In the previous paragraph we saw how the flow of atmospheric pressure in relation to the active loudspeaker can be visualized as a waveform. Waveforms can actually be very complicated, but fortunately every single one of them, and I mean every single one, can be considered as an extension of a very simple wave form: the sinusoid, which at its simplest can be expressed by the following formula:
Equation 1.1. Sinusoid equation
The following figure represents a sinusoid graph:
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 semi-wave and a negative semi-wave. It is measured in Hz (1/sec), a single 1Hz frequency wave completes one cycle every second. The following figure shows a 5KHz frequency sinusoid:
5Hz frequency Sinusoid
Defined as the distance between two corresponding points (for example two successive maximums) along the wave form. Its amount can be calculated by the following formula:
Equation 1.3. Wavelength of a sinusoid
Where:
c= speed of sound in the vehicle in question (in air it moves at 344 m/sec).
To begin to get an idea of the dimensions we're dealing with let's consider a wave with a frequency of 1Hz travelling through the air. In the previous formula we'd find that:
Equation 1.4. Calculation of the speed of sound
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 whose frequencies are greater than 20-30 Hz- wavelengths of 15-18 metres.)
The following figure shows the wavelength of a sinusoid:
Wavelength of a sinusoid
Amplitude is the measuring unit of the maximum gap from the position of equilibrium. Greater amplitudes correspond to higher volumes. Two different kinds of amplitude-measurements exist. The first is a measurement of an absolute kind, and is called Peak Amplitude. This is effectively the measurement of the point at which the wave reaches its greatest amplitude. The second is a measurement of amplitude as it is perceived by the human ear. In this case we call it 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 is always a relation between two wave forms. To understand this concept we need to explain how a sinusoidal wave form is created. To do so we'll use the following figure:
Phase graphs
Let's imagine that point A moves anti-clock-wise along the circumference starting from the point at 0 degrees. If ( ) is the angle, the projected segments of point A on the x and y axis' shall respectively be:
Thus what you see on graph (a) is nothing but the length of the projection of point A on the axis of the (y) co-ordinates, when the angle changes. Now imagine that point A rotates clock-wise. Its projection on y shall be negative at the beginning and shall have the flow of 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 upon frequency. The equation that links phase and time is:
Equation 1.6. Relationship between phase and time
Example 1.1. Relationship between delay and phase
To give an example of its usefulness we can calculated the necessary delay for 100Hz frequency sinusoids to arrive out of phase by 90o
Let's now substitute the values in 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 344m/sec. The more dense the vehicle 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 ] . A sound that is diffused through a vehicle has a diffusion speed that depends upon the characteristics of the vehicle itself. Every vehicle has its own speed of sound calculated at the constant temperature of 23/24 Co. This serves as a reference value seeing that when temperature varies, the characteristics and thus the speed of sound within the vehicle varies. When a vehicle is warmed up, kinetic energy is passed on to its particles. When the latter make contact with the front of a wave, the particles of the vehicle respond more quickly to the stimulus and thus transmit the received sound energy more readily. In other words- the vehicle's speed of sound increases. On average we have an increase (decrease) of 0.6 m/sec for every degree Centigrade increased (decreased) in the vehicle's temperature.
Related topics on Wikipedia
Sinusoid graph