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.

A B

• A and B are points on the medium.

CD

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.

A B

CD

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.

A B

CD

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.

A B

CD

Formula Description

Displacements y of A and B with time

A B

y(x,t) = A cos[(2/T)t – (2/)x]

y(x,t) = A cos(t–kx) = A cos (kx–t)

yA yB

ty

+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

x2

2y

t2v21=

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

x2

2y

t2v21=