Which of the following harmonics and amplitude levels are combined to compose a square wave?
A Even harmonics of the fundamental frequency and negative amplitudes that increase proportionally with the harmonic.
B Odd harmonics of the fundamental frequency with positive amplitudes that decrease proportionally with the harmonic.
C Odd harmonics with exponentially decreasing amplitudes that alternate between negative and positive values.
D Odd and even harmonics with amplitudes that decrease proportionally with the harmonics and alternate between negative and positive values.
The correct answer and explanation is:
The correct answer is D.
Explanation: In acoustics, harmonics are integer multiples of a fundamental frequency, and they play a significant role in shaping the sound wave produced by various instruments. When we refer to a series of odd and even harmonics with amplitudes that decrease proportionally with the harmonics, we are typically describing a specific type of waveform, often seen in the Fourier series representation of complex sounds.
Odd and even harmonics with decreasing amplitudes often occur in waveforms like square waves and triangular waves. The relationship between the harmonics and their amplitudes follows a predictable pattern: in the case of a square wave, the odd harmonics are present, and their amplitudes decrease as the inverse of the harmonic number. For a triangular wave, both odd and even harmonics are present, with amplitudes decreasing as the inverse square of the harmonic number.
When amplitudes alternate between negative and positive values, this is indicative of a wave with a symmetry that changes direction with each harmonic. This is often seen in waves with alternating positive and negative components, which might occur due to phase shifts or the particular nature of the waveform.
In many real-world examples, the harmonics will not only alternate between positive and negative values but will also gradually diminish in amplitude as the harmonic number increases. This effect is often used in synthesizing sounds to replicate musical instruments or voices, with the goal of creating rich, complex timbres that include both harmonic and anharmonic components.
To summarize, the key characteristics of the described wave are alternating odd and even harmonics, with the amplitudes decreasing proportionally and alternating between positive and negative values. This behavior is typical of complex waveforms like square or triangular waves in signal processing.