Objective Wave Motion This laboratory examines the principle on which most musical instruments operate and allows the student to observe standing waves, hear resonance and calculate the velocity of the waves. Background Music is based on the relative pitch and quality of sound waves produced by voice or musical instruments. Nearly all sounds emanating from musical instruments are the result of standing waves. Two wave trains of the same frequency traveling in the same medium in opposite directions may produce standing waves. This can be demonstrated by fastening one end to the string to a post and moving the other end up and down in simple harmonic motion. For a given frequency the number of loops is determined by the tension in the string. As the wave, moving to the right, created by the up and down motion of the free end meets the reflected wave, moving to the left, they interact to form nodes and antinodes. The nodes are points of no motion, and the antinodes are points of maximum amplitude. The fundamental principle of wind instruments is the vibration of an air column. Varying the length of the air column brings about the various notes produced by most wind instruments. A vibrating tuning fork held in front of the open end of a long pipe with a plunger at the other end can be made to resonate to the frequency of the fork. This is accomplished by moving the plunger such that the sound wave traveling down the pipe and the wave reflected off the plunger constructively interfere and set up standing waves. The length of the pipe for standing waves depends on the frequency of the vibrating tuning fork. Materials needed String vibrator String 2-Meterstick weight pan weights Resonant pipe Tuning forks (384 Hz & 480 Hz) Mallet Meterstick Clamps and support rods Wave Motion 1
Procedure Part I. Standing Waves on a String Figure 1 With the wave generator and string setup, you will adjust the tension in the string (by varying the weight on the weight pan at the end of the string) to produce standing waves. 1. Measure and record the length, L, of the vibrating string with the 2-meterstick from the tip of the metal vibrator to where the string touches the top of the pulley. 2. Add or subtract mass on the weight pan until a standing wave is produced and a clear set of loops are observed. 3. Record the total mass hanging on the end of the string (including the hanger mass). 4. Record the number of loops observed on the string. Figure 1 shows a standing wave with 2 loops. 5. Repeat Steps 2-4 to produce 2 more different standing wave patterns (# s of loops). Method I: Wave Velocity on a String 2L 6. Calculate and record the wavelength (λ) of the standing waves by using λ =, where n n is the number of loops observed on the string and L is the length of the string. 7. The wave generator vibrates at a constant frequency (f) of 120 Hz. Use this frequency and the wavelengths for your standing waves to calculate the velocity of the waves on the string using the wave equation, v = fλ. Wave Motion 2
Method II: Wave Velocity on a String 8. Calculate and record the velocity of the wave using the tension on the string (T) and the T mass per unit length (µ) of the string using the equation v =. µ (For this string, µ = 3.614 x 10-4 kg/m and for the Tension, use T = Mg, where M is the total mass hanging on the end of the string and g is 9.8 m/s 2.) 9. Calculate a percent difference in velocity (v) from the 2 different methods. Value 1 Value 2 % difference = *200% Value 1 + Value 2 Part II. Resonant Pipe 1. Push the plunger all the way to the end so that it is visible at the end of the pipe. Strike one of the tuning forks and place it near the opening. Slowly pull the plunger stick outward until a resonance (loudest sound) is observed. ** You WILL need to frequently strike the tuning fork (to keep it vibrating) while searching for resonance. 2. Once a resonance is located, carefully insert the meter stick into the pipe to measure and record the length (L 1 ) from the opening of the pipe to the plunger face. 3. Place the same tuning fork at the pipe opening and continue to pull the plunger outward until a second resonance occurs. Measure and record this new length (L 2 ) from the opening of the pipe to the plunger face. 4. Repeat Steps 1 3 two more times using the same tuning fork. [Note: All your L 1 values should be similar as should all your L 2 values.] 5. Calculate the wavelengths (λ) of the sound waves by using λ = 2(L 2 L 1 ). 6. Calculate the velocities of sound (v) in air using the wave equation, v = fλ. 7. Calculate the percent error for each of your velocities of sound using 344 m/s as the accepted value for the velocity of sound in air. Observed Value Accepted Value % error = *100% Accepted Value 8. Calculate the average wavelength (λ avg ) and average speed of sound (v avg ). 9. Repeat Steps 1 7 using the second tuning fork. Wave Motion 3
Part I Report Sheet Data Analysis Length of string (L) m Frequency (f) = 120 Hz g = 9.8 m/s 2 µ = 3.614 x 10-4 kg/m Method I Method II # loops (n) λ (m) v from f & λ Mass (kg) v from T % difference Calculations: Part II First tuning fork: f = Hz L 1 (m) L 2 (m) λ (m) v (m/s) % error Calculations: λ avg = m v avg = m/s Second tuning fork: f = Hz L 1 (m) L 2 (m) λ (m) v (m/s) % error Calculations: λ avg = m v avg = m/s Wave Motion 4
Post-lab questions: 1. How did the mass on the end of the string effect the number of loops you observed? 2. How did the velocity of the wave on the string change with number of loops? 3. How did the wavelengths change for the tuning forks as the frequencies increased? 4. What would you have to do in order to get the next resonant sound out of the pipe? 5. How did your averages for the speed of sound in Part II for each tuning fork compare with the actual speed of sound (344 m/s) for our room temperature? Comment on the accuracy and precision of your results. 6. What are some of the possible sources of error in both parts of this experiment? Wave Motion 5
Summary/Conclusions: Wave Motion 6