Download Experiment 1 - 6. Waves-Where does It Come from and Where does

Physics Laboratory
Last update: 2015. 08. 31
Experiment 1 - 6. Waves-Where does It Come from
and Where does It Go?
-Generation and Propagation of Waves-
Purpose of Experiment
Wave is the spacial propagation of a physical quantity that oscillates as a function of time.
There are many kinds of waves with respect to the oscillating physical quantities such as
elastic waves(deformation), surface waves(displacement of water molecules), sound or sound
of ultrasound( density of media), electromagnetic waves(electric field and magnetic field).
Waves can be divided into transverse waves(oscillating direction is perpendicular to the
propagation direction) and longitudinal waves(oscillating direction is parallel to the
propagation direction). When we describe waves, we use distance of which the form of
oscillation repeats (wavelength), and time(period), or equivalently, wave number, and
frequency is used which are the reciprocals of the formers. The speed of a wave is
multiplication of wave number and frequency.
Waves change propagating direction when they meet obstacles. Reflection and refraction are
representative examples. But if the size of the obstacle or the size of a slit in the obstacle is
comparable with the wavelength of the wave, diffraction which is an unique phenomena of a
wave occurs. Also, when two waves passed through different paths meet at a point,
interference occurs. To make interference, the two waves must be coherent waves which
means that the phase difference between the two waves doesn’t change with time.
In this experiment, by using a surface wave generator, produce coherent waves to
investigate its propagating properties and interference effects.
-from general physics book of Halliday & Resnick
The picture above shows that surface waves produced by the surface wave generator
diffracts at the slit. Though the method of generating waves, media, propagation mechanisms,
frequency, wavelength, and speed of waves are different, you can observe same phenomena
in electromagnetic waves such as light, microwaves, x-rays and sound waves.
Outline of Experiment


Observe method of surface wave generation, propagation of waves, propagation speed.
Investigate interference of surface waves from two coherent point sources.
·
Investigate interference phenomena when surface waves from linear source passes
through a small obstacle or a slit.
Other than these, it also recommended to put many form of obstacles in the way of the
waves and observe the propagation of the waves.
Experimental Method
These equipments are prepared in the laboratory. (Parentheses mean the number of them.)
A water tank(1)
A sponge to absorb surface waves (little)
A surface wave generator(1)
Wave generator(1, including a connection line)[video]
A small bead for the use of point source (2)
A light source (1)[video]
30 cm ruler (1)
A protractor (1)
Paper (little)
obstacles (acrylic rod long 2, short 1)
obstacles (a needle, a pen and so on 1, , prepare those for yourself)
a water beaker (1ι a plastic beaker)
If you need more equipments, inquire to your teaching assistant or the experiment preparation
room (19-114), or prepare them yourself. And you need to investigate stroboscope too.
Bellows are the recommended and standard experimental methods
1) Set up surface wave generator after filling water in water tank about 2/3 full. Place the
water tank level and place sponges in the edges to absorb surface waves so that it does not
produce reflected waves.[notice: this procedure may already done]
2) Oscillate two circular shaped surface wave sources in phase and measure the interference
of the surface waves[video]
① Turn off the surface wave generator and disassemble the plastic rod from the wave
generator. Adjust the height of water for the two plastic beads to sink about 1/3. Be sure
that the two beads sink at the same level.
② Turn on the wave generator and adjust frequency of the light source and intensity of
the oscillator for the shadow of surface waves to be seen clearly.
In this time, place white paper under the water tank for you to see the shadow clearly.
Also if you use stroboscope, you can record wave patterns easily because it provides you a
fixed image of the moving shadow.
[Tip: Try to adjust frequencies of wave generator and light source frequency of stroboscope.]
Think reasons why shadows are produced.
Mark bright points(constructive) or dark points(destructive) on the paper to calculate path
difference. And compare this with the theoretical consideration of interference.
③ Change the distance of the two circular surface wave sources and repeat the
measurement ¨è. Be careful not to apply too strong force. Can you see the differences?
If you can, think the reasons of the differences and compare the measurement result
with the theory.
④ Measure the wavelength of the surface waves. [Caution : if the light source does
not produce parallel rays, the distance between shadows and the real wavelength can be
different. Can you calibrate this? ]
⑤ Obtain velocity of the surface wave. Surface of the wave can be obtained by period
(=1/frequency) and the measured wavelength.
⑥ Float small paper or pieces of sponge on the water to observe that the propagation
velocity due to the surface waves is slower that the velocity of the wave. If possible,
measure the velocities at the surface.
⑦ Use an acrylic rod of a semi sphere bottle(fill water not to float) as obstacles and
observe how these influence the surface waves.
3) Measure interference phenomena of surface waves reflected or diffracted by obstacles.
① Adjust the linear wave generating rod to sink 3~5 mm in water as does 1). Place the
rod at the center of the water tank and parallel to the edges.
② turn on surface generator and adjust period and intensity of the oscillator to make
shadow of surface waves be clear.
③ Place obstacles that you prepared for yourself like a needle and observe the changes
of the waves. Change number, sizes, distances, and position of obstacles.
④ Mark bright points of the interference pattern on the paper. [Caution: if the light
source does not produce parallel rays, the distance between shadows and the real
wavelength can be different. ]
⑤ What can you say from this?
Additionally, observe the influence of obstacles such as a mechanical pencil, lead of a
pencil when the obstacles are smaller than the wavelength of the surface wave. And observe
how the impulse exerted on the surface propagates. Surface wave propagates at last via the
interaction of water molecules. What interaction can be there?
Background Theory
The general mathematical representation of waves propagating to +x direction at time t and
x is, if the displacement of the wave is y, given by
y(x,t) = ym sin(kx-ωt)
position
(1)
where w is angular frequency, k a wave number and ym an amplitude.
And angular frequency and wave number are related with respect to frequency f and wavelength λ by
ω = 2πf
(2)
k = 2π/λ
(3)
The velocity of wave v is x/t where x and t satisfy kx-ωt = const.. Here, kx-wt is phase of sine
function. Therefore,
v = x/t = ω/k = fλ
(4)
.Strictly speaking, this velocity is called phase velocity. To transfer information via waves,
monochromatic waves are not enough and wave packets which is superposition of many waves with
different frequencies like pulse are needed. The velocity of propagation of this bundle of waves is
called group velocity and this can be formulated by
vg = dω/dk = v + kdv/dk = v - λdv/dλ
(5)
. In equation (4), a medium where the ratio of angular frequency ω and wave vector k, ie phase
velocity is constant regardless of frequency or wavelength is called a nondispersive medium, and in a
nondispersive medium, vg = v. But for some media, there are some cases where the phase velocities
are depend on the wavelengths, this medium is called a dispersive medium, and vg ≠ v. In a dispersive
medium, the shape of a grouped wave in which many different waves of different wavelength changes
as the wave propagates.
Now, the superposition of the two coherent(same frequencies, different phases) waves
y1(x,t) = ymsin(kx-ωt)
(6)
and
y2(x,t) = ymsin(kx-ωt+φ)
(7)
are
y(x,t) = y1(x,t) + y2(x,t)
= ymsin(kx-ωt) + ymsin(kx-ωt+φ)
= 2ymcos(φ/2)sin(kx-ωt+φ/2)
= Ymsin(kx-ωt+φ/2)
(8)
. Here, for convenience, frequencies, wavelengths, amplitudes of the two waves are the same and the
phase difference of the two waves is φ. The frequency of the superposed wave in formula (8) is the
same with that of original waves, and the amplitude is depends on φ, which meets the relation below.
Ym = 2ymcos(φ/2)
In other words, interference occurs.
(9)
If we think of interference with the reflected waves, formula (8) becomes
y(x,t) = y1(x,t) + y2(x,t)
= ymsin(kx-ωt) + ymsin(kx+ωt+φ)
= 2ymcos(ωt+φ/2)sin(kx+φ/2)
(8')
, which is a standing wave. In this case, where kx + φ/2 = nπ (n =0,1,2,...) are always dark points(y=0),
and where kx + φ/2 = (n+1/2)π (n =0,1,2,...) are the points with maximum amplitude. Usually, the
former is called a node and the latter is called a antinode. Therefore, the distance between nodes or
antinodes is △x=π/k, which is half of original wavelength λ. Though the frequency of the standing
wave is different from that of original wave, the positions where the maximum and minimum
amplitude occur don’t changes.
Now let’s think about interference of two coherent waves. If we regard the two wave sources as point
sources, the two surface waves are circular waves. As shown in the picture, the interference of waves
at a distance r form the center of the two sources and at an angle θ is determined by path difference
from the two sources. In other words, when the phase differences φ with respect to path difference is
2nπ (n is integers), magnitude of the composed wave, from (9), becomes maximum. The path
difference is, if θ is small,
r1 - r2 = d sinθ
(10)
so the condition for antinode(bright points) is
d sinθ = nλ
(n = 0,1,2,3, .....)
(11)
And the condition for nodes is, when the phase difference with respect to path diffence φ is
(n is integer),
d sinθ = (n+½)λ
(n = 0,1,2,3, .....)
(2n+1)π
(12)
The case transmitting of two slits (slit size ~ wavelength) distant d away can be interpreted like above.
General physics book of Halliday & Resnick
More thinking

Velocity of surface waves (group velocity)
Velocity of surface wave v generated in the surface wave generator is , where h is the height
of the water and if the amplitude is negligible with respect to the h, approximately
v = (gh)1/2
(A1)
. Here g is gravitational acceleration. This, of cource, can be induced by physical laws, but
this also can be obtained by dimensional analysis easily. In other words, the related physical
quantities are the height of the water h, density ρ, gravitational acceleration g. Velocity of the
surface wave may be concerned with these quantities and the only way to compose these to
have dimension of velocity is to make √gh . This dimensional analysis is sometimes used to
find relations between physical quantities. By using this method, you can find only the
proportionality relations, of course, you can not find out the proportionality constant. In (A1),
as velocity of the wave is independent of its wavelength, this wave is a nondispersive wave.
The velocity of surface waves is faster when the height of the water is high. In a beach, you
can see high tides come, which is originated by this reason. The wavelength of surface wave
of frequency f is
λ = v/f = (gh)1/2/f
(A2)
. Compared with this velocity, the real velocity of water is very slow. This fact was mentioned
by Leonardo da Vinci). He compared surface waves with the wave produced by wind in
wheat fields. In this case, the wave of the wheat field propagates but wheat stand on its place.
In case of surface waves, about 1% slow drift occurs in the direction of the waves. When
surface wave propagates, water molecules rotates circularly in the vertical plane. It rotates in
the direction of the wave at crest, at trough in the opposite direction, and goes up in the front
of the crest, and goes down in the back side of the crest. This motion of water molecules
vanishes rapidly as goes inside the water. Therefore, when thunderstorm occurs in the sea, the
submarine under the sea is safe.
Reference
• Stroboscope
• Analysis method of measured data
• Analysis method via graph
• Christiaan Huygens – the pioneer who invented pendulum clock
• Thomas Young – interference of light(wave)
• Joseph Fourier – denied Fourier series
※site with various apps about simple harmonic oscillator
http://www.scienceall.com/content/c072/physics/mechanical_energy.htm
http://www.scienceall.com/content/c072/physics/oscillation_spring_horizon.htm
http://www.scienceall.com/content/c072/physics/oscillation_spring_vertical.htm
http://www.scienceall.com/content/c072/physics/oscillation.htm
※ site with apps about lissajous
http://www.scienceall.com/content/c072/algorithm/lissajous.htm