Describing Waves traveling disturbances § 14.1–14.2

What’s a Wave? Oscillation –object moves cyclically Wave –medium moves cyclically –disturbance travels, medium does not

Group Whiteboard Work 2.A wave generator produces 10 pulses each second. The pulses travel at 300 cm/s. a.What is the period of the waves? b.What is the wavelength of the waves?

Wave Pulse Why does the pulse move? What determines its speed? What happens inside the medium?

Points to Ponder The particles of the string change their motion as the wave travels. What force accelerates them? ac b d What are the velocity and acceleration of the string particles at the following positions? Why? a.middle (leading edge) b.crest c.middle (trailing edge) d.trough

Types of Waves Motion of the medium is perpendicular to the direction the wave travels: transverse wave (example: string wave) Motion of the medium is parallel to the direction the wave travels: longitudinal wave (examples: sound wave, slinky wave) Animation

Wave Speed Speed of disturbance traveling through the medium Generally not the speed of the oscillating medium itself!

Periodic Waves repeat in time and space § 15.2

Wavelength: crest-crest distance Trough: low point Period: crest-crest-timing Features of a Wave Crest: high point crest trough

Periodic Wave Parameters Angular frequency =  (rad/s) Cycle frequency f =  /2  (cycle/s) Repeat time = period T = 1/f (s/cycle) Repeat distance = wavelength (m/cycle) Angular wavenumber k = 2  / (rad/m) Wave speed v = /T = f =  /k (m/s)

Poll Question Doubling the frequency of a wave while keeping its speed constant will cause its wavelength to A.increase. B.decrease. C.stay the same.

Poll Question Doubling the frequency of a wave while keeping its wavelength constant will cause its speed to A.increase. B.decrease. C.stay the same.

Poll Question Doubling the wavelength of a wave while keeping its speed constant will cause its period to A.increase. B.decrease. C.stay the same.

Propagation Speed in a rope § 14.2

Purely transverse wave In a rope, string, or spring: Speed increases with tension F Speed decreases with density  v = F/ 

Wave Functions oscillations extended § 14.3

Think Question The waves travel to the right.  In which direction is A moving right now? A.A is momentarily stationary. B.Upward.  C.Downward.  AB A and B are points on the medium. C D

Poll Question The waves travel to the right.  A and B are points on the medium. In which direction is B moving right now? A.B is momentarily stationary. B.Upward.  C.Downward.  AB C D

Think Question The waves travel to the right.  A and B are points on the medium. In which direction is C moving right now? A.C is momentarily stationary. B.Upward.  C.Downward.  AB C D

Think Question The waves travel to the right.  A and B are points on the medium. In which direction is D moving right now? A.D is momentarily stationary. B.Upward.  C.Downward.  AB C D

Formula Description Displacements y of A and B with time AB y(x A,t) = A cos(  t) y(x A + /4,t) = A cos(  t–2  /4) yAyA yByB t y +A −A generalize to any x

Formula Description y(x A,t) = A cos(  t) y(x A + /4,t) = A cos(  t–2  /4) y(x,t) = A cos(  t–kx) same as y(x,t) = A cos(kx–  t) where –k( /4) = –2  /4 k = 2  /

Parameters  = 2  /T = angular frequency (rad/s) k = 2  / = wave number (rad/m)

Graphing y (x,t) Each tells only part of the story! y (x = 0) y (t = 0) xx tt T snapshot of the wave at one time displacement of the medium at one place

3D Graph xx tt crest trough T slope = v Displacement-position-time

Traveling the Other Way xx tt crest trough T slope = v Displacement-position-time

Wave (Phase) Velocity Where is the wave at any time? Continuity of single y-value (crest, trough, etc.) How does location x giving some y change with time? y = A cos(kx –  t) = constant y kx −  t = constant phase =  x =  t/k +  /k Phase velocity =  x/  t =  /k= /T

Wave Equation General solution: y = f(x – vt) Phase travels with velocity v (Disclaimer: Physical waves don’t have to follow this equation, but folks may forget this detail.) 2y2y x2x2 2y2y t2t2 v2v2 1 =

What Does It Mean? Acceleration of the medium is directly proportional to its curvature, so Restoring force is directly proportional to distortion. (stiffness matters) 2y2y x2x2 2y2y t2t2 v2v2 1 =

What Does It Mean? curvature = (1/v 2 ) a = (1/v 2 ) F/m mv 2 = F/curvature = stiffness v 2 = stiffness/mass (Note similarity to  2 = k/m.) 2y2y x2x2 2y2y t2t2 v2v2 1 =