Print

Frequency Analysis of Sound Waves

Theory

Sound waves can be analyzed in terms of their amplitude and frequency. The loudness of a sound corresponds to the amplitude of the wave, and is measured in decibels. The frequency of a sound wave affects the pitch of the sound we hear. Most musical sounds are composed of a superposition of many frequencies called partial tones, or simply partials. The lowest frequency for a given sound is called the fundamental frequency. For a vibrating object like a tuning fork, this is also the natural resonant frequency of the sound source. Partial tones that are whole multiples of the fundamental frequency are called harmonics. A tone that has twice the frequency of the fundamental is called the second harmonic, which is one octave higher than the fundamental. For example, middle C in the equal-tempered chromatic scale has a pitch of 262 Hz, and the same note one octave higher has a frequency of 524 Hz. The combination of multiple frequencies is what gives a sound its characteristic quality. In this lab, you will analyze the sounds from several different sources and examine the qualities that make each sound unique.
Figure 1

Figure 1

      • Note Letter
        name
        Frequency
        (Hz)
        Frequency
        ratio
        Interval
        do C 264
        9
        8

        Whole
        re D 297
        10
        9

        Whole
        mi E 330
        16
        15

        Half
        fa F 352
        9
        8

        Whole
        sol G 396
        10
        9

        Whole
        la A 440
        9
        8

        Whole
        ti B 495
        16
        15

        Half
        do C' 528
        Table 1: Diatonic C Major scale


      • Note Frequency
        (Hz)
        Frequency
        ratio
        Interval
        C 262
        122

        Half
        C♯ or D♭ 277
        122

        Half
        D 294
        122

        Half
        D♯ or E♭ 311
        122

        Half
        E 330
        122

        Half
        F 349
        122

        Half
        F♯ or G♭ 370
        122

        Half
        G 392
        122

        Half
        G♯ or A♭ 415
        122

        Half
        A 440
        122

        Half
        A♯ or B♭ 466
        122

        Half
        B 494
        122

        Half
        C' 524
        122
        Table 2: Equal-tempered chromatic scale

Resonance

When two waves of the same wavelength encounter each other by traveling in opposite directions in the same medium, they combine under certain conditions to yield standing waves. Standing waves are patterns whose points of maximum and minimum (zero) amplitude are fixed so that, apart from a rapid vibration of the medium, the pattern appears to be standing still. If a string is fixed at both ends, each wave on the string will be reflected every time it reaches either end of the string. In general, the multiply-reflected waves will not all be in phase, and the amplitude of the wave pattern will be small. However, at certain frequencies of oscillation, the reflected waves are in phase with the incoming waves, resulting in a standing wave with a large amplitude. These frequencies are called resonant frequencies.

Standing Waves on a String

As stated earlier, if a string is fixed at both ends, then each end must be a node (a point of no displacement). The simplest standing wave pattern that occurs is shown below (called the fundamental mode of oscillation), along with two other modes. The wavelength of the fundamental mode is twice the string length.
Figure 2

Figure 2

Standing Waves in a Half-Open Tube

The standing waves produced by sound waves in air in a half-open tube are longitudinal waves with a node at the interface between media where reflections take place (between water and air in this experiment) and an antinode at the open end of the tube. Patterns of the fundamental (first harmonic) and third harmonic modes of oscillation are shown in the figure below. (Only odd harmonics appear in a half-open tube due to the conditions that there must be a node at the closed end and an antinode at the open end.) From the figure, we see that in the length L from the open end of the tube to the reflective surface, there are an odd number of quarter wavelengths. This can be expressed as:
( 1 )
L = (n/4)λ, where n = 1, 3, 5 ...
 
Figure 3

Figure 3

Procedure

Part A. Waveform Analysis with the computer interface

Throughout this lab, you will use a microphone and Vernier LabPro interface connected to your CCI laptop to collect sound data, and the Vernier LoggerPro software to analyze the sound waves with frequencies that range from 20 Hz to 16,000 Hz (approximately the range that humans can hear).

Part 1. Tuning Fork

Part 2. Mono chord

Part 3. Human voice

Part 4. Theme and Variations

Part B. Standing Waves in an Open Tube

In this investigation, a tuning fork will be used to produce a standing wave in a half-open tube that can be varied in length. The resonance tube will consist of a tall vertical cylinder whose effective length can be changed by changing the depth of water in it. When the vibrating fork is held over a tube of the proper length, the incident waves and those reflected at the opposite end will establish a standing wave. When this situation occurs, an observer nearby will detect a marked amplification of the sound. Calculate the wavelength using equation 1
L = (n/4)λ, where n = 1, 3, 5 ...
 
. From your calculated wavelength of the wave produced, and the known frequency of the fork, compute the velocity of the sound wave using v = fλ. Compare with the value predicted by the equation for the velocity of sound waves in air:
( 2 )
v = (1.4RT/M)1/2
 
where R is the universal gas constant (8.31 J/mol · K), M is the approximate molecular weight of air (0.029 kg/mol for dry air, ~0.0287 kg/mol for humid air) and T is the absolute temperature of the air in Kelvin. Record the room temperature.

Discussion

What differences in waveforms did you observe for the monochord and tuning forks? For the human voice? The complexity of the waveform for the voice reveals why it is so difficult to reproduce. How did your empirical value for the speed of sound compare with the predicted value? What other phenomena did you observe?