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Fig. 24-A
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Using a procedure called Fourier ( foo ree AYE ) analysis,{ it is found that the above square wave can be represented by the following in nite series: ! 4 1 1 1 sin x sin 3x sin 5x sin 7x 8-A y  3 5 7 where sin x is the fundamental sinusoidal component, the angle x being in radians. The equation shows that the square wave is composed of odd harmonics only 3x; 5x; 7x, and so on). Note that the higher the order of the harmonic, the lower is its amplitude. Also, because of its symmetry and its position relative to the x-axis, the wave has no constant term (no dc component). If, now, you were to take the time to actually calculate a number of values of y, using eq. (8-A),{ then plot the values of y versus x, you would get the result shown in Fig. 25-A, for x 08 to x 3608, where degrees radians)(180/) (section 5.4).
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* The function can include a possible CONSTANT term (often referred to as the dc component). { Named for Joseph Fourier (1768 1830), French mathematician. { Using the rst four terms of eq. (8-A).
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Appendix
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Fig. 25-A
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It s obvious that the sum of just the rst four terms of the series doesn t produce a very good square wave. If, for example, we had taken the sum of the rst eight terms of the series (up to the 15th harmonic) the result would have been considerably improved.
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Note 19.
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LOGARITHMS are EXPONENTS. By de nition, the LOGARITHM of a number is the POWER to which a xed number, called the base number, must be raised to, to equal the number. In theoretical work, the base number is taken to be the irrational number denoted by  (section 6.5). In certain practical work, however, it s more meaningful to use the number ten as the base number.* The OBJECT of this note is to show, in just a general way, why eq. (315) does make sense in a practical way. To do this, let us begin with the previous de nition that If x is any positive number, then log x is the POWER that 10 has to be raised to, to equal x; that is, by de nition, x 10log x Note that x will always be greater than the exponent log x. To see this relationship more clearly, consider rst the table of values to the left of Fig. 26-A, in which the calculator values of log x have been rounded o to two decimal places.
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* The notation ln x denotes the logarithm of x to the base , while log x denotes the logarithm of x to the base 10.
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Appendix
Fig. 26-A
We see that, as x increases in value, log x also increases in value. Note, however, that, as x increases in value, the amount of CHANGE produced in log x depends not only upon the amount of change in x but also upon the particular value of x; for example, from the table we see, for equal unit increases in the value of x, that log x increases by the amount of 0:30 0:18 0:12 . . . 0:05
if x increases from x 1 to x 2 x 2 to x 3 x 3 to x 4 . . .
x 9 to x 10
and so on, showing that the greater the value of x, the slower is the rate of increase in the value of log x with respect to x. This is evident from inspection of Fig. 26-A. Let us now turn to the relationship between ACOUSTIC POWER ( sound power ) and the sensation of LOUDNESS, as registered by the human ear. It should be noted that the ear is capable of responding to an ENORMOUS RANGE of acoustical power; for example, in the case of a full symphony orchestra, the sound power produced during the loudest passages can be 10 million times the sound power produced during the softest passages. The ear can handle such a tremendous range of sound power because, as the power increases, the sensitivity of the ear automatically decreases, so that the RATE OF INCREASE in the sensation of loudness decreases as the power increases, in the same manner that the rate of increase in the value of log x decreases as x increases, as shown in Fig. 26-A. Hence the e ect that sound power has on the ear for two di erent power levels is not proportional to the power ratio itself but, instead, is approximately proportional to the LOGARITHM of the power ratio. It is this fact that led to the de nition of eq. (315), and is responsible for the statement that the ear hears logarithmically. In closing this discussion, we should note that, for prescribed conditions, eq. (315) can be written in terms of a VOLTAGE RATIO. To show this, we rst need to prove that if x is any positive number raised to a power n, then log xn n log x 9-A