Upload
vuongdan
View
266
Download
0
Embed Size (px)
Citation preview
1
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
1
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, electromagnetic wave, “light”, quantum wave…. Mechanical wave: Propagation of small motion (“disturbance”) in a medium.
Physical quantity to be propagated: displacement.
Recall: Displacement is a vector. 2
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-mechanical) waves. e.g Electromagnetic waves are always transverse.
3
Electromagnetic Waves are Transverse
x
y
z
E
B c
Two polarizations possible 4
2
Seismic Waves (motion in earthquakes)
Longitudinal
Transverse
Transverse
Transverse
5
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:
lmTvΔΔ
≡= µµ
;
6
Reflection of wave on string from hard and soft walls
7
Hard wall reflected wave is inverted
Soft wall reflected wave is not inverted
Reflection and Transmission of Waves
8
3
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 have any form
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
9
An Exercise to Explain Wave Speed A wave is described by function y=f(x-vt).
At time t1 in position x1, how large is the quantity y? y= f(x1-vt1) = y1
At a later time t2=t1+Δt, what is y at position x2=x1+vΔt?
y2= f(x2-vt2) = f(x1+vΔt -v(t1+Δt))= f(x1-vt1) = y1
How to interpret the result? Between t1 and t1+Δt, the value y1 has
transmitted from position x1 to x1+vΔt speed = (x1+vΔt - x1 )/(t1+Δt - t1 ) =v i.e v>0 moving right; v<0 moving left;
10
Linear Wave Equation Linear wave equation
2
2
22
2 1ty
vxy
∂∂
=∂∂
certain physical quantity
Wave speed
Sinusoidal wave
)22sin( φπλπ
+−= ftxAy
A:Amplitude
f: frequency φ:Phase
General wave: superposition of sinusoidal waves
v=λf k=2π/λ ω=2πf
λ:wavelength
11
Parameters For A Sinusoidal Wave Snapshot with fixed t: wave length λ=2π/k Snapshot with fixed x: angular frequency = ω frequency f =ω/2π Period T=1/f Amplitude =A Wave Speed v=ω/k v=λf, or v=λ/T Phase angle difference
between two positions Δφ =-kΔx
Snapshot:Fixed t
Snapshot:Fixed x 12
4
Standing Waves When two waves of the same amplitude, same
frequency but opposite direction meet standing waves occur.
€
y1 = Asin(kx −ωt)
€
y2 = Asin(kx +ωt)
€
y = y1 + y2 = 2Asin(kx)cos(ωt) Points of destructive interference (nodes)
€
kx = nπ; x =nπk
=nλ2
; n =1,2,3...
Points of constructive interference (antinodes)
€
kx = (2n −1) π2
; x = (2n −1) λ4
; n =1,2,3...13
Standing Waves (cont)
Nodes and antinodes will occur at the same positions, giving impression that wave is standing
14
Standing Waves (cont) Standing waves with a string of given length L are
produced by waves of natural frequencies or resonant frequencies:
€
λ =2Ln
; n =1,2,3...
€
f =vλ
=nv2L
=Tµ
n2L
15
Sound travels at 340 m/s in air and 1500 m/s in water. A sound of 256 Hz is made under water. In the air,
A. the frequency remains the same but the wavelength is shorter.
B. the frequency remains the same but the wavelength is longer.
C. the frequency is lower and the wavelength remains the same.
D. the frequency is higher and the wavelength remains the same.
E. both the frequency and the wavelength remain the same.
5
Sound travels at 340 m/s in air and 1500 m/s in water. A sound of 256 Hz is made under water. In the air,
A. the frequency remains the same but the wavelength is shorter.
B. the frequency remains the same but the wavelength is longer.
C. the frequency is lower and the wavelength remains the same.
D. the frequency is higher and the wavelength remains the same.
E. both the frequency and the wavelength remain the same.
We can hear sounds that are produced around a corner but cannot see light that is produced around a corner because
A. light travels only in straight lines whereas sound can travel in a curved path.
B. sound has more energy than light.
C. sound has shorter wavelengths than light.
D. sound has longer wavelengths than light.
E. None of these is correct.
We can hear sounds that are produced around a corner but cannot see light that is produced around a corner because
A. light travels only in straight lines whereas sound can travel in a curved path.
B. sound has more energy than light.
C. sound has shorter wavelengths than light.
D. sound has longer wavelengths than light.
E. None of these is correct.