The formation of standing waves in a vibrating system is not a simple subject to understand; full comprehension will require a conscientious effort on the part of the student. In Melde’s experiment it was demonstrated that standing waves form when (a) two waves of equal wavelength are traveling in opposite directions in a media and (b) when the physical parameters are adjusted in such a way that the length of the strain is an integral number of half-wavelengths; i.e., L = nl /2.

This experiment is quite analogous to Melde’s experiment except that longitudinal sound waves in an air-column are substituted for the transverse waves in the string of the earlier exercise.

The apparatus for this experiment consists of a long tube containing air at atmospheric pressure and closed at one end by an ear-phone or speaker. This speaker is fed by an audio frequency oscillator and serves as a source of sound waves of variable frequency and loudness. Within the tube is a microphone which can be moved by a long push rod. This microphone serves as a detector of sound waves, and its output is displayed on a cathode ray oscilloscope (CRO). Your instructor will discuss the proper use of both the oscillator and the oscilloscope at the beginning of the lab period.

Sound waves produced at S move down the tube and strike the detector D, where part of the sound energy is reflected and part of is absorbed resulting in the electrical signal seen on the CRO. As the detector is moved away from the source, the CRO signal in general decreases indicating that the sound wave is being partially absorbed in its passage down the tube. However, at certain RESONANCE POINTS the signal increases to a maximum and the sound produced by the tube is appreciably greater. AT resonance the reflected energy from the detector passes down the tube and reflects again off the source in phase with the new incident sound wave thus tending to increase the amplitude; the result is a strong standing wave produced by the two (incident and reflected) traveling waves. This situation occurs when the separation of source and detector is some integral number of half wavelengths of the sound wave.

Figure 1a shows the standing wave pattern when the detector is at the first resonance point. The curve represents the displacement of the air molecules at various points in the tube. At points marked AN (anti-node) the air motion is maximum and at point N (nodal point) the displacement of air molecules is zero. Figure 1b represents the situation when the detector is at the second resonance point. Obviously the detector has moved 1/2l in going from one resonance point to the next.

When you have completed this experimental activity, you should be able to: (1) explain how standing waves are produced in an air column; (2) define resonance as applied to a vibrating system and state the boundary conditions which lead to resonance in an air column; (3) conduct an experiment to determine the velocity of sound in air: and (4) write and use the equation:

which relates the velocity of sound in a gas to the parameters of the gas.

Set the audio oscillator (OSC) at 1000 Hz and turn on the power for both the oscillator and the oscilloscope. Take a few minutes to become familiar with the basic controls of each device. In general, you should keep the OSC amplitude fairly low because of the nose in the room. The low level of the signal is then amplified by increasing the sensitivity of the oscilloscope.

With the frequency at 1000 Hz, determine the wavelength of the sound produced by measuring the separation of resonance points. Observe the decrease in amplitude as the probe moved further from the source. Calculate the velocity of sound in air from the wave equation:

The frequency can be read directly from the oscillator dial and checked against the sweep-time control of the CRO. Repeat for several other values of frequency. Make a data table with frequency, wavelength and sound velocity as the entries. Is the velocity of sound a function of frequency? Support your answer on the basis of your observations.

Without recording any numerical data, conduct a systematic investigation to determine the upper and lower frequency limits for which this method works using this apparatus. Try to determine the reasons for which the system fails at the lower limit and at the upper limit (the reasons are different). Summarize your results briefly; however, you need to be specific in your statements.

Now we will use the apparatus in a slightly different manner to determine the speed of sound in air. Set the source and detector about one foot (30 cm) apart; the exact distance doesn’t matter. Now tune the oscillator (OSC) until a resonance is clearly observed. Fine tune the OSC until the resonance is maximized, then record the frequency as f₁. It should be apparent to you that in this resonance condition the distance between the source S and the detector D is some integral number (say N₁) of half-wavelengths, or:

Now, keeping the source and detector in the same place, so that L is unchanged, slowly increase the frequency until the NEXT resonance is found. Again, fine tune and record this new frequency as f₂. Since this is the next higher frequency, there must be one more half-wavelength than before, so that:

Solving equations (3) and (4) together, we get:

which can be solved for N₁. With a careful measurement now of L, then you can use equation (3) to solve for v, the wave velocity. Try this method and report your results. Did N₁ turn out to be an integer? If not, why not?

The ideal gas law is:

where N’ is the number of moles of the gas present, R is the ideal gas constant, and T is the absolute temperature of the gas. N’ may be written as N/N_o where N is the total number of molecules present and N_o is Avogadro’s number (the number of molecules in one gram molecular mass of the gas). If we now write M/m for the number of molecules, where M is the total mass of gas and m is the molecular mass, the gas law can be written as:

Now, if we divide through by V, the volume, we get:

Or, since M/V is the density of the gas, we have the ratio P/r for the gas is given by:

Use the Handbook of Chemistry and Physics, or your textbook, to find values for P and r for air at 0^oC, and calculate (P/r ) in dyne-cm/gram for that temperature. Then use equation (9) to calculate (P/r ) for the current laboratory temperature. Be sure to show your calculations in the report.

In your text it is shown that the velocity of a sound wave in air is given by equation (1) where g is the ratio of the specific heats of the gas. Calculate g from your experimental value of v, compare it with the standard value of 1.41, and calculate the percent error.

Report your results and state any conclusions you can draw from this lab.

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