# Determining the Speed of Sound in Air in a Resonance Tube ## Objective of the Experiment - The experiment aims to measure the speed of sound in air by creating stationary (standing) waves inside a tube partially filled with water - A tuning fork of known frequency is used as the sound source, and the tube acts as a resonance chamber **Variables:** - **Independent variable** = Height of air column in the tube (changed by adjusting water level) - **Dependent variable** = Length of air column $L$ at which resonance (loudest sound) occurs - **Control variables:** - Water temperature (affects speed of sound) - Tuning fork frequency (must remain constant throughout)

Resonance occurs when the natural frequency of the air column matches the frequency of the tuning fork, producing maximum amplitude vibrations

## Required Apparatus | Equipment | Purpose | |-----------|---------| | Tuning fork with known frequency | Produces sound waves of constant frequency | | Small rubber hammer | Strikes the tuning fork to make it vibrate | | Tube open at both ends | Creates the air column where standing waves form | | Clamp stand and boss | Holds the tube vertically in position | | Water | Adjusts the effective length of the air column | | Rubber band or marker pen | Marks the water level at resonance points | | Cylindrical beaker | Contains the water and supports the tube | ## Experimental Method **Setup:** - Secure the tube vertically using the clamp stand - Fill the beaker approximately halfway with water - Lower the tube into the water so about one-quarter of the tube is submerged - The water surface acts as a closed end; the top of the tube is an open end **Procedure:** 1. Strike the tuning fork gently with the hammer to make it vibrate 2. Hold the vibrating fork horizontally just above the open end of the tube (approximately 1-2 cm above) 3. Gradually raise the tube out of the water by loosening the clamp, which increases the air column length 4. Listen carefully for an increase in sound intensity - when resonance occurs, the note becomes noticeably louder 5. When maximum loudness is heard, mark the water level with a rubber band or pen - record this position as $L_1$ 6. Continue raising the tube slowly until the second resonance point is detected - mark and record this as $L_2$ 7. If the tube is long enough, identify and record the third resonance point $L_3$ ## Physics of Resonance in the Tube **Standing Wave Formation:** - The tuning fork generates progressive sound waves that travel down the tube - These waves reflect off the water surface (closed end) - The incident and reflected waves superpose to create a stationary wave pattern - At certain air column lengths, constructive interference produces resonance **Conditions for Resonance:** - At the water surface (closed end): a **node** forms (point of zero displacement) - At the open end: an **antinode** forms (point of maximum displacement) - The air column length must equal an odd multiple of quarter wavelengths for resonance **Resonance positions:** - First resonance (fundamental): $L_1 = \frac{\lambda}{4}$ - Second resonance: $L_2 = \frac{3\lambda}{4}$ - Third resonance: $L_3 = \frac{5\lambda}{4}$ Where $\lambda$ is the wavelength of the sound wave.

The sound becomes dramatically louder at resonance because the standing wave stores energy rather than transferring it away from the tube

**What you hear as the tube is raised:** As you increase the air column length, the sound intensity varies continuously, reaching maximum values at specific positions. The graph below shows how the relative amplitude (loudness) changes with air column length:

The peaks in this graph correspond to the resonance positions where the air column length equals odd multiples of $\frac{\lambda}{4}$. ## Calculating the Wavelength **Between consecutive resonance points:** $$L_2 - L_1 = \frac{3\lambda}{4} - \frac{\lambda}{4} = \frac{2\lambda}{4} = \frac{\lambda}{2}$$ **Therefore, the wavelength is:** $$\lambda = 2(L_2 - L_1)$$ **For improved accuracy:** - Measure the distance between $L_2$ and $L_3$ to get another value: $\lambda = 2(L_3 - L_2)$ - Calculate a mean wavelength from multiple measurements - Using separation between non-adjacent resonance points reduces percentage uncertainty ## Determining the Speed of Sound **The wave equation relates speed, frequency and wavelength:** $$v = f\lambda$$ Where: - $v$ = speed of sound in air (m s$^{-1}$) - $f$ = frequency of the tuning fork (Hz) - $\lambda$ = wavelength calculated from resonance positions (m) **Steps:** 1. Use the wavelength calculated from: $\lambda = 2(L_2 - L_1)$ 2. The frequency $f$ is known from the tuning fork specification 3. Substitute both values into $v = f\lambda$ to find the speed of sound The accepted value for the speed of sound in air at room temperature is approximately 340 m s$^{-1}$.

Temperature significantly affects sound speed - ensure water is at room temperature for results comparable to standard values

## Sources of Error and Improvements **Systematic Errors:** - The tuning fork must be held at the same height above the tube for each measurement - Strike the tuning fork with consistent force each time - varying the amplitude affects the clarity of resonance - Ensure the tuning fork is held horizontally and doesn't touch the tube walls, which would dampen vibrations **Random Errors:** - Use a thin, precise line when marking the water level to improve measurement accuracy - Adjust the tube height slowly and smoothly - moving too quickly may cause you to overshoot the resonance point - The point of "maximum loudness" is subjective and varies between observers - Repeat the experiment multiple times (at least 3 trials) and calculate a mean to reduce random uncertainty - A resonance tube with a built-in scale eliminates errors from transferring length measurements **Improvements:** - Use vernier callipers or a metre rule with millimetre divisions for precise length measurements - Conduct the experiment in a quiet environment to hear resonance more clearly - Measuring across more resonance points (e.g., $L_1$ to $L_3$) reduces percentage uncertainty - Calculate the speed from different pairs of resonance positions and find the mean ## Safety Considerations - Never allow the vibrating tuning fork to strike the tube - the impact could crack or shatter the glass - Use water at room temperature only - avoid hot water that could cause burns or damage equipment - Keep all electrical equipment away from the water to prevent short circuits or electric shock - Place the beaker on a stable, level surface to prevent spills - If using a glass tube, inspect it for cracks before use and handle carefully ## Expected Results For a typical experiment with: - Tuning fork frequency: $f = 512$ Hz - First resonance at: $L_1 = 16.5$ cm - Second resonance at: $L_2 = 49.5$ cm - Third resonance at: $L_3 = 82.5$ cm **Visualizing the resonance spacing:** The chart below shows the three resonance positions, demonstrating the regular spacing of $\frac{\lambda}{2}$ between consecutive resonances:

**Calculation:** $$\lambda = 2(L_2 - L_1) = 2(49.5 - 16.5) = 2 \times 33.0 = 66.0 \text{ cm} = 0.660 \text{ m}$$ $$v = f\lambda = 512 \times 0.660 = 338 \text{ m s}^{-1}$$ This is close to the accepted value of approximately 340 m s$^{-1}$ for sound in air at 20°C.

The wavelength of audible sound in air ranges from about 17 mm to 17 m, making resonance tubes a practical method for wavelengths in the tens of centimetres