00 Wave phenomena in a ripple tank - Phywe Systeme GmbH

Related topics
Generation of surface waves, propagation of surface waves,
reflection of waves, refraction of waves, Doppler Effect.
Principle
Water waves are generated by a mechanical oscillator. A circular
wave pattern is used to investigate the dependency of
the wave length on the oscillator’s frequency and to demonstrate
the Doppler effect. With the aid of plane waves the
dependency of the waves’ velocity of propagation on the
depth of the water can be investigated. Moreover, the reflection
of waves as well as the refraction of waves can be illustrated
at objects such as a plate, a prism, a concave lens and
at a convex lens.
Equipment
Ripple tank with LED-light source, complete 11260.99 1
Ext. vibration generator for ripple tank 11260.10 1
Connecting cord, 32 A, 500 mm, red 07361.01 1
Connecting cord, 32 A, 500 mm, blue 07361.04 1
Demo set for ripple tank 11260.20 1
Software „Measure Dynamics“ 14440.61 1
Tasks
1. Use the single wave exciter to generate circular waves. By
using a ruler the wave length can be determined. The measurement
is repeated for different frequencies.
2. The external vibration generator is connected to the ripple
tank device and circular waves are generated. By moving
the external vibration generator, the Doppler Effect is investigated.
Fig. 1: Overview of the experimental setup.
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3. Plane waves are generated by the integrated vibration
generator. Place a plane plate in the bassin to create a
zone of lower water depth and measure the wave length
difference in front of and above the plate.
4. Observe the refraction of plane water waves at several
objects (plate, prism, concave and convex plate).
5. By using two barriers and a concave / convex reflector
show the reflection of water waves.
Setup and Procedure
Task 1: Dependence of wave length on frequency
Set up the experiment as shown in Fig. 2. Mount the camera
with its attachment to the drawing-table (Fig. 3), connect it to
a computer and start the respective software. For further information
about using the software, please refer to the operating
instructions.
Set the frequency f of the vibration generator (Fig. 4) to 15 Hz
and select the amplitude in a way that a clear wave image can
be seen on the drawing-table.
You should also see the wave image in the display of your
computer. Turn on stroboscope illumination to obtain a standing
wave image. By placing a ruler on the drawing-table, measure
the wave length l. (Note: one wave length includes one
bright and one dark stripe.) To improve the measurement’s
accuracy, measure a large distance between two bright or
between two dark stripes and then divide the measured value
by the number of wave lengths n that are included in this interval.
Repeat the measurement for three more frequencies
between 20 and 30 Hz.
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Write down the measured values and calculate the product
c = l · f.
Before proceeding to Task 2, take a snapshot of a wave image
with the ruler lying on the drawing-table. This picture is important
for the calibration process in Task 2.
Fig. 2: Arrangement for generating circular waves.
Fig. 3: Ripple tank with attached camera.
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Wave phenomena in a ripple tank
Task 2: Doppler effect
Mount the single wave exciter to the external vibration generator
and connect it with two connecting cords to the ripple
tank device. Since the integrated vibration generator is not
needed in this experiment, unscrew its head and turn it to the
side. Position the external vibration generator as shown in
Fig. 5.
Fig. 4: Keypad of the ripple tank device.
Fig. 5: Arrangement for demonstrating the Doppler effect. The
external vibration generator with single wave exciter
is placed to the rear of the ripple tank.
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Select a frequency f between 15 and 25 Hz that you have
already used in Task 1 and the amplitude in a way that you can
see a clear wave image. Move the vibration generator with a
slow and nearly constant velocity in a sideway direction and
observe the wave image. While moving the generator, take a
snapshot of the wave image.
Repeat this procedure with a faster movement of the vibration
generator.
Then start the PHYWE software MEASURE DYNAMICS. First,
open the file of the wave image that you have taken in Task 1
(exciter at rest). Before you can use a picture for any measurements,
you have to calibrate it. This is done by clicking on
“Measure” S “Scale” S “Calibration”. For further information
about the correct use of MEASURE DYNAMICS, please refer to the
manual.
Document the calibration data since these values are needed
for any other picture that you will take with the camera. Then,
open the first Doppler image and calibrate it as previously
described.
After the calibration, measure the wave length in front of (l 1 )
and behind (l 2) the single wave exciter with “Ruler” in the
“Measure”-menu. “In front of” and “behind” the wave exciter
is meant when looking in the direction of the movement. As in
Task 1, measure a large distance between two bright or two
dark stripes and then divide the measured value by the number
of wave lengths n that are included in this interval.
Note your values for f, l 0 , l 1 and l 2 , where l 0 is the wave
length of the circular waves at frequency f without movement
(Task 1).
Proceed the same way with your second picture of the
Doppler effect (faster movement).
Disconnect the external vibration generator from the ripple
tank device, turn the integrated vibration generator back to its
starting-position and fix its head.
Fig. 6: Arrangement for demonstrating the dependence of the
wave velocity of propagation on the depth of water. The
plane waves that are generated by the plane wave
exciter propagate also above the plate with
altered wave length l 1 .
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Task 3: Dependence of wave velocity of propagation on water
depth
Replace the single wave exciter of the integrated vibration
generator by the plane wave exciter. With the aid of the adjusting
screws, adjust the wave tray horizontally to get the same
water level all over the tray. Adjust the plane wave exciter in
such a way, that it is exactly parallel to the water surface. This
adjustment is important since otherwise no clear wave images
of plane waves would be possible. Then, set up the experiment
as shown in Fig. 6. The plane plate is used to create a
zone of lower water depth. Make sure that it is covered completely
with water.
Select a frequency f of the vibration generator between 18
and 25 Hz and the amplitude so, that you can observe a clear
wave pattern. Start the synchronised stroboscope illumination
with a frequency difference ∆f = 0. You will now see a standing
wave image. Use the flask to suck out of the wave tray as
much water as you see a remarkable change in the wave
length l above the plane plate. Note: the plane plate must still
be covered completely with water.
Take a snapshot and use this image to measure the wave
length in the deeper (l 0 ) and in the lower water (l 1 ) with MEA-
SURE DYNAMICS the same way as you did in Task 1 and 2. Do
not forget to calibrate the picture!
Leave the plane plate in the wave tray.
Task 4: Refraction of water waves
In this task, you will investigate the refraction of water waves
at several objects. First, set up the experiment as shown in
Fig. 7. Make sure that the plane plate is still covered completely
with water. Optionally, add two drops of washing-up
liquid to the water in the wave tray. This might be helpful to
achieve a complete covering of the objects.
Fig. 7: Arrangement for demonstrating the refraction of water
waves at a plane plate. The plane water waves that are
generated by the plane wave exciter are refracted at
the plane plate . On leaving the plane plate they are
refracted back towards their initial direction .
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Select a frequency f between 20 and 25 Hz and the amplitude
so, that you can see a clear wave image. Turn on the stroboscope
illumination and set the frequency difference ∆f >0 to
observe the propagation of the water waves in front of, above
and behind the plane plate in slow motion. You should see the
refraction of the water waves on entering and on leaving the
plane plate (Fig. 7).
Fig. 8: Arrangement for demonstrating the refraction of water
waves at a prism. The plane waves that are generated
by the plane wave exciter are refracted on entering
the zone of lower water depth above the prism and
are further refracted towards the same direction on
leaving the prism .
Fig. 9: Arrangement for demonstrating the refraction of water
waves at a convex plate. The plane waves that are generated
by the plane wave exciter are refracted at hte
convex plate and run into a focus behind the
plate.
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Wave phenomena in a ripple tank
After that, remove the plane plate out of the water tray and use
the prism to set up the experiment according to Fig. 8. Make
sure, that the prism is completely covered with water.
Use the same settings as above. Make sure, that you can see
a clear wave image and the refraction on entering and on leaving
the prism. Otherwise, it can be useful to change the amplitude.
You should observe a wave image as shown in Fig. 8.
Now, replace the prism by a convex plate to set up the experiment
as shown in Fig. 9. Make sure, that the plate is covered
completely with water.
Select a frequency f between 15 and 25 Hz and the amplitude
so, that you can see a clear wave image, which is similar to
Fig. 9. Use continuous illumination or the stroboscope mode
with ∆f = 0. You should see the water waves running into a
focus behind the plate.
After that, replace the convex plate by a concave plate
(Fig. 10) and repeat the experiment.
Observe the refraction of the water waves on leaving the concave
lens. You should see the divergent water waves behind
the lens (Fig.10).
Task 5: Reflection of water waves
Use the 190 mm and the 71 mm barrier to set up the experiment
as shown in Fig. 11.
Select a frequency f between 20 and 25 Hz and the amplitude
so, that you can see a clear wave image. The barrier ‚
shades the region S from the direct waves generated by the
wave exciter so that the reflected waves exclusively can be
observed in this region.
First, observe the wave image for an angle of 45° between the
plane reflector and the water waves (Fig. 11). Then, observe
the wave image for different positions of the plane reflector.
Fig. 10: Arrangement for demonstrating the refraction of water
waves at a concave lens. The plane waves that are
generated by the plane wave exciter are refracted
at the concave lens and leave the lens as divergent
waves .
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After that, remove the two barriers from the wave tray and use
the concave reflector to set up the experiment as shown in
Fig. 12.
Generate a sequence of single plane waves by pushing the
button “Pulse” at the keypad (cp. Fig. 4). With this method you
can determine the focal point of the concave reflector. Use the
camera to record this determination of the wave propagation.
Then, run this video with MEASURE DYNAMICS and measure the
distance l between the concave reflector and the focal point.
Note: In this setup, continuously generated wave trains would
result in a complex wave pattern where the original plane
waves overlay the reflected waves running into focus.
After that, make sure that the focus of the reflector is lying on
the extension of the vibration generator’s arm (cp. Fig. 12).
Then, exchange the plane wave exciter for the single wave
exciter and position it exactly at the focal point. Generate several
single circular waves with “Pulse” and obServe the wave
image.
Note
By turning the concave reflector around it can be also used as
a convex reflector. When using the convex reflector, you are
able to observe that the plane waves are reflected as divergent
circular waves after hitting the reflector.
Theory and Evaluation
Task 1:
This experiment reveals two important issues:
The higher the frequency f, the smaller the wave length l.
The phase velocity of water waves c = l · f is nearly constant.
The same results occur when you are dealing with light waves.
Therefore, water waves are particularly suitable for demonstrating
the properties of light waves and waves in general.
Fig. 11: Arrangement for demonstrating the reflection of plane
waves at plane barriers. The water waves that are
generated by the plane wave exciter are partly
shaded by the barrier in order to enable an observation
of only those water waves that are reflected by
the plane barrier .
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In theory of propagation of water waves, the following relation
holds:
v c · k 3 c v
k
where v is the angular frequency, k is the wavenumber and
c is the phase velocity. For v and k also the following is valid:
v 2pf , k 2p
l .
On inserting these values into (1), one obtains the well-known
formula
c l · f .
Since we are dealing with water surface waves, the phase
velocity c is also dependent on gravity, surface tension and
water density. The respective relationship between these
magnitudes is given by the dispersion relation
v 2 gk sk3
r
where g is the acceleration of gravity, s is the surface tension
of water and r is the density of water. With (1), formula (4)
leads to
c 2 k 2 gk sk3
r
1 c2 g sk
k r
1 c 2 gl 2ps
2p lr .
Fig. 12: Arrangement for demonstrating the reflection of plane
waves at a concave reflector. The plane water waves
that are generated by the plane wave exciter hit the
concave reflector and are reflected as circular
waves. These circular waves run into a focal point .
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(1)
(2)
(3)
(4)
(5)

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On inserting the values for the surface tension of water
s = 72.5 · 10 -3 Nm -1 (20 °C) and its density r =10 3 kgm -3 , the
acceleration of gravity g =9.81ms -2 and a measured wave
length cm l =1.44 cm (20 Hz), one gets
In a reference measurement, we got the following results (see
table 1):
By calculating the average value of l · f, c results in
c =0.281 ms -1 .
The deviation of the measured value from the theoretical value
calculated above can be explained by the fact, that there is
always an inaccuracy on measuring on the drawing-table due
to the error in the projection of the wave image to the drawing-table.
The wave image that appears on the paper of the
drawing-table is enlarged compared to the real wave image in
the wave tray.
Task 2:
It can be clearly seen that the waves emitted in the direction
of the generator movement are shortened while the waves
running in the opposite direction are lengthened. Perpendicular
to the direction of movement the wavelength remains
unchanged.
This phenomenon can be explained by the following:
A fixed wave generator, which vibrates with frequency f 0 emits
a continuous wave train with wavelength l = c/f 0 (c = phase
velocity of the wave in the medium). If the wave generator
moves with velocity n, it travels a distance nT during the period
T. The wavelength l 1 of the wave produced by the moved
generator is shortened by this distance in front of the generator
and is lengthened by the same distance behind the generator
in accordance with
or
The negative sign in this formula applies in the direction of
movement in front of the generator, the positive sign applies
behind the generator.
Table 1
c 20.0225 0.0316 m
s
l 1 l 0 ± nT
l1 l0 a 1 ± n
b .
c
Wave phenomena in a ripple tank
0.233 m
s .
(6)
In our sample measurement at a frequency of 20 Hz we got
the following results:
Slow movement
From Task 1 (see Table 1) we got l 0 = 1.44 cm and
c =0.288 ms -1 = 28.8 cms -1 . The measured wave length in
front of the generator was l 1f = 1.13 cm and behind the generator
l 1b = 1.70 cm. From (6) follows: ƒl 0 – l 1f ƒ = ƒl 0 – l 1b ƒ.
Here:
and
ƒ1.44 cm – 1.13 cmƒ = 0.31 cm
ƒ1.44 cm – 1.70 cmƒ = 0.26 cm.
The difference between the two values is caused due to the
limitations of the used method.
We use formula (6) and l 1f = 1.13 cm to calculate the velocity
of the movement:
Faster movement
The measured wave length in front of the generator was
l 1f = 0.8 cm and behind the generator l 1b = 2.05 cm. This
leads to:
and
1.44 cm · n
1.13 cm 1.44 cm
28.80 cms 1
1.44 cm · n
3 0.31 cm
28.80 cms1 1 n 6.20 cms 1 0.062 ms 1
ƒl 0 – l 1f ƒ = ƒ1.44 cm – 0.8 cmƒ = 0.64 cm
ƒl 0 – l 1bƒ = ƒ1.44 cm – 2.05 cmƒ = 0.61 cm.
On using equation (6) and l 1f = 0.8 cm we calculate the velocity
of the movement the same way as above and get:
n 12.80 cms 1 0.122 ms 1 .
The Doppler effect is well known in our everyday life. When an
ambulance moves in someone’s direction one can hear a
change in the sound of its siren: the pitch of the sound gets
higher. When the ambulance moves away from this person the
pitch of the sound gets lower. The faster the ambulance
moves the higher the pitch (or lower, respectively). This phenomenon
can be shown in this experiment (the moving generator
represent the moving ambulance): the smaller the wavelength,
the higher the pitch of the sound.
f in Hz nl in cm n l in cm c = l · f in cms -1
10 6.7 2.5 2.68 26.9
15 6.5 3.5 1.86 27.9
20 7.2 5 1.44 28.8
30 7.2 7.5 0.96 28.8
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Task 3:
The experiment shows that the wavelength and thus the
velocity of the wave’s propagation is larger in deep water than
in shallow water. As a reference, the following results were
obtained (Table 2):
Table 2
f in Hz nl in cm n l in cm c in cm/s
deep water 22 4.2 3.5 1.20 26.40
shallow water 22 3.6 3.5 1.03 22.63
Since the water level is only a fraction of the wave length
l
(water depth d< 2 ), the phase velocity c strongly depends on
the water depth d. On decreasing water depth d , the phase
velocity c also decreases.
The behaviour of water waves at the boundary between a
zone of large water depth, and reduced water depth, is analogous
to the behaviour of light waves at the boundary between
air and glass. The propagation velocity of light waves is lower
in glass than in air. The same effect was observed in this
experiment where the propagation velocity of water waves is
lower in the zone of shallow water than in the zone of deeper
water.
The refractive index is here defined as the ratio of the propagation
velocity in deep water to the propagation velocity in
shallow water. In our sample measurement we got a refractive
index of 1.17. (A more detailed treatment of the refraction
index is performed in Task 4.)
In principle, higher refractive indices can be achieved by further
lowering the water level. However, the smaller the water
depth the larger the attenuation of the waves so that ultimately
they only penetrate a few centimetres into the zone of shallow
water. Precise observations and quantitative measurements
are then no longer possible. The behaviour of shallow
water zones is therefore analogous to the behaviour of glasses
with high absorption. The refraction of water waves can
therefore never be demonstrated without large absorption
losses.
b
Fig. 13 Geometrical description of the refraction of a plane
wave at the interface of two different water depths.
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Task 4:
Plane plate
When the front of the plane wave enters the boundary of the
shallow water zone, bending of the wave front occurs. A
change in the propagation direction of the waves towards the
normal at the point of incidence can be observed (Fig. 7). On
leaving the shallow water zone the wave is refracted by the
same angle in the opposite direction: Behind the plate, the
wave front is once more bended and ends up roughly parallel
to the initial wave front.
Prism
When the wave front enters the zone above the triangular
plate (prism) a bending of the wave crests and troughs can be
seen. The wave front is refracted towards the base of the
prism. On leaving the area of the shallow water zone the
waves are bent further towards the same direction (Fig. 8).
In both cases, a change in the wavelength above the plate and
the prism can be seen (Task 3). As displayed in Fig. 13, the
principle of the refraction of water waves at the boundary
between two different water depths is shown:
For the relationship between the angle of incidence a and the
angle b, the angle of the refracted wave, the following relationship
is taken directly from Fig. 13
The quotient
sina
sinb l0>0b0 l1>0b0 l0 .
l1 n01 l0
l1 c0>f c1>f c0 c1 (c 0 = propagation velocity in deep water, c 1 = propagation
velocity in shallow water) is called the refraction index for the
crossover from deep to shallow water.
Summarising, the refraction law is obtained in the more familiar
form from optics:
sina
sinb c0 n01 .
c1 The bending of the water waves on entering and leaving the
shallow water zone corresponds to the refraction of light on
passing through a plane-parallel plate and refraction in a
prism.
Convex plate
The plane waves leave the shallow water zone of the convex
plate as circular waves. They are convergent behind the plate
and run into a focus (Fig. 9).
Concave plate
You should have observed that the plane waves leave the concave
plate as divergent circular waves (Fig. 10.)
Due to the low propagation velocity of the water waves in the
shallow water zone, the water waves are refracted above the
convex and concave plate in the same way as light waves are
refracted in a convex or concave lens. The characteristic wave
patterns are formed as a result as displayed in Fig. 9 and
Fig. 10.
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Task 5:
Plane reflector
At an angle of 45° between the plane reflector and the propagating
water waves, the waves are reflected perpendicular to
its initial direction (90°; cp. Fig. 11). This means that the angle
of incidence is equal to the angle of reflection. On varying the
position of the plane reflector, one can recognise that this law
of reflection, which is known from the geometrical optics
(angle of incidence equals angle of reflection), is also valid for
water waves.
The law of reflection, which could be verified in this experiment,
can be explained by Huygens’ Principle. Huygens’
Principle states that every point of the reflector can be seen as
a circular wave exciter that oscillates with the same phase as
the waves that are generated by the plane wave stimulator.
The resulting interference is the reason for the characteristic
wave image (cf. Fig. 11).
Concave reflector
You should have observed that plane waves are reflected at
the concave reflector as circular waves. These circular waves
run into a focus (Fig. 12). Circular waves, which are generated
in this focus are reflected at the concave reflector as plane
waves.
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Wave phenomena in a ripple tank
As a sample measurement of the distance l between the focus
and the reflector we measured to be
l = 7.62 cm.
This distance is about half the radius of the concave reflector.
This experiment illustrates the unification of parallel beams in
a focal point of a concave mirror, as well as the parallel
bundling of beams that come from the focus of a concave mirror.
As a conclusion, the experiment shows the possibilities of
using surface water waves to depict waves phenomena. Many
phenomena, which are known from optics or from dealing with
sound waves, for example, can be shown and explained by
using water waves. This is why water waves are often used to
demonstrate the behaviour of waves in general.
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