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Aviation History
1917
1917 - 1338.PDF
lapse of a period depending upon the value of h, and that for other points along the path the time-lag, or period elapsing before the value of R proper to each point is heard, increases the further the point is from C. Accordingly, the ordinates of the dotted curves require displacement to the right, the dis- placement of each being proportional to the time lag. As a result we get the full-line curves which are slightly steeper than the dotted curves above the line xx, and not quite so steep below. With h =o, the sudden drop would be heard imme- diately it occurs. With h =100, the unit value of R is heard •JV second later, and with h =i ,000, i« second later, the source being then 40 ft. and 400 ft. past the pedal point. The full curves give a graphic representation of the manner in which the note heard falls, and show clearly that the further the pedal point is away from the observer, the more gradual is the fall of the note. It is of interest to notice, also, that the full-line curves intersect at a common point on the zero time ordinate. This indicates that the value of R for the impulse which reaches the observer at the instant the source is actually at the pedal point, an impulse really emitted some time pre- viously, is always the same, no matter what the distance of the pedal point. Referring to Fig. 1, let £ be a point such that the impulse emitted there reaches the observer at the instant the source reaches C. Then we have = -y or S =y. Substituting this value for £ in the expression"R = r V V -vs- we get R — -y^ -2- which is independent of h. With v -=440 we get R =1.19, the value of R for all the full-line curves where these intersect one another and BD. Since V"--v2 ~ \V+v)\V-v) we see that this value of R is the product of the maximum and C C O C minimum values of R, i.e., _x_-=—=1.19- We can now examine the variations in R when the source' moves uniformly round a circular path, say in anti-clockwise direction. Fig, 3, the observer being at 0 exterior to, but in the same plane as, the circle. At the point P, <f> =90° so that R =1. As the source moves round the circle, <p decreases until the point Q, where the tangent from o touches the circle, is reached, when <j> =0 and R has its lowest possible value V ... y . As the motion continues <f> increases, agam becoming 90* at T and then obtuse, as at V. The increase continues until the point of contact Y of the tangent OY is reached, Vwhen # = 180° and R has its maximum value y _^, after which <p decreases. It will be noticed that at the points Q, Y re- spectively, the source is moving directly away from, and directly towards, the observer. We thus see that R decreases along YPQ and increases along QTY and also (since c/> is acute along PQT and obtuse along TYP) that R is less than unity during the first half of the movement and more than unity during the second half. The value of, R at any point of the path can be calculated from the expression R - y cos This has been done and the results plotted, giving the curves shown in Fig. 4. In calculating these curves the following values have been used, viz.:—V =1,100 ft. per sec, v =440 ft. per sec, radius PW = 1,000 ft. and OP =PW. The period in which the source moves once round the circle has been taken as the unit of time in plotting. With the above di- mensions the angle QWY is 1200, so that the period during which R falls is half the period during which R rises. The dotted curve, Fig. 4, shows R plotted against time of emission. The full-line curve (a) shows R plotted against time of recep- tion, the curve being obtained by displacing each ordinate of the dotted curve towards the right an amount corresponding to the time lag due to the lengths of the corresponding paths, such as PO, QO, between source and observer. As the dis- tance of the observer from the centre of the circle is increased, the points Q, Y move further apart towards the extremities of the diameter of the circle at right angles to WO. The periods of rise and fall in the value of R thus become more nearly equal. Also, the time lag becomes greater but more uniform, and therefore produces greater displacement but less distortion in the curve. On the other hand, as the distance of the observer decreases, the points Q, Y come nearer to the point P, so that the period of fall becomes shorter and of rise longer. When the observer is on the circumference of the circle, the points 0, P, Q, Y, coincide. The fall in the value of R therefore takes place suddenly, whilst there is a con- DECEMBER 20, 1917, tinuous rise as the source travels round the circle from the observer and back again. <f» increases continuously from zero to 180° as the source moves round the circle, and then, as the source passes through P, falls suddenly from 180° to zero. If we put 0 for the angle such as SWP swept out from WP by the line SW, joining the source and the centre of the circle, as the source is always moving at right angles to SW and the line PS joining the observer to the source is always at right Q angles to the bisector of the angle 9, it follows that $ — — for all positions of the source. The distance of the observer from the source is the diameter of the circle multiplied by0 sin , and this distance divided by the velocity of sound gives the corresponding time-lag. The curve (b) gives the varia- tions in R as heard by an observer at P, the sudden fall being shown by the vertical line YZ, Fig. 4. When the observer is inside the circle the maximum and minimum values of <p are found at the points where the circle is intersected by a line through the observer at right angles to the line joining the ob- server to the centre of the circle. With the observer at 0', the middle point of PW, these points coincide with the points Y, Q at which the maximum and minimum values occur with the observer at O. The maximum and minimum values of <p are always supplementary, and the minimum value is always equal to the corresponding value of 0. Thus with the dimensions already given with respect to Fig. 3 6 —QWP =6o° so that <j> varies between 6oc and 120°, and the maximum and minimum values of R are respectively 1.25 and51,100 —220 1,100 -f 220 = .83. The curve (c) shows the variations of if plotted against time of reception, the observer being at 0'. As the observer is placed nearer and nearer to the circumference the maximum and minimum values of <p become more nearly equal to i8oc and zero respectively. On the other hand, as the observer approaches the centre of the circle the maximum and mini- mum values of <p approximate to 900. With the observer at the centre, </> =90° and is constant, that is to say the note emitted is heard with its true frequency. When the variations in if are plotted against both time of emission and reception on the same diagram, We may use the diagram as follows. Assume it is required to find the varying values of R for a note emitted between any two points of the path, of the source, such as those corresponding to the points m, t, Fig. 4. Draw the dotted curve ordinates tnn, tu; through n, u draw the horizontals no, uv, meeting the curve (a) at 0, v ; and draw the ordinates op, vw. Then mp gives the time lag of the initial impulse and tw the time lag of the final impulse. Thus the note emitted during the time mt is heard during the time pw. The ratio of the average frequency of the note heard to the average frequency of the note emitted is therefore — , and this is equal to the average pw value of R for the note under consideration. — is equal to pw the average height of the figure povw, so that by obtaining ml and pw, estimation of the average height is avoided. When ov is short or nearly straight, however, the height of the middle point of ov may be taken as the average height. We have seen that with given values of V and v the value of R depends on <j>. When <p is constant the note heard is of cdnstant pitch, not necessarily the pitch of the note emitted. In general.for </> to be constant when the source describes a path in the plane of the observer, it is of theoretical interest to note that the path is an equiangular or logarithmic spiral. From this point of view the circular path with the observer at the centre is a particular form of the spiral in which <j> = go\ and the straight line path passing through the observer a particular form of the spiral when </> = 0 or 1800. In conclusion, it should be borne in mind when comparing theoretical results with observed phenomena that the untrained ear is liable to confusion between the varying intensity of the note due to proximity or remoteness of the source, and the variation of pitch. Moreover, in the case of an aeroplane as the source, the sound heard is not a simple note but contains a multiplicity of noises due to vibration of various parts of the machine, rotation of propeller, and engine exhaust, and -- these sound components are seldom constant for more than • comparatively short intervals. * -. • ::rp m m m as •>; ••"'« End of the Bristol Strike. . - _ BRISTOL aeroplane workers, to the number of betwees 2,000 and 3,000, came out on strike on December 13th for am increase in wages. As a result of negotiations -with the Ministry of Munitions, the men returned to work on December 17th, pending a further conference with the Ministry. ".'/ 1338
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