Resonance Tube Lab

Resonance 1 Williams Lab 1: Tube Staci Williams Kevin Schesing, Nicole Harty, Caitlin Kubota Section 015 2 Performed February 2, 2010 Due February 13, 2010 3 Theory: 2.1 Air As A Spring Williams Gas is a springy material, and when placed in a cylinder with pistons on each side it can be compressed as pistons push in, raising the pressure inside. There will be a net force from the pressure to push the piston back out. Since gas has mass it can support oscillations and waves. 2.2 Traveling Sound Waves in Air When a cone of a speaker moves out, it compresses air next to is and imparts an outward velocity to the air molecules around it, in addition to the random thermal velocities of air molecules. The molecules nearest to the speaker will collide with those near them and impart those molecules into motion, propagating away from the speaker producing sound. Similar statements apply to when the cone is moved in as well. If speaker cone vibrates sinusoidally, a traveling wave will be emitted form the speaker and the wave relation f = v < = wavelength, f = frequency of wave, v = velocity of wave> is satisfied.

AS the motion of the wave molecules move along the direction of the propagation of the wave are called longitudinal waves, which is contrasting to transverse waves which are on strings. The waves as the elements of the string move transverse to the direction in which the waves travel. In traveling waves the displacement of air satisfies the wave equation. V = (P/) < v = velocity of wave, = specific heats at constant pressure/ ” constant volume = Cp/Cv, P = air pressure, = air mass density>. With the ideal gas law it can be written as V = (RT/M ) < R = molar gas constant, T = absolute temperature, M = Molar mass>. For a given gas the speed will be proportional to the square root of the temperature giving the equation vrms = (3RT/M) < vrms ~ thermal speed of the gas molecules>. The speed of sound in gas is close to the thermal speed of molecules in gas, so the velocity of propagation is essentially the thermal speeds of the molecules giving this equation V = 331.5 + .606T m/s . 2.3 Traveling Sound Waves in a Tube Sound waves are able to travel in a tube of a constant cross section much similar to how they travel in open air.

The tube is assumed to have rigid walls that will not flex under pressure variations, as well as be smooth so that there is not much attenuation of the wave, allowing the speed of the waves to be nearly the same as in open air. 2.4 Standing Sound Waves in a Finite Tube Traveling sound waves in a finite closed tube will reflect at the ends, allowing for resonance to occur at certain conditions called resonant frequencies (normal modes). Resonance will occur when the reflected waves at both ends reinforce one another. The “pressure” of the air in the wave is the change of pressure from the average value, with the “displacement” of air to be its displacement from the equilibrium position, with both pressure and displacement varying sinusoidally in space and time. Points where pressure is maximum are called pressure antinodes, and zero are called pressure nodes. Likewise, points where displacement is maximum are called displacement antinodes and zero displacement are called displacement nodes. In standing sound waves pressure nodes occur at displacement antinodes and pressure antinodes occur at displacement nodes.

An open end of a finite tube will be a pressure node because of the normal air pressure outside of the tube, making the point same a displacement antinode, while the end of a closed tube must be a displacement node and a pressure antinode. Frequencies can be calculated for tubes with both ends closed, one end closed and one end 4 opened, and both ends open. Resonance wavelengths can be calculated y fitting standard waves into the tube so that boundary conditions are settled. The lowest resonance frequency is called the fundamental frequency or the 1st harmonic. The nth harmonic is n multiplied by the fundamental frequency, and not all harmonics must be present. Data and Calculations:

4 Measuring Wavelength (m) .708 .412 .582 .759 .350 .268 .384 .501 .618 .736 .233 D3 (m) D4 (m) D5 (m) D6 (m) D7 (m) Frequency (Hz) 500 1000 1500 D1 (m) .159 .059 .038 D2 (m) .513 .248 .152 Velocity (m/s) 343 343 343 Theoretical (m) .686 .343 .229 Percent Error (%) 3.21 2.04 1.75 Sample Calculations 1500 Hz: .513m – .159m = .354m .354m * 2 = .708m (.708m – .686m)/ .686m * 100% = 3.21 % Since wavelength observed, multiply by 2 5 Pulsed Experiments 5.1 Speed of Sound X = .55m (Distance from piston to speaker) t = .0015 sec. (pulse time) V = X / t = .55/.0015 = 366 m/s (velocity of sound) 366 -343/343 * 100 % = 5.83% 5.2 Boundary Conditions .2 cm needed to change reflected pulse Error Analysis: There was very little error present during the experiment when we calculated the wavelength, all of which had a percent of error 3.21 percent or less. The small error that was encountered could be 5 Williams attributed to human error, in such a case that the distance was falsely read, or that the graph was not zoomed in enough to see exactly where the maximum intensity occurred.

The percent error decreased as the amount of data points we were able to take went up, suggesting that if more data points were available, the percent error would be less. In the experiment where we found the speed of sound a possible error may have arisen due to the mike not being fully erect towards the other side of the tube, potentially creating false resonance/pulse. Another factor that may have caused error is that the end of the tube was not completely sealed, which means sound waves could stream out or in, diminishing or increasing the frequency. Conclusion: Measuring Wavelengths For a frequency of 500 Hz the speaker is about a quarter of a wavelength away from a minimum or maximum. Comparison to the string experiment… The wavelengths change with frequency in the way I expect that as the frequency increases, the wavelength decreases allowing for more data points to be detected in the tube. This experiment adequately demonstrated how to calculate the wavelength using points of maximum intensity of the SWS software. Speed of Sound The reflected pulse in this experiment was inverted.

While moving the piston slowly toward the mike with the scope running it is seen that the reflected pulse had a lower amplitude than that of the original pulse. This experiment allowed for the calculation of the speed of sound. This data figured is off of the expected value, but it is close to the expected value, showing that if a better point would have been chosen, the result would have been better than the results that were attained. Boundary Condition The tube must be cracked .2cm to change the reflected pulse. It allows enough of the pulse to escape allowing for a change in the amplitude. Questions: 1. open/open FN = NV/2L —— F 1 = V/2L open/closed FN = NV/4L —— F 1 = V/4L 2. V = F = V/F F = V/2L = V*2L/V = 2L = 2L PV = NRT, P = NRT/V = M/V V = (NRTV/VM) V = (RT/M) 3. V = (P/) 6