Experiment 11:
The Speed of Sound
Objective:
To measure the speed of sound at Room Temperature.
Equipment:
A few tuning forks, a mallet, a resonance tube apparatus, a one meter long ruler, 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 from equilibrium called amplitude for a sound wave can only occur at the open end(s) of a tube (if the tube length is suitable). This is because of the fact that at an open end of a tube, air molecules are free to oscillate back and forth along the tube end. At a closed end air molecules are not free to perform oscillations. In other words, closed ends form nodes and open ends can form antinodes if tubes have the right lengths for antinodes to occur.
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 length and not up and down as shown. These figures only indicate where maxima and minima occur.
As can be seen from the figures, the length of a tube must be multiples of ( ¼ λ ) for maxima at the open ends to occur. In an open end tube, if the length of the tube is an even multiple of ( ¼ λ ), then the open ends are at maximum states of oscillation, and a load sound is heard. The tube is said to be in resonance. In a closed end tube, the length of the tube must be an odd multiple of ( ¼ λ ) for resonance to occur (See the above figures.)
In this experiment, a closed end 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 a load sound is heard.
This certain length happens to be an odd multiple of ¼ λ . 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 peak 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:
Obtain a resonance tube apparatus and fill its canister with sufficient water. When the tube is full, the canister bottom must be about the top level of the tube. At this position, the canister should be almost empty with may be 1 inch of water in it (Fig. 1). As the canister is lowered, water will flow from the tube into it and fills it up. Water level changes in the tube by raising or lowering the canister. This mechanism establishes a means of having a variable length closed pipe.
Now use the mallet and hit a tuning fork of known frequency ( 1024 Hz, for instance) with it to make it vibrate. This should be done away from the pipe’s opening so that it doesn’t hit the glass tube and break it. Hold the oscillating fork at about ½ to
Fig. 1
1 inch from the pipe’s opening and keep lowering the water level by lowering the canister until you hear the fork sound amplify and become its loudest. At its loudest (resonance), the distance from the water level in the tube to the tube’s opening is a proper length of the tube for which resonance occurs. This first length is 1( ¼ λ ). Keep lowering the water level until you here the next resonance. The second length (measured from the water level to the opening) is 3 ( ¼ λ ). As we know a closed pipe whose length is an odd multiple of ( ¼ λ ) of the sound wave performs resonance for that sound wave. Use the two measured lengths to calculate λ in each case as shown below:
L1 = 1(¼ λ) λ = (4/1)L1 (1)
L2 = 3(¼ λ) λ = (4/3) L2 (2)
The calculated λ from the above two equations should result in the same value. There will be a little difference due to experimental errors anyway. Average the two values for λ and then multiple by 5( ¼ λ ) to estimate the right length for the third resonance occurrence. Name this length L3 expected. Continue the experiment to find the position and length for the third resonance. Measure the length for this 3rd resonance and name it L3 Measured. Calculate a %Difference for L3. Also find λ from L3.
L3 = 5(¼ λ) λ = (4/5) L3 (3)
The next step is to find the mean value for λ from the past three equations. Once λ is determined, equation v = f λ may be used to find the measured value for v, the speed of sound.
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. Use the room temperature in this equation to find the accepted value for the speed of sound.
Repeat the above steps for another tuning fork of a different frequency.
Data:
Given:
Equation v(T) = [ 331 + 0.6T ] m/s, to calculate the accepted value of v.
Frequencies of tuning forks used are:
f1 = ? Hz (Part 1) f2 = ? Hz (Part 2)
Measured:
Room Temperature T = ? ˚ C L1 = ? m L2 = ? m L3 = ? m
Calculations:
To be performed by students
Comparison of the Results:
Calculate a %error on the speed of sound.
Conclusion: To be explained by students
Discussion: To be explained by students