Wave Velocity Dynamics: String Tension and Sound Propagation

Standing Waves & Speed of Sound: Experimental Analysis

Purpose of the Experiment

• To investigate the relationship between the frequency of vibration and tension in waves on a vibrating string.
• To measure the speed of sound experimentally.

Part 1: Standing Waves on a Vibrating String

Procedure for Part 1

• Vary the tension on the string by hanging different weights at its ends. Use 150g, 200g, and 250g.
• Once a standing wave is achieved, record the following information in a table: mass, weight (tension) on the string, frequency, wavelength, wave speed, and the square root of the tension.
• Relevant formulas: λ = 2L / n, f = 1 / T (where T is the period), v = λf, and v = √(T / μ).
• Create a graph of wave speed (v) versus the square root of the tension (√T) using a computer.
• Apply a linear fit to the graph and analyze what the slope of this line represents.

Data and Analysis (Part 1)

Experimental Data Table

Interpretation of Results

• The linear fit of the graph (wave speed vs. square root of tension) yielded a slope (m) of 23.59. Theoretically, this slope represents the inverse of the square root of the string's linear mass density (1/√μ).
• Conclusion on Wave Speed: As the tension in the string increases, the wave speed also increases. This relationship is clearly demonstrated by the formula v = √(T / μ).
• The determined value for the linear mass density (μ) of the string is 0.0045 kg/m.

Conclusion for Part 1

• In a string where a standing wave is produced by applying a frequency, increasing the tension leads to a noticeable increase in wave speed. This dependency is clearly shown in the wave speed formula.
• The slope of the v vs √T graph represents the inverse of the square root of the linear mass density of the string (m = 1 / √μ). Therefore, the linear mass density (μ) can be theoretically calculated as μ = 1 / m².
• Wave velocity of the string increases with increasing tension.
• The wavelength increases with increasing tension while maintaining a constant frequency, which implies that the wave velocity of the string increases (v = λf).

Part 2: Measuring the Speed of Sound

Procedure for Part 2

• Choose three different frequencies of sound waves from the wave generator to measure the speed of sound.
• For each frequency, locate two consecutive resonance points by moving the rod carrying the microphone inside the tube.
• Measure the distance between these two consecutive resonance points.
• Determine the wavelength (λ) by multiplying the measured distance by 2.
• Calculate the value of the speed of sound (v) for each set frequency using the formula: v = λf.

Data and Analysis (Part 2)

Experimental Data Table

Average Speed of Sound

• The average experimental speed of sound calculated is 357.4 m/s.

Conclusion for Part 2

• The average speed of sound determined in the laboratory (357.4 m/s) is a reasonably good value when compared to the accepted literature value (approximately 343 m/s at 20°C).
• While the experimental value is close, small variations in readings and measurement precision may account for the observed difference. This experiment provides a practical understanding of how these variables are calculated and measured.