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Physics 202, Lecture 20
Today’s Topics
Wave Motion General Wave Transverse And Longitudinal Waves Wave speed on string Reflection and Transmission of Waves Wave Function Sinusoidal Waves Standing Waves
General Waves
Wave: Propagation of a physical quantity in space over time q = q(x, t) Examples of waves: Water wave, wave on string, sound wave, earthquake wave, electromagnetic wave, “light”, quantum wave…. Mechanic wave: Propagation of small motion (“disturbance”) in a medium. Physical quantity to be propagated: displacement. Recall: Displacement is a vector.
Transverse and Longitudinal Waves
If the direction of mechanic disturbance (displacement) is perpendicular to the direction of wave motion, the wave is called transverse wave. If the direction of mechanic disturbance (displacement) is parallel to the direction of wave motion, the wave is called longitudinal wave. see demos. In general, a wave can be a combination of the above modes. The definition can be extended to other (non- mechanic) waves. e.g Electromagnetic waves are always transverse.
Electro-Magnetic Waves are Transverse
x y z E B c Two polarizations possible
Seismic Waves
Longitudinal Transverse Transverse Transverse
Wave On A Stretched Rope
It is a transverse wave See demos. The wave speed is determined by the tension and the linear density of the rope:
l
T m
v
= μ ≡ μ
Reflection and Transmission of Waves Wave Function
Waves are described by wave functions in the form: y(x,t) = f(x-vt) y: A certain physical quantity e.g. displacement in y direction f: Can any forms x: space position. Coefficient arranged to be 1 t: time. Its coefficient v is the wave speed v>0 moving right v<0 moving left
Standing Waves (cont)
Nodes and antinodes will occur at the same positions, giving impression that wave is standing
Standing Waves (cont)
Standing waves with a string of given length L are produced by waves of natural frequencies or resonant frequencies:
λ = 2 L
n
; n = 1 , 2 ,3...
f = v
= nv
2 L
= T
n
2 L
Forced (driven) Oscillation
If in addition there is a driving force with its own frequency ω: F 0 cos(ωt), the equation becomes: This equation can be solved analytically. At large t, the solution is: with At large t, the frequency is determined by driving ω When ω=ω 0 , amplitude is maximum resonance
2 0 cos(^ )
2
F t
dt
dx
kx b
dt
d x
m =− − + ω
x = A cos(ω t + φ )^20222 0 ( ) ( 2 )
m b A F^ m − +
ω ω
Resonance Amplitude
2 022 2 0
m
b
A F^ m
ω ω
