Problem 20 - College Physics

Question

\(\bullet\) Voiceprints. In this chapter, we have been concentrating on sinusoidal waves. But most waves in the real world are far more complicated. However, many complicated waves can be created by adding together sine waves of varying amplitude and frequency. When a singer, for example, sings a note, the pitch we hear is the fundamental frequency at which his or her larynx is vibrating. But the larynx also vibrates in other frequencies (the overtones) at the same time. So the sound we hear is a superposition of the fundamental frequency plus all the overtones. This set of all the frequencies (with their respective amplitudes) is called the person's voice print. (a) To see how this works, carefully graph a sine wave of frequency 440 Hz (concert \(A ),\) with time on the horizontal axis and displacement on the vertical axis. Let the amplitude be 1 unit. On the same set of axes, graph the first overtone of 880 \(\mathrm{Hz}\) , but with an amplitude of \(\frac{1}{2}\) unit. (b) Now add the two waves to find their superposition. Notice that the shape is no longer a Isine wave.

Step-by-Step Solution

Answer

Superposing a 440 Hz sine wave and its 880 Hz overtone creates a more complex waveform.

Step 1: Define the Sine Wave Equations

The fundamental sine wave for this problem has a frequency of 440 Hz and an amplitude of 1. This wave can be represented as \( y_1(t) = \sin(2\pi \times 440 \times t) \) where \( t \) is time in seconds. The first overtone has a frequency of 880 Hz and amplitude \( \frac{1}{2} \), which can be written as \( y_2(t) = \frac{1}{2} \sin(2\pi \times 880 \times t) \).

Step 2: Graph the Fundamental Sine Wave

To graph \( y_1(t) \), plot \( \sin(2\pi \times 440 \times t) \) with the horizontal axis representing time in seconds and the vertical axis representing displacement. This will produce a sinusoidal wave oscillating with a frequency of 440 Hz, completing 440 cycles in one second.

Step 3: Graph the First Overtone

Plot \( y_2(t) = \frac{1}{2} \sin(2\pi \times 880 \times t) \) on the same axes used in Step 2. This wave has a frequency of 880 Hz, which means it completes double the number of cycles compared to the fundamental wave within the same time frame. The peak of this wave reaches only 0.5 units, due to its amplitude.

Step 4: Add the Two Waves

To find the superposition of the two waves, calculate the sum \( y(t) = y_1(t) + y_2(t) = \sin(2\pi \times 440 \times t) + \frac{1}{2} \sin(2\pi \times 880 \times t) \). This new wave represents the combined effect of the fundamental wave and the first overtone.

Step 5: Graph the Superposed Wave

Graph the superposed wave \( y(t) = \sin(2\pi \times 440 \times t) + \frac{1}{2} \sin(2\pi \times 880 \times t) \) on the same set of axes. The resulting graph will show a wave that is more complex and not a simple sine wave. This complexity is due to the interaction between the fundamental frequency and its overtone.

Step 6: Conclusion: Observe the Resulting Waveform

The resulting waveform is a combination of the original sinusoidal waves, revealing a pattern different from a pure sine wave. This represents the complexity of real-world waveforms, such as those found in a person's voice print.

Key Concepts

sine waves

Sine waves are the building blocks of most periodic phenomena in both nature and technology. They can be described using a simple mathematical function:

• A sine wave represents smooth repetitive oscillation, like a pendulum swinging back and forth or sound waves produced by a tuning fork.
• The equation for a sine wave is typically written as \( y(t) = A \sin(2\pi f t + \phi) \), where:
  • \( A \) is the amplitude, the height of the wave's peaks.
  • \( f \) is the frequency, shown in cycles per second (Hz).
  • \( t \) represents time.
  • \( \phi \) is the phase shift, which denotes how delayed the wave is compared to a standard sine wave.

Sine waves are pristine in their pure form. When combined, they generate complex waves through superposition, a principle that synthesizes multiple frequencies into one wave. This concept is incredibly relevant in understanding sound generation and processing, including musical tones and acoustic technologies.

fundamental frequency

The fundamental frequency is the lowest frequency of a periodic waveform and is often referred to as the "pitch" of a sound.

• In the context of a musical note, like concert A which vibrates at 440 Hz, this frequency is the basic vibration rate of an object or system and can be thought of as the frequency that provides the primary musical note we hear.
• The wave representing the fundamental frequency can be perceived as the backbone of sound, upon which other additional frequencies (such as overtones) are stacked.

Understanding this frequency is crucial for various applications. For instance, in creating music, the precise control of fundamental frequencies ensures that the right pitch is produced. It serves as the reference point from which musicians tune their instruments and forms the core of a waveform's identity in acoustics.

overtones

Overtones are the other frequencies that occur in conjunction with the fundamental frequency in any vibrating system.

• They are often integral multiples of the fundamental frequency: the first overtone is twice the fundamental, the second is three times, and so on. This results in a harmonic series.
• For example, if a wave has a fundamental frequency of 440 Hz, its first overtone would be 880 Hz. These overtones contribute to the unique sound or timbre of a voice or instrument, giving each sound its characteristic tone and color.
• Different instruments have different overtone patterns, which is why a piano sounds different from a violin even if they're playing the same note.

By adding overtones to the fundamental frequency, we achieve a richer, more complex wave. The combination and intensity of these overtones shape the sound's timbre and are thoroughly studied in fields like music acoustics and sound engineering.

voiceprint

A voiceprint is essentially a unique fingerprint of someone's voice. It's a visual representation of the spectrum of frequencies produced when a person speaks or sings.

• It displays various frequencies (including the fundamental frequency and overtones) present in a sound wave and their respective amplitudes.
• Voiceprints are used in voice recognition systems and to analyze and identify individuals based on their vocal attributes.
• In speech therapy and linguistics, studying voiceprints helps in understanding voice functioning and in diagnosing voice disorders.

Visualizing sound through a voiceprint uncovers the intricacies in vocal characteristics. Each person has a distinct voiceprint due to unique physical attributes of their vocal tract, enabling technologies to authenticate and even mimic voices in advanced applications.