Transcript Electric Potential

Wave Motion
•Sinusoidal waves
•Energy and power in sinusoidal waves
Physics 1B03summer-Lecture 9
Sine Waves
For sinusoidal waves, the shape is a sine function, eg.,
f(x) = y(x,0) = A sin(kx)
A
(A and k are constants)
y
x
-A
Then y (x,t) = f(x – vt) = A sin[k(x – vt)]
Physics 1B03summer-Lecture 9
Sine wave: y (x,t) = A sin[kx – wt]
A
y
l
v
x
-A
l (“lambda”) is the wavelength (length of one complete wave);
and so (kx) must increase by 2π radians (one complete cycle)
when x increases by l. So kl = 2p, or
k = 2π / λ
Physics 1B03summer-Lecture 9
Rewrite:
y = A sin [kx – kvt]=A sin [kx – wt]
The displacement of a particle at location x is a sinusoidal
function of time – i.e., simple harmonic motion:
y = A sin [ constant – wt]
The “angular frequency” of the particle motion is w=kv; the
initial phase is kx (different for different particles).
Review: SHM is described by functions of the form
y(t) = A cos(wt+f) = A sin(p/2 –f –wt), etc., with
ω = 2πf
“angular frequency”
radians/sec
frequency:
cycles/sec (=hertz)
Physics 1B03summer-Lecture 9
Quiz
y
A
a
e
b
-A
c
x
d
Shown is a picture of a wave, y=A sin(kx- wt), at time t=0 .
i) Which particle moves according to y=A cos(wt) ?
ii) Which particle moves according to y=A sin(wt) ?
iii) Sketch a graph of y(t) for particle e.
Physics 1B03summer-Lecture 9
The most general form of sine wave is y = Asin(kx ± ωt – f)
amplitude
“phase”
y(x,t) = A sin (kx ± wt –f )
angular wavenumber
k = 2π / λ
(radians/metre)
phase constant f
angular frequency
ω = 2πf
(radians/second)
The wave speed is v = 1 wavelength / 1 period, so
v = fλ = ω / k
Physics 1B03summer-Lecture 9
Thus, y = A sin (kx ± ωt)
or
y = Asin(kx ± ωt - f)
in general
A : amplitude
k = 2π/λ : “angular wavenumber” (radians/metre)
ω = 2πf : “angular frequency” (radians/second)
f : “phase constant” (radians)
Note: Wave speed, v = 1 wavelength / 1 period
v = f λ = ω/k
Physics 1B03summer-Lecture 9
Wave Velocity
The wave velocity is determined by the properties of the medium:
Transverse waves on a string:
FT
tension
v wave 

mass/unit length

(proof from Newton’s second law)
Electromagnetic wave (light, radio, etc.):
v = c  2.998108 m/s (in vacuum)
v = c/n (in a material with refractive index n)
(proof from Maxwell’s Equations for E-M fields)
Physics 1B03summer-Lecture 9
Exercise
What are w and k for a 99.7 MHz FM radio wave?
Physics 1B03summer-Lecture 9
Particle Velocities
Particle displacement, y (x,t)
Particle velocity, vy = dy/dt (x held constant)
(Note that vy is not the wave speed v – different
directions! )
Acceleration,
d2y
ay 
 2
dt
dt
dvy
Physics 1B03summer-Lecture 9
“Standard” sine wave:
y  A sin( kx  wt -  )
dy
vy 
 wA cos( kx  wt -  )
dt
dv y
ay 
 -w 2 A sin( kx  wt -  )
dt
 -w 2 y
maximum displacement, ymax = A
Same as before
maximum velocity, vmax = w A
for SHM !
2
maximum acceleration, amax = w A
Physics 1B03summer-Lecture 9
Example
y
x
string: 1 gram/m; 2.5 N tension
vwave
Oscillator:
50 Hz, amplitude 5 mm
Find: y (x, t)
vy (x, t) and maximum speed
ay (x, t) and maximum acceleration
Physics 1B03summer-Lecture 9