Acoustics Wavelegth ad speed of soud Speed of Soud i Air GENERATE AND MEASURE STANDING SOUND WAVES IN KUNDT S TUBE. Geerate stadig waves i Kudt s tube with both eds closed off. Measure the fudametal frequecy as a fuctio of the legth of the Kudt s tube. Measure the frequecies of the fudametal ad overtoes for a fied legth of tube. Determie the speed of propagatio of the wave from the resoat frequecies. UE1070320 06/16 UD Fig. 1: Measuremet set-up. GENERAL PRINCIPLES It is possible to geerate stadig waves i Kudt s tube by producig waves of a suitable resoat frequecy from a loudspeaker at oe ed of the tube, which are the reflected by the cap at the other ed. If the legth of the tube is kow, it is possible to determie the speed of propagatio of the waves from the resoat frequecy ad the umber of the harmoics. Soud waves propagate i air ad other gases by meas of rapid chages i pressure ad desity. It is easiest to describe them o the basis of the soud pressure, which is superimposed o top of atmospheric pressure. As a alterative to the soud pressure p, the soud velocity v ca also be used to describe a soud wave. That is the average velocity of gas molecules at a give poit i the oscillatig medium at a poit i time t. Pressure ad velocity of soud are liked, for eample by a Euler equatio of motio: p v (1) 0 t 0: desity of gas I Kudt s tube, soud waves propagate alog the legth of the tube, i.e. they ca be described with the help of a oedimesioal wave equatio, which applies to both soud pressure ad velocity: (2), p, t p t 2 c t, v, t v t 2 c t c: speed of soud This eperimet studies harmoic waves, which are reflected at the ed of the Kudt s tube. or 1 / 5
FUNCTION GENERATOR FG100 Offset Start/Stop Trig. I/Out I/Out Cotrol Voltage Sweep Frequecy Amplitude 0 V 10 V Output 12 VAC 2 A UE1070320 Fig. 2: Schematic of eperimet set-up To fid the solutios to the wave equatio, the superpositio of the outgoig ad reflected waves eeds to be take ito accout: (3) v 2i f t 2i f t 0 0 p p e p e 2i f t 2i f t 0 0 v v e v e p0>, v0>: amplitudes of outgoig wave, p0<, v0<: amplitudes of returig wave f: frequecy, : wavelegth I this case (4) f c By substitutig these solutios ito equatio (1) ad cosiderig the outgoig ad returig waves separately, the followig ca be derived: (5) p0 v0 Z or p0 v0 Z. The quatity L p (6) Z c 0 is kow as the soud impedace ad correspods to the resistace to the waves from the medium itself. It plays a key role i cosideratios of the reflectio of a soud wave by walls with a impedace of W. The followig the applies: v0 Z W (7) rv v Z W 0 1 1 p 0 ad r Z W p p 1 1 0 Z W I this eperimet W is much higher tha Z so that we may assume rv = 1 ad rp= -1. If the reflectig wall is selected, for simplicity s sake, to be at = 0, the spatial compoet of the soud wave ca be derived from equatio (3) as follows: 2i 2i p p0 e e e (8) 2 2 p0 cos e ad 2i f t 2i f t Oly the real compoets of these terms have ay actual physical relevace. They correspod to stadig soud waves which have a pressure ati-ode at the ed wall (i.e. at = 0), while the soud velocity at that poit has a ode i its oscillatio. The velocity is phase shifted ahead of the pressure by 90. Soud waves are geerated by a loudspeaker at a distace L from the wall. These waves oscillate with frequecy f. At this poit, too the pressure has a ati-ode ad the velocity has a ode. Such boudary coditios are oly fulfilled whe L is a iteger multiple of half the wavelegth: (9) L 2 From equatio (3) the, the frequecies must fulfil the followig coditio for resoace: c (10) f. 2 L LIST OF EQUIPMENT 1 Kudt s tube E 1017339 (U8498308) 1 Microphoe probe, log 1017342 (U8498282) 1 Microphoe bo @230V 1014520 (U8498283-230) or 1 Microphoe bo @115V 1014521 (U8498283-115) 1 Fuctio geerator FG 100 @230V or 1 Fuctio geerator FG 100 @115V 1009957 (U8533600-230) 1009956 (U8533600-115) 1 USB oscilloscope 250 MHz 1017264 (U112491) 1 Escola 100 aalog multimeter 1013527 (U8557380 2 HF patch cords, BNC/4-mm plugs 1 Pair of safety eperimet leads, 75 cm 1002748 (U11257) 1002849 (U13812) 1 HF patch cord 1002746 (U11255) SET UP AND PROCEDURE Set up the soud tube o the supplied feet (fig. 1). First isert the capillary disc ito the tube ad the the ed cap with the sockets for the speaker. Set them up such that they meet flush iside the tube. Isert the log microphoe probe through the hole i the ed cap with the heatig rod sockets as far as it will go ad the push it through the guide disc. Screw the probe disc oto the log microphoe probe. Isert the ed cap with the heater sockets ad with the log microphoe probe attached ito the tube. 2i 2i v v0 e e e 2 2 i v0 si e 2i f t 2i f t 2 / 5
UE1070320 Note: If ecessary apply some glycerie or soap to the sealig gaskets to make isertio easier. Clamp the movable scale ito its holder o the base uit ad move it util the zero mark coicides with the frot edge of the capillary disc. Coect the output sockets of the fuctio geerator to the speaker sockets. Coect the log microphoe probe to the Chael A iput of the microphoe bo. Coect the chael A output of the microphoe bo to the aalog multimeter by meas of a BNC/4-mm cable. Coect the plug-i power supply to the microphoe bo ad plug it ito the mais. I this eperimet the frequecy f of the sigal to the speaker will be cotiuously varied while the microphoe probe measures the soud pressure at the reflectig ed wall. Resoace occurs whe the microphoe sigal reaches its maimum amplitude. Fudametal frequecy as a fuctio of legth By movig the microphoe probe i or out, set up a legth L = 80 cm betwee the capillary disc ad the probe disc. Match up the amplitude of the fuctio geerator ad the measurig rage of the multimeter as appropriate, eve durig the eperimet if ecessary. Take ote of the maimum power limit for the speaker (Urms = 6 V ma.). Set the frequecy to 150 Hz ad gradually icrease it. Fid the frequecy at which the multimeter displays the first maimum ad eter the value ito Table 1. The frequecy f1 foud i this way correspods to that of the fudametal oscillatio whe L = 80 cm. Adjust the legth L betwee the capillary disc ad the probe disc i steps of 10 cm dow to L = 10 cm ad repeat the measuremet for each of these steps. I each case, use the resoat frequecy f1 foud for the previous step as the ew start frequecy. Frequecy sca By movig the microphoe probe i or out, set up a legth L = 60 cm betwee the capillary disc ad the probe disc. Use a HF cable to coect the chael A output of the microphoe bo to chael CH1 of the USB oscilloscope. Coect the pair of sockets labelled Cotrol voltage iput/ramp output o the fuctio geerator directly to chael CH2 of the USB oscilloscope. Press the Sweep o the fuctio geerator ad cofigure the followig parameters: Freq. start stop it. cotiuous mode Time 100 Hz 6000 Hz 4.00 sec Set the marker for the horizotal trigger positio o the USB oscilloscope all the way to the left ad move the y positio marker for CH 1 dow by half a divisio from the middle ad that for CH 2 right to the bottom, the cofigure the followig parameters: Time/div CH1 CH2 Trigger Mode Sweep Source Slope Level 400 ms 1.00 V DC 2.00 V DC Edge Auto CH2 850 mv Start the fuctio geerator sweep ad wait util the full frequecy spectrum is displayed o the USB oscilloscope. Stop the measuremet by pressig the Stop butto o the USB oscilloscope ad take a photograph of the trace o the scree. Frequecies of fudametal ad harmoics for a fied legth By movig the microphoe probe i or out, set up a legth L = 60 cm betwee the capillary disc ad the probe disc. Match up the amplitude of the fuctio geerator ad the measurig rage of the multimeter as appropriate, eve durig the eperimet if ecessary. Take ote of the maimum power limit for the speaker (Urms = 6 V ma.). Slowly icrease the frequecy from 250 Hz to 3500 Hz. Eter the frequecies where the multimeter displays maimum values ito table 2. 3 / 5
UE1070320 SAMPLE MEASUREMENT AND EVALUA- TION Table 1: Resoat frequecies of fudametal oscillatio ( = 1) as measured ad wavelegth as calculated usig equatio (9) for various legths L. L / m f1 / Hz 1 = 2 L 0.8 218 1.60 m 0.7 250 1.40 m 0.6 292 1.20 m 0.5 350 1.00 m 0.4 436 0.80 m 0.3 583 0.60 m 0.2 884 0.40 m 0.1 1768 0.20 m 2000 1000 f 1 / Hz 0 0 1 1 / m Fig. 4: Graph of frequecy agaist wavelegth for fudametal frequecies ( = 1) with variable legth L. Table 2: Resoat frequecies of fudametal oscillatio ad harmoics as measured ad wavelegth as calculated usig equatio (9) for a fied legth L = 0.6 m. f / Hz 2L 1 289 1.20 m 2 582 0.60 m 3 876 0.40 m 4 1164 0.30 m 5 1453 0.24 m 6 1746 0.20 m 7 2039 0.17 m 8 2331 0.15 m 9 2623 0.13 m 10 3206 0.12 m 11 3494 0.10 m Fudametal frequecy as a fuctio of legth Accordig to equatio (9) the wavelegths of the resoaces f1 for the fudametal wave are as follows: (11) 1 2 L. Calculate wavelegths usig equatio (11) ad eter them ito Table 1. Plot the resoace frequecies measured ad the wavelegths calculated (Table 2) i a graph of f agaist (Fig. 4). Fit a hyperbola to the measuremet poits: 1 (12) f 1 a. 1 The fittig of the curve cofirms equatio (4). The speed of soud i air is equal to the gradiet a: m (13) c a 353. s Fig. 3: Oscilloscope trace of frequecy sca, L = 0.6 m. The value differs by about 2% from that quoted i literature c = 346 m/s at T = 25 C. 4 / 5
UE1070320 f / Hz 4000 3000 2000 1000 Frequecy sca The times t where the maima i the frequecy sca occur, ca be determied with the help of the cursors of the USB oscilloscope. These times do ot match the periods of oscillatio T which correspod to the resoat frequecies f. The resoat frequecies eed to be calculated from the frequecy sca parameters as follows: t t T 4 s. (17) start stop start f f f f 100 Hz 5900 Hz sweep From this the speed of soud ca agai be determied by the meas described above. 0 0 1 / m Fig. 5: Graph of frequecy agaist wavelegth for fudametal ad harmoics whe L = 0.6 m. Frequecies of fudametal ad harmoics for a fied legth Accordig to equatio (9), the resoat frequecies determied f should have wavelegths 2L (14). Calculate the wavelegths usig equatio (14) ad eter them ito Table 2 Plot the resoat frequecies measured agaist the calculated wavelegths (Table 2) i a graph (Fig. 5). Fit a hyperbola to the measuremet poits: 1 (15) f a. The fittig of the curve cofirms equatio (4). The speed of soud i air is equal to the gradiet a: m (16) c a 356. s The result differs by about 3% from the value quoted i literature c = 346 m/s at T = 25 C. 3B Scietific GmbH, Rudorffweg 8, 21031 Hamburg, Germay, www.3bscietific.com Copyright 2016 3B Scietific GmbH