Experiment 11:
The Speed of Sound
Objective:
The objective is to measure the speed of sound at Room Temperature.
Equipment:
A few tuning forks, a mallet, a resonance tube apparatus, and a calculator
Theory:
Sound is a longitudinal wave that travels at a speed of v = 331m/s at STP conditions. A longitudinal wave is one that oscillates parallel to its propagation direction. The way an induced disturbance travels in a slinky is a longitudinal wave motion. At a frequency ( f ) that we speak or sing (Audio range: 200Hz – 2000 Hz), the wavelength ( λ ) of a sound wave is in the range of (150cm – 15cm ). This may be verified by using the wave speed formula:
v = f λ
Resonance of Sound Waves in Open and Closed Tubes:
In music, a tube open at both ends is called an open tube, and a tube open at one end only forms a closed tube as shown below:
Maximum deviation of a wave from equilibrium is called amplitude. For a sound wave, maximum deviation can only occur at the open end (s). If the tube has the right length, this happens and is associated with an intensified loudness called resonance. This is because of the fact that at an open end, air molecules are free to oscillate back and forth. At a closed end, air molecules are not free to perform wide oscillations. In other words, closed ends can form nodes and open ends can form antinodes.
The following figures show how maximum and minimum oscillations occur at open and closed ends for a certain wavelength at different tube lengths:
Note that, for simplicity, representation of transverse waves are used to show states of maximum and minimum oscillation at open and closed ends. Sound waves; however, are longitudinal and oscillate back and forth parallel to the tube's length and not up and down as shown. These figures only indicate where maxima and minima occur.
As can be seen from the above figures, the length of a tube must be multiples of λ/4 for an antinode (maximum) to occur at its any open end.
For an open tube (left figure), if the tube's length is an even multiple of λ/4, each open end forms an antinode and resonance is heard.
For a closed tube (right figure), when the tube's length is an odd multiple of λ/4, resonance occurs.
In this experiment, a closed tube will be used. It will be seen that when the length of the tube is an odd multiple of a certain length, the tube is in resonance and intensified sound is heard.
At this point, it is suitable to repeat the definitions of wavelength and frequency.
Wavelength: Wavelength,λ , is defined as the distance from one peak to the next one on a wave. Of course, in general, wavelength is the distance between two successive points on a wave that are in the same state of oscillation.
Frequency: Frequency, f, is the number of waves (full λs) generated per second.
Procedure:
1) Obtain a resonance tube apparatus. The apparatus is just a long pistoncylinder system that allows a variable length closed pipe, as shown below.
Fig. 1
2) Hit a tuning fork of a known frequency with the rubber end of the mallet to make it oscillate. This should be done away from the pipe’s opening so that it doesn’t hit the glass tube causing it to break. While one student is holding the oscillating fork at about ˝ to1 inch from the pipe’s opening, another student may pull the piston's handle out and increase the pipe length until resonance is heard. A few trials and adjustments may be needed to locate the best length of the pipe at which the loudest possible sound can be heard.
3) Read the pipes length from the meterstick and record it as L_{1}. Note that L_{1} = 1λ/4.
4) In a similar manner locate the position of the 2nd resonance. As you know the second resonance should occur at about L_{2} = 3λ/4.
5) Locate the position of the 3rd resonance, as well. The 3rd resonance should occur at about L_{3} = 5λ/4.
6) Calculate the wavelength (λ) in two different ways as shown below:
L_{1}= 1λ/4 ; L_{2} = 3λ/4. L_{2}  L_{1} = 2λ/4 = λ/2, or, λ = 2(L_{2}  L_{1}).

L_{1}= 1λ/4 ; L_{3} = 5λ/4. L_{3}  L_{1} = 4λ/4 = λ, or λ = L_{3}  L_{1}.

The reason for finding the difference L_{2}  L_{1} is that both measured L_{1} and L_{2} must be corrected for the diameter of the pipe's opening. One empirical rule states that when 0.6D (D = pipe's internal diameter) is subtracted from each of L_{1} and L_{2}, the λ found from each of L_{1} and L_{2} gives great results in the calculation of the speed of sound. By subtracting L_{1} from L_{2}, the 0.6D's are cancelled automatically, and there is no need to measure the inner diameter of the tube and subtract it from each of L_{1} and L_{2}. 
7) Once λ is determined, equation v = f λ may be used to find the measured value for v, the speed of sound. Of course, f is the frequency of the tuning fork used.
8) For accepted value of v, the empirical formula v(T) = [ 331 + 0.6T ] m/s may be utilized. In this equation, T is the ambient temperature in ˚C. Read the temperature from the thermometer in the room you are in.
9) Repeat the above steps for two other tuning forks of different frequencies.
Data:
Given:
Equation v(T) = [ 331 + 0.6T ] m/s, to calculate the accepted value of v.
Frequencies of tuning forks used are:
f_{1} = Hz, f_{2} = Hz, and f_{3} = Hz.
Measured:
Room Temperature: T = ˚C,
Trial  f
(Hz) 
L_{1} (m) 
L_{2} (m) 
L_{3} (m) 
λ=2(L_{2}L_{1}) (m) 
λ=(L_{3}L_{1}) (m) 
λ_{avg.} (m) 
v=fλ_{avg} _{m/s} _{Measured} 
v=331+0.6T m/s Accepted 
% error 
1  
2  
3 
Calculations:
To be performed by students.
Comparison of the Results:
Calculate a %error on the speed of sound for each case.
Conclusion: To be explained by students.
Discussion: To be explained by students.