Wave phenomena in a ripple tank

Transcription

1 Wave phenomena in a ripple tank LEP 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 Ext. vibration generator for ripple tank Connecting cord, 32 A, 500 mm, red Connecting cord, 32 A, 500 mm, blue Demo set for ripple tank Software Measure Dynamics 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. 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. Fig. 1: Overview of the experimental setup. PHYWE series of publications Laboratory Experiments Physics PHYWE SYSTEME GMBH & Co. KG D Göttingen P

2 LEP Wave phenomena in a ripple tank 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. 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. 2: Arrangement for generating circular waves. Fig. 4: Keypad of the ripple tank device. Fig. 3: Ripple tank with attached camera. 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. 2 P PHYWE series of publications Laboratory Experiments Physics PHYWE SYSTEME GMBH & Co. KG D Göttingen

3 Wave phenomena in a ripple tank LEP 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. 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. 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. 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. PHYWE series of publications Laboratory Experiments Physics PHYWE SYSTEME GMBH & Co. KG D Göttingen P

4 LEP Wave phenomena in a ripple tank 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. 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. 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. 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. 4 P PHYWE series of publications Laboratory Experiments Physics PHYWE SYSTEME GMBH & Co. KG D Göttingen

5 Wave phenomena in a ripple tank LEP 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. 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 c 2 g k sk r 1 c 2 gl 2ps 2p lr. (1) (2) (3) (4) (5) 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. 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. PHYWE series of publications Laboratory Experiments Physics PHYWE SYSTEME GMBH & Co. KG D Göttingen P

6 LEP Wave phenomena in a ripple tank On inserting the values for the surface tension of water s = 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 c m s m s. l 1 l 0 ± nt l 1 l 0 a 1 ± n c b. The negative sign in this formula applies in the direction of movement in front of the generator, the positive sign applies behind the generator. (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 cms cm n cm cms 1 1 n 6.20 cms ms 1 ƒl 0 l 1f ƒ = ƒ1.44 cm 0.8 cmƒ = 0.64 cm ƒl 0 l 1b ƒ = 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 cms 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. Table 1 f in Hz nl in cm n l in cm c = l f in cms P PHYWE series of publications Laboratory Experiments Physics PHYWE SYSTEME GMBH & Co. KG D Göttingen

7 Wave phenomena in a ripple tank LEP 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 shallow water 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 (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. 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 l 0>0b0 l 0. l 1 >0b0 l 1 n 01 l 0 l 1 c 0>f c 1 >f c 0 c 1 (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 c 0 c 1 n 01. 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). b Fig. 13 Geometrical description of the refraction of a plane wave at the interface of two different water depths. 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. PHYWE series of publications Laboratory Experiments Physics PHYWE SYSTEME GMBH Göttingen, Germany P

8 LEP Wave phenomena in a ripple tank 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). 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. 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. 8 P PHYWE series of publications Laboratory Experiments Physics PHYWE SYSTEME GMBH Göttingen, Germany