Frequencies Of Simple Sound And Complex Sound
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Frequencies Of Simple Sound And Complex Sound

(a) The sound produced by a tuning fork (figure 113.4) is considered a simple sound. In general, the sound wave of a simple sound can be represented by the following simple sinusoid:
y = asin2πft --------------(1)
Where a is the amplitude of the sound wave, f is the frequency and t is time.
What is the amplitude and frequency of the simple sound represented by 10sin(π/2)16t?

(b) Sounds from musical instruments and the human voice are complex sounds so they are not representable by only the simple sinusoid of equation (1). However, intelligible sounds (simple or complex) are periodic even if they are not sinusoids. Consider the following complex sound:
y = 0.07sin480πt + 0.05sin760πt + ... -------------(2)
(i) What are the frequencies of the fundamental, first harmonic and first partial?
(ii) What is the frequency of the second harmonic?
(iii) Why is the frequency of a complex sound always that of the first harmonic?
(iv) What is the difference between natural frequency (resonant frequency) and fundamental frequency?
(v) Can humans hear an infrasonic or an ultrasonic sound?

The strings: S7P4A44 (Motion - Oscillatory).

The math:
Pj Problem of Interest is of type motion (oscillatory).

(a) General sinusoid of simple sound: y = asin2πft
Given sinusoid:10sin(π/2)16t?
Comparing both sinusoids:
2πf = (π/2)16
So, f = 8/2 = 4 cycles per second.
Amplitude = 10.

(b)i Fundamental, first harmonic and first partial mean the same thing. Fourier's theorem says that any periodic function is a sum of simple sine functions representable by equation (1) and that the frequencies of the simple sine functions are all integral multiple of one frequency (the lowest frequency in the series) called the fundamental frequency. The simple sine function with the fundamental frequency is called the fundamental or first harmonic. The second simple sine function is called the second harmonic and so on.
So, 0.07sin480πt contains the fundamental frequency
So, 480 = 2f; f = fundamental frequency = 240 cycles per sec.

(ii) So, the frequency of the second harmonic is 2(240) = 480 cycles per sec.
So, the frequency of the third harmonic is 3(240) = 720 cycles per sec
Frequencies of subsequent harmonics are similarly calculated. The harmonics beyond the first harmonic are also called overtones.

(iii) The cycles of the overtones are contained within the cycle of the fundamental tone.

(iv) All objects have a natural frequency (resonant frequency) at which they vibrate. They create standing waves (waves that do not appear to be moving) when vibrating at their natural frequencies or multiples of their natural frequency. The lowest frequencies in the series of these multiples is the fundamental frequency.

(v) Sound waves with frequencies below 20 hertz are called infrasonic. Sound waves with frequencies above 20,000 hertz are called ultrasonic. Infrasonic and ultrasonic sounds are naturally inaudible to the human ears.
Range of artificially unaided human hearing is 20 hertz - 20,000 hertz.

Blessed are they that have not seen, and yet have believed. John 20:29