White Noise, Harmonics, and Speed of Sound — Study Notes (Part 1)

Setup and Equipment

Experiment uses Kunst's tube to measure speed of sound in air and various gases (nitrogen, carbon dioxide, helium); potential additions include argon and possibly other gases.
The tube is about 1.54 meters long.
Speaker is on one side of the tube; microphone on the opposite side and must be securely plugged in. If the microphone moves, results become unsatisfactory.
Gas connections are available on one side of the setup; there is a switch and a tube that leads to the gas supply. Gases can be introduced into the tube for measurements later in the experiment.

First objective: measure the speed of sound in air.
Lab apparatus: Kunst tube with length ~1.54 m. Measurements are controlled from a computer using LabVIEW programs.
Available LabVIEW files:
- tune file: generates a monochromatic frequency to drive the speaker.
- white noise file: generates broad-spectrum noise.
- ok fit peak file: records the microphone signal, converts it from the time domain to the frequency domain, identifies peaks corresponding to harmonics, and plots harmonic frequency peaks; these are used to extract the base frequency.
Data handling:
- Results are stored in a folder labeled Data1 (referred to as data one folder in the transcript).
- The trend line (in Excel) is used to extract the slope, which corresponds to the base frequency.
- From the base frequency and the corresponding wavelength (derived from the lowest frequency mode in the tube), the speed of sound is calculated.
Core equation:
- The speed of sound is the product of frequency and wavelength: v = f \lambda
Practical note: use the measured frequency and the associated wavelength to compute the speed of sound in air.

There are two main programs used to generate sound for the experiment:
- Tune program: produces a monochromatic tone; in the example, about 1.53 kHz (1,530 Hz).
- White noise program: produces a broad-spectrum sound (white noise).
Demonstration of tune:
- When you start the tune program (the arrow control), it outputs about 1.53 kHz and you should be able to hear a tone.
- The display shows a high-resolution view where the tone appears as a nearly pure sine wave (monochromatic sound).
- The frequency can be adjusted; for example, changing to 1.4 kHz results in a slightly different tone.
Demonstration of white noise:
- White noise is softer and contains a combination of several wavelengths; you cannot distinguish a single frequency within the audible range.
Program structure:
- White noise program: simple, creates a signal that contains multiple frequencies across a broad spectrum; outputs to the speaker.
- Tune program: similar structure but outputs a single, specific frequency.
Visual and signal concepts:
- The programs simulate sound generation: a broad-spectrum signal (white noise) versus a single-frequency signal (monochromatic).

Time-domain observation (monochromatic case):
- When running a single-frequency tone (e.g., 1.4 kHz), the time-domain signal shows a regular, periodic waveform.
- The example window shows a time scale of 0.01 seconds (i.e., 10 ms).
- To estimate frequency from the time-domain signal, count the number of pulses (cycles) in the window; for 1 kHz, one cycle is ~0.001 s (1 ms).
- In the transcript: a 1.4 kHz tone corresponds to a period of about 0.714 ms, which fits within the observed time window.
- There are small disturbances when switching between signal packages, but the tone remains largely sinusoidal.
Frequency-domain observation (Fourier analysis):
- A Fourier transform converts the time-domain signal into the frequency-domain signal, showing a peak at the tone frequency for a monochromatic signal.
- If you change the tone frequency (e.g., from 1.4 kHz to 1.6 kHz), the frequency-domain peak shifts accordingly.
- The tune/monochromatic program is used to validate that the time-to-frequency conversion in the analysis is correct.
Practical observation: by adjusting the tone frequency in the tune program, you can see the corresponding shifts in the frequency-domain spectrum and verify the analysis pipeline.

When switching back from monochromatic tone to white noise:
- The signal becomes broadband; multiple frequencies are present simultaneously.
- The audible result is a fuller, less tonal sound characteristic of noise.
Summary of differences:
- Monochromatic (tune): single, identifiable frequency; clean sine wave in time domain; sharp peak in frequency domain.
- White noise: broad range of frequencies; time-domain signal appears less periodic; frequency-domain analysis shows a spread of energy across many frequencies.

Beyond air, measurements can be performed with several other gases connected to the Kunst tube via the gas line and switch.
Expected gases mentioned: nitrogen (N2), carbon dioxide (CO2), helium (He); possible additions include argon (Ar) and perhaps other gases.
Practical implications:
- Gas choice affects the speed of sound due to gas properties (density, temperature, and molecular composition).
- Ensuring a proper gas connection and secure valve/switch positions is essential for consistent data.

Data flow:
- Generate signal with LabVIEW (tune or white noise).
- Record microphone signal and convert to frequency spectrum using the ok fit peak tool.
- Identify harmonic peaks and determine the base frequency by plotting peaks and applying a trend line in Excel.
- Use the fundamental frequency f and the corresponding wavelength λ to compute the speed of sound via v = f \lambda.
Key relationships:
- Harmonics and base frequency: peaks occur at multiples of the base frequency; the slope from the Excel trend line corresponds to the base frequency.
- Time-domain to frequency-domain transformation: Fourier transform relates the time signal x(t) to the frequency spectrum X(f). In integral form, the Fourier transform is X(f) = \int_{-\infty}^{\infty} x(t) e^{-i 2 \pi f t} dt.
Practical workflow notes:
- Data are collected in a folder (Data1) and opened later for analysis.
- The speed of sound is computed using the measured base frequency and the appropriate wavelength derived from the tube geometry for the lowest fundamental mode.
- Always verify the time-to-frequency conversion with a known frequency (monochromatic tune) before relying on the white-noise data for more complex analyses.

Speed of sound in a medium: v = f \lambda
Fourier transform (time to frequency): X(f) = \int_{-\infty}^{\infty} x(t) e^{-i 2 \pi f t} dt
Relationship notes:
- Frequencies are measured in Hz (s^{-1}); wavelengths in meters (m); speed in meters per second (m/s).
Time-domain frequency estimation tip:
- For a monochromatic tone at frequency f, the period is T = \frac{1}{f} and a full cycle occupies approximately T seconds; in a window of length 0.01\text{ s}, you can count roughly how many cycles occur to approximate the frequency.