Enjoying this Course?In the previous paragraph we saw how atmospheric pressure in proximity of a loudspeaker can be plotted on a graph as a waveform. Waveforms can actually be very complicated, but fortunately every single one of them can be considered as an extension of a very simple waveform: the sinusoid, which can be expressed by the following generic formula:
Equation 1.1. Sinusoid equation
The following figure represents a sinusoid graph:
and the following is the sound coming from a sinusoid with a 1 KHz frequency:
Sinusoid [f=1 KHz, φ=0o] [Track 1]
![Sound theory - Sinusoid [f=1 KHz, φ=0o] [Track 1]](/images/en/figures/layout/icona_suono.jpg "Sound theory - Sinusoid [f=1 KHz, φ=0o] [Track 1]")
Sinusoids have a set of properties:
Frequency (f)
Period (T)
Wavelength (λ)
Amplitude (A)
Phase (φ)
Speed (v)
Frequency is literally the number of cycles made by a wave in one second. A cycle is composed of a positive half wave and a negative half wave. It is measured in Hz (1/sec). A 1Hz frequency wave completes one cycle every second. The following figure shows a 5 Hz frequency sinusoid:
5 Hz frequency Sinusoid
Wavelength is the distance between two corresponding points (for example two successive maxima) along the waveform. Its value can be calculated by the following formula:
Equation 1.3. Wavelength of a sinusoid
Where:
c = speed of sound in the medium in question (through air sound propagates at 344 m/sec).
To begin to get an idea of the quantities we're dealing with let's consider a wave with a frequency of 1Hz travelling through air. Using the previous formula we'd find that:
Equation 1.4. Calculation of the wavelenght
In other words, at every cycle the wave spreads out for 344 m, two football stadiums! (As we will see, the human ear begins to perceive sounds when frequencies are greater than 20-30 Hz, which means wavelengths of 15-18 metres).
The following figure highlights the wavelength of a sinusoid:
Wavelength of a sinusoid
Amplitude is the unit that measures the distance between the equilibrium point and the maximum point of the waveform. Greater amplitudes correspond to higher volumes. Two different kinds of amplitude measurements exist. The first is an absolute quantity, and is called peak amplitude. This is effectively the measurement of the point at which the wave reaches its greatest amplitude. The second measures amplitude as it is perceived by the human ear and it is called effective amplitude (RMS, Root Mean Square), and has the following formula:
Equation 1.5. Effective amplitude
The following figure shows the amplitude of a sinusoid:
Amplitude of a sinusoid
This unit always represents a relation between two waveforms. To understand this concept we need to explain how a sinusoidal waveform is created. To do so we'll use the following figure:
Phase graphs
Let's imagine that point A moves anti clockwise along the circumference starting from the point at 0 degrees. If α is the angle, the projected segments of point A on the X and Y-axis' shall respectively be:
Thus what you see on the graph (a) is nothing but the length of the projection of point A on the Y-axis, as the angle changes. Now imagine that point A rotates clockwise. Its projection on the Y-axis shall be negative at the beginning and shall have the rate shown in figure (b). Now we can further interpret the frequency by saying that it is the number of times that point A makes a complete cycle in one second. The maximum amplitude shall always be at 90° regardless of the frequency. More generally speaking, we can say that phase does not depend on frequency. The equation that relates phase and time is:
Equation 1.6. Relationship between phase and time
Example 1.1. Relationship between delay and phase
This equation is quite useful. Let's take a look at an example and calculate the necessary delay for two 100 Hz frequency sinusoids to arrive out of phase by 90o
Let's now put these values into the equation and work out as follows:
Equation 1.7. Calculation of delay between two sinusoids
Previously we mentioned that the speed of sound through air is approximately 344 m/sec. The more dense the medium is, the quicker sound diffuses. Later we'll take a closer look at this phenomenon and how it is at the heart of refraction [Refraction ] . The diffusion speed of a sound depends on the medium's density. Every medium has its own speed of sound stated at the constant temperature of 23/24 oC. This serves as a reference value seeing that when temperature varies, the characteristics and thus the speed of sound within the medium varies. When a medium warms up, kinetic energy is transferred to its particles. When the latter make contact with the wave, the medium's particles respond more quickly to the stimulus thus transmitting the received sound energy more promptly. In other words: the medium's speed of sound increases. On average we have an increase (or decrease) of 0.6 m/sec every time the medium's temperature increases (or decreases) by one degree centigrade.
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Sinusoid graph