Describing Waves traveling disturbances § 15.1–15.3

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

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

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)

Board 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.

Board 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.

Board 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.

Wave Functions oscillations extended § 15.3

Point 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

Point 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

Point 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

Point 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,t) = A cos[(2  /T)t – (2  / )x] y(x,t) = A cos(  t–kx) = A cos (kx–  t) yAyA yByB t y +A −A

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

Wave (Phase) Velocity Where is the wave at any time? Advance 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 = dx/dt =  /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 =