By Alvin Halpern, Erich Erlbach
AKA: Schaum's define of Preparatory Physics II : electrical energy and Magnetism, Optics, smooth Physics
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Additional resources for Beginning Physics II: Waves, Electromagnetism, Optics and Modern Physics
Figure 2-1(a) and (b) shows graphs of the exponential function y \ 10x. The Logarithmic Function As x increases continuously from [O to O, y is always positive and increases continuously from 0 to O, passing through y \ 1 when x \ 0. Thus, for any positive y there is a unique number x for which CHAP. 5 x (b) (a) Fig. 2-1 y \ 10x holds. The logarithmic function ““ to the base 10 ÏÏ is then deÐned as : log y \ x. 6) 10 For example : log (1) \ log (100) \ 0 ; log (10) \ log (101) \ 1 ; log (100) \ log (102) \ 2 ; 10 10 10 10 10 10 log (1000) \ log (103) \ 3.
001 in the case of Fig. 2-3. Equal upward spacings correspond to a multiplicative factor (10 in our case) while equal downward spacings correspond to decreases by the same factor. Natural Exponential and Logarithm There is a number with special mathematical properties, called the naperian base, e, that is particularly useful for exponentials. The number e, like )2 and n, is a real number that cannot be expressed as a fraction. 7183. The exponential function using e is : y \ ex, sometimes called the ““ natural ÏÏ exponential function, and is often written as : y \ exp (x).
1-16(a) to (c). The wave envelope is shown just below the pipe, as a transverse representation of the to and fro motion of the molecules from their equilibrium positions at various points along the pipe. Since the distance between successive anti-nodes is always a half wavelength, we see that a whole number of half-wavelengths Ðt into the length, L , of the pipe, or the allowed wavelengths are : j \ 2L /n n The allowed frequencies are then given by : n \ 1, 2, 3, . . 14a) f \ v /j \ n(v /2n) n \ 1, 2, 3, .
Beginning Physics II: Waves, Electromagnetism, Optics and Modern Physics by Alvin Halpern, Erich Erlbach