1. Physics 1B03summer-Lecture 9
Wave Motion
Sinusoidal waves
Energy and power in sinusoidal waves
Physics 1B03summer-Lecture 9

2. Then y (x,t) = f(x – vt) = A sin[k(x – vt)]
Sine Waves
For sinusoidal waves, the shape is a sine function, eg.,
f(x) = y(x,0) = A sin(kx)
(A and k are constants) y A x -A
Then y (x,t) = f(x – vt) = A sin[k(x – vt)]
Physics 1B03summer-Lecture 9

3. Physics 1B03summer-Lecture 9
y (x,t) = A sin[kx – wt]
Sine wave: y l A 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

4. frequency: cycles/sec (=hertz)
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

5. Physics 1B03summer-Lecture 9
Quiz a A e b x d -A c
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

6. 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 )
phase constant f
angular wavenumber
k = 2π / λ (radians/metre)
angular frequency
ω = 2πf (radians/second)
The wave speed is v = 1 wavelength / 1 period, so
v = fλ = ω / k
Physics 1B03summer-Lecture 9

7. Physics 1B03summer-Lecture 9
Thus, y = A sin (kx ± ωt) or
y = Asin(kx ± ωt - )
in general
A : amplitude
k = 2π/λ : “angular wavenumber” (radians/metre)
ω = 2πf : “angular frequency” (radians/second)
 : “phase constant” (radians)
Note: Wave speed, v = 1 wavelength / 1 period
v = f λ = ω/k
Physics 1B03summer-Lecture 9

8. Physics 1B03summer-Lecture 9
Wave Velocity
The wave velocity is determined by the properties of the medium:
Transverse waves on a string:
(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

9. Physics 1B03summer-Lecture 9
Exercise
What are w and k for a 99.7 MHz FM radio wave?
Physics 1B03summer-Lecture 9

10. 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,
Physics 1B03summer-Lecture 9

11. Physics 1B03summer-Lecture 9
“Standard” sine wave:
maximum displacement, ymax = A
maximum velocity, vmax = w A
maximum acceleration, amax = w 2 A
Same as before for SHM !
Physics 1B03summer-Lecture 9

12. Physics 1B03summer-Lecture 9
Example y
string: 1 gram/m; 2.5 N tension x
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