the rocket produces the same second pulse of sound, an intensity of which measured by the monitoring station is I 2. Find r 2, velocities v 1y and v 2y of the rocket at the heights r 1 and r 2, respectively, as well as the time Δt elapsed between the two measurements. (See related problem )

(a) Derivation of the wave equation: y(x,t) is a transverse displacement. Restoring force exerted on the segment Δx of spring: F is a tension force. μ = Δm/Δx is a linear mass density (mass per unit length). Newton’s 2 nd law: μΔx a y = F y, a y = ∂ 2 y/∂t 2 Slope= F 2y /F=∂ y/∂x Slope = -F 1y /F=∂y/∂x Exam Example 34: Wave Equation and Transverse Waves on a Stretched String (problems – 15.53) Data: λ, linear mass density μ, tension force F, and length L of a string 0<x<L. Questions: (a) derive the wave equation from the Newton’s 2 nd law; (b) write and plot y-x graph of a wave function y(x,t) for a sinusoidal wave traveling in –x direction with an amplitude A and wavelength λ if y(x=x 0, t=t 0 ) = A; (c) find a wave number k and a wave speed v; (d) find a wave period T and an angular frequency ω; (e) find an average wave power P av. Solution: (b) y(x,t) = A cos[2π(x-x 0 )/λ + 2π(t-t 0 )/T] where T is found in (d); y X 0 L A (c) k = 2π / λ, v = (F/μ) 1/2 as is derived in (a); (d)v = λ / T = ω/k → T = λ /v, ω = 2π / T = kv (e)P(x,t) = F y v y = - F (∂y/∂x) (∂y/∂t) = (F/v) v y 2 P av = Fω 2 A 2 /(2v) =(1/2)( μF) 1/2 ω 2 A 2.

Principle of Linear Superposition. Wave Interference and Wave Diffraction Constructive interference at the time of overlapping of two wave pulses. Energy is conserved, but redistributed in space.

Energy is conserved, but redistributed in space. Destructive interference at the time of overlapping of two wave pulses:

Diffraction is the bending of a wave around an obstacle or the edges of an opening. Direction of the first minimum: sin θ = λ / D for a single slit, sin θ = 1.22 λ / D for a circular opening.

The phenomenon of beats for two overlapping waves with slightly different frequencies

Reflection of Waves and Boundary Conditions Example: Transverse waves on a stretched string.

Traveling and Standing Waves. Transverse Standing Waves. Normal (Natural) Modes. When a guitar string is plucked (pulled into a triangular shape) and released, a superposition of normal modes results. Traveling waves (in ±x direction): y(x,t) = A cos (±kx - ωt) = = A cos [ k (±x - vt) ] Standing wave: y(x,t) = A [cos (kx + ωt) – cos (kx - ωt)]= = 2A sin (kx) sin (ωt) Amplitude of standing wave A SW = 2A 2A SW =4A λ n = 2L/n

Longitudinal Standing Waves Tube open at both ends: f n = nf 1, n= 1, 2, 3, …; L=n λ 1 /2 Tube open at only one end: f n = nf 1, n= 1, 3, 5, …; L=n λ 1 /4. Only odd harmonics f 1, f 3, f 5, … exist.