EXAMPLES OF VOWEL ANALYSIS. 139 



55.4, 29.0, 0.1, 4.8, 15.2, 28.6, 29.0, 15.9; those for which they differ are 

 0.6, 21.0, 21.3, 10.5, 0.2, 14.7, 36.8, 48.5, 37.2, 2.8, 2.7, 24.8, 31.8, 25.4, 

 12.2, 8.5, 14.4, 1.8, 3.5. Taking the former as plus and the latter as minus, 

 we obtain + 235.8. Divided by half the number of ordinates used, namely, 

 by 18, this gives + 13.1, as the value for a^. Using the pattern for 6, 

 in the same way, we obtain + 6.4 . T he amplitude of the first partial, 

 or fundamental, is therefore Ci = v'a^ + 6^ = 14.6. 



The phase of a component may be calculated in two ways. By 



equation (11) on page 85 we have directly tan q = — -^ from which we 



obtain q by means of a table of tangents. A method that is often more 

 convenient is the following one. The curve for a component may be 

 represented as the sum of a cosine and a sine curve, or 



y = c.sin ( — t — q) = a. cos — t + b.sm — t. 



For i=0, 



y=c.sm (— q) = a. 



This gives 



a 



Since tan a; = tan (180° + a;) and sin z = sin (180° — x), there will 

 always be two values obtained for q by the above equations. The required 

 one can be found by considering that 



c.sin (—q) = — c.sin q = a. 



Since c is always positive, sin q must have the sign opposite to that 

 of a. We thus have the following rule: When a is positive, q must lie 

 between 180° and 360°; when a is negative 5 must he between 0° and 180°. 



From q we obtain r = -^T and r = -^;, (p. 84), where T is the" period of 



2?! 271 



the vibration and X its length. 



For the first component we have Oi = + 13.1, 61 = 4- 6.4, whence 

 tan qi = — 2.1, qi = 295°. Since the wave-length for the first partial 



is 360mm., r = ^ = ?||360 = 295mm. 

 Zt: odO 



In the adjacent table of results the first column gives the serial num- 

 ber of each partial. Its wave-length is obtained from the wave-length 

 of the fundamental 360mm. by dividing by 1, 2, 3, etc. The coefficients 

 a and b give the amplitudes of the cosine and the sine series respectively 



