Describing Waves traveling disturbances § 15.1– 15.3
Jan 16, 2016
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=