Physics 31210 Spring 2006 Experiment 9 TRAVELING WAVES Reference: Halliday, Resnick & Walker, 7th Ed., Sections 16-1 to 5, Sections 17-1 to 4 I. Introduction: Waves of all kinds, propagating through many different media, have certain characteristics in common. The frequency f and wavelength λ of the waves are related to the traveling wave speed V through the expression V = λf In this experiment you will study a particular type of traveling wave. The waves are sound waves in air at frequencies above the audible range (ultrasound). Sound waves are a very important example of longitudinal waves and consist of a series of high- and low-pressure regions traveling through the medium so that particles of the medium vibrate along the direction of motion of the wave. This experiment provides a unique opportunity to determine separately the frequency f, the wavelength λ, and the wave speed V of the same traveling wave and to check the relation given above. The waves are radiated when a transducer, labelled "source" in the diagram below, is fed an alternating current of the appropriate frequency from an electrical oscillator (function generator). In turn, these waves induce electrical signals in other transducers, labelled "receivers". WAVEFRONT ARRIVING AT RECEIVER 1 IN PHASE WITH WAVEFRONT ARRIVING AT RECEIVER 2 OSCILLOSCOPE SOURCE RECEIVER 1 RECEIVER 2 The two received signals are displayed as curves of voltage vs. time on a dual channel oscilloscope as shown schematically in the figure on the next page. (This experiment will provide you with a brief introduction to the oscilloscope, a most important laboratory instrument. At this time you will need to know only the most basic adjustments of the oscilloscope, which will be explained and demonstrated during the laboratory period.) Consider receiver 1 to be fixed and receiver 2 to be movable. The only purpose of receiver 1 throughout the experiment is to provide a reference time to the oscilloscope which could also have been provided direct from the function generator. When receiver 2 is moved, only its displacement relative to its initial position or to receiver 1 is relevant. II. Equipment: 8-1
Optical Bench with 3 clamps, set of 3 ultrasonic transducers with brackets and cables, sine wave oscillator (function generator), dual-trace oscilloscope, ruler or meter stick with caliper jaws. CHANNEL 1 (A) CHANNEL 2 (B) III. Procedure: Record the air temperature T air (in o C) in the room. Place the transmitter (source) transducer near one end of the optical bench. Place both receivers about 20 cm away, toward the other end of the optical bench. Make sure that the transmitter is connected to the function generator, that receiver 1 is connected to channel 1 (or A) of the oscilloscope, and that receiver 2 is connected to channel 2 (or B). First you will determine the frequency at which the transducers operate best. They only work well near their resonant frequency which is typically about 37 khz. Start with the function generator at this frequency and at maximum sinusoidal output. The oscilloscope time base (time/div or sec/div) should be set initially at 20 µsec/div. (Make sure the red variable control on the time base of Tektronix oscilloscopes is set at the click stop.) Note: If you use an incorrect value for the time base in this experiment, it is a MISTAKE which could look like a serious systematic error in your results! Set the vertical sensitivity at maximum (lowest numerical value, knob fully clockwise). Adjust the frequency of the signal generator to maximize the size of the receiver signals, reducing the oscilloscope sensitivity if necessary as the signals get bigger and reducing the voltage output of the function generator if needed to keep the signals on the scope reasonable in size. When you have completed the process you should see "clean" sine-wave traces. If distortions are seen, reduce the function generator output further; if the signal is "fuzzy" try increasing the output. When you have optimized the frequency setting, do not change the frequency for the rest of the experiment! Any change can make your data useless. A. Frequency. Each lab partner should measure the period T of the oscillations by reading the time for several complete cycles on the oscilloscope scale. A distance on the horizontal scale can be converted to a time based on the time base setting. Adjust the time base setting if necessary (but 8-2
do NOT use the variable control) so several complete oscillations can be seen. When making time measurements using the sine wave displayed on the oscilloscope screen, it is often convenient to adjust the vertical position knob so as to center the sine curve vertically on the horizontal line which has small divisions, as this permits the most accurate determination of where the sine curve crosses the time axis. Estimate the error in your distance measurement on the scale and convert this error into an error δt in the period. Calculate the frequency f = 1/T. This should agree (within 5% or so) with the oscillator dial setting. If it does not, ask your TA to check the frequency. B. Wavelength. Adjust the position of the scope traces so that trace 2 is in a definite known position with respect to the reference trace 1. Measure the distance from receiver 2 to the source transducer with the caliper ruler and estimate the error. Now slowly move receiver 2 away from the source. Notice that as you move receiver 2, the corresponding scope trace moves to the right (later in time). For this part of the measurement, fix your attention on one particular time marker (one of the vertical reference lines on the scope screen). To determine the wavelength λ of the sound waves, move receiver 2 so that an integral number N (say 10 or 20) of full cycles passes the fixed time marker. Record this number of cycles. Measure and record the final distance from receiver 2 to the transmitter and the error. The displacement x between the final and initial positions of receiver 2 is the integral number of wavelengths Nλ. Each lab partner should perform these measurements and estimate the errors in the distances. Note that the wavelength measurement does not utilize the oscilloscope except to count the number of waves (the time scale is not read). C. Wave Speed. Measure the speed of sound V First move receiver 2 back near receiver 1 and line up the patterns on the scope screen. Measure this initial distance from receiver 2 to the transmitter with the caliper ruler and estimate the error. Now slowly move receiver 2 away from the source as before while keeping track of one particular peak (or other characteristic point on the trace) as it moves to the right across the screen (i.e. moves to a later time) as far as possible. A slower time base setting will give you a longer time over which you can shift the trace. Record the time shift t (and the error) of the trace on the scope screen corresponding to the shift x in position of the receiver. Measure and record the final distance from receiver 2 to the transmitter and the error. The displacement x between the final and initial positions of receiver 2 is the distance the sound waves travel in time t. This provides a direct measurement of the wave speed V = x/ t. Repeat making measurements of time shift t (and its error) and displacement x (and its error) in the same way for at least two other different displacements of receiver 2. Again record the air temperature T air (in o C) in the room. From your values of the air temperature before and after you made your measurements, make an estimate of the air temperature at which your measurements are made and the error in this temperature. (Note: The room thermometers are accurate to about ±1.0 C o but the room temperature may vary by more than this at various locations around the room.) IV. Analysis: 8-3
A. Frequency. Calculate the frequency f and the error δf in the frequency of the sound waves from your measurements. Assume that the inherent accuracy of the oscilloscope time base is ±3% in this and other error determinations. In calculating δf include contributions from both your earlier estimate of the error δt in the period and from the accuracy of the oscilloscope time base. B. Wavelength. Calculate the wavelength λ and the error δλ in the wavelength of the sound waves from your measurements. C. Wave Speed. For each of your 3 or more sets of measurements of displacement x and time shift t, calculate the wave speed V = x/ t and the error δv in the wave speed of the sound waves. Using these values of V, calculate the mean value for V, the standard deviation σ of the measurements, and the standard deviation in the mean value. How do these standard deviations compare with your error estimates δv for the individual determinations ofv? What is YOUR best estimate of the error for your final value of V? What is YOUR best value forv and its uncertainty based on these results? Give reasons for your answer. Calculate the wave speed V and the error δv in the wave speed of the sound waves from the relationship V = λf using your determinations of f and λ and their errors. The speed of sound in air can be determined from the formula V = γrt/m where γ = 1.40, R is the gas constant (8314 J kmol -1 deg -1 ), T is the absolute temperature (in K) and M is the molecular mass (29 kg/kmol) for air. T in K = T in o C + 273.15. Use this information and your measurement of T air to calculate the sound wave speed V under the conditions in which you measured it. Calculate an error δv for this value of V based on your estimate of δt air. V. Discussion of Results. You now have several values for the traveling wave speed V of sound waves in air under the conditions when you made your measurements. Compare your various determinations of V. Are they consistent with one another? How did you reach your conclusion? If they are not consistent, can you think of any possible reasons for discrepancies? What do you think is your "best" value for V? Why? Do you regard your results as providing a verification of the relation V = λf? Why? The frequency range for sound waves within the range of sensitivity of the human ear is commonly listed as 20 Hz to 20 khz. What would be the corresponding range of wavelengths for sound waves in air for the conditions under which you made your measurements? If one wanted to perform a similar experiment using sound waves of about 100 Hz, how would things have to be modified? NOTE: Before you leave the laboratory you must make sure you have all of the measurements required for the determinations of the frequency, wavelength, and velocity of the sound waves and their errors. Do not forget to record the temperature before and after your measurements. If you have time, you may want to make quick calculations of f, λ, V = λ f, and V = x/ t. to make 8-4
sure your measurements are reasonable. Each lab partner must have a complete data sheet initialed by your TA. The rest of the work can be done outside of the laboratory. 8-5