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A rigorous method of analysing data of the rocket-grenade experiment
Research notes
Despite severe interference of the type mentioned good records have been obtained (apart from zero drift) even when the receiver was left unmonitored for three weeks. The rotation of the sky could be followed, the smooth maximum corresponding to the euhnination of the Sagittarius region. A representative portion of the recordings is illustrated in Fig. 2. Continuous records were taken between 18 November 1952 and 30 January 1953 giving a total of 1700 hours. Although this work has had to be postponed it was felt that the instrumentation described here might be of assistance to other experimenters. M. LA~TINEUR
Institut D'Astrophysique, Paris Cavendish Laboratory, Cambridge
J. ]). WHITEHEAD I~EFERENCE
LOVELLA. C.B.
1955 Joddrell Bank Radio Astronomy Symposium
A rigorous method of analysing data of the rocket-grenade experiment (Received 27 June 1956)
BETWEEN July 1950 and September 1953, the Signal Corps Engineering Laboratories' (SCEL) rocket-grenade experiment was carried out successfully on twelve occasions with Aerobee rockets at White Sands. The data from these firings have now been analysed, and results of temperatures and wind velocities between 30 and 80 km altitude have recently been published (STRoVD et al., 1956). Any attempt at determining the speed of sound in the open atmosphere has to take account of the possibility of the results being seriously influenced by the motion of the atmosphere. In the early determinations of the speed of sound in air, made by timing a sound wave between two points some miles apart, the effect of wind was mainly eliminated by arranging simultaneous transmissions in both directions. The procedure has been similar in the abnormal sound propagation experiments of C~ARY and other workers, the contributions from wind and temperature being separated out by timing sound waves travelling in various different azimuthal directions. Another way, by which the presence of wind may be detected, is its effect on the direction of arrival of the sound waves at the reception point. This method has been adopted in SCEL's experiments, using an L-shaped array of five microphones spaced 1000 feet apart on the ground roughly vertically beneath the grenade tmrsts. The theoretical problem of separating out the effects of wind and temperature in the rocket-grenade experiment has been treated by SCEL by a method of successive approximation (FERENCEet al., 1956). Firstly, the wind is found using temperatures that are uncorrected for wind, and then the speed of sound (and hence temperature) is found using these wind values, etc. The longest part of the calculation appears to be finding the wind velocity, as this requires the solution of 19 independent equations by successive approximation in order to obtain the two unknown components of wind velocity (WEISNER, 1956). The object of the present note is to point out that a theory is now available (GROVES, 1956a), which enables the separate contributions from wind and temperature to be calculated directly from the experimental data. Besides the usual assumption of a horizontally homogeneous atmosphere, the theory is based on the assumption that the s
349
Research n o t e s
atmosphere departs only slightly from being a uniform medium of propagati(m, i.e. that the variations with height in the speed of sound and in the wind velocity are small compared with the speed of sound. On presently available evidence this assumption appears to be justified up to 90 km altitude and possibly even higher, as variations in wind- and soundvelocities with height are of the order of about 50 m/see, and so are small compared with the speed of sound ( ~ 3 2 0 m/see). This assumption replaces the assumption made by SCEL t h a t winds and temperature are constant in any horizontal layer defined by two eonseeutive grenade bursts, and Jeads to a more refined mathematical treatment of the problem. Let (x, y, z) be the co-ordinates of a grenade burst with respect to Oxyz-axes, where O is the reception point on the ground at which the microphone array is situated, and let t be the travel time of the sound wave to O. Let a and b be the velocities with which the ineoming sound wave passes 0 in the directions Ox and Oy respectively. In terms of this single set of observations, it can be shown (GRoves, 1956a) t h a t
v(~) ~
f0
v(~) d~ = - - z { ( y + z2/b'r)/t + small-order terms}
V(z) ~ C(z) ~-
~,(~) d~ = --_,{(x + ~/~.)/t + small-order terms}
f:
(1)
c(~) d~ ~ z{[1 + (z/k~-)2]l/°'z/t + small-order terms}
where u(~), v(~), are the Ox, Oy-eomponents of wind velocity and c(~) is the speed of sound at height ~, and - r = t + x/a + y/b l / k ~ = 1/a 2 + 1/b 2.
From observations on Y grenades, bursting at heights z = z 1 . .. z x, values can therefore be calculated for the functions U(z), V(z) and C(z) at z = z1. . . . zx. B y either graphical or numerical differentiation of these values, empirical determinations of u(z), v(z), c(z) are obtained. With N grenades, U(z), V(z), C(z) m a y be expressed as polynomials of degree N - - 1 in z, and then u(z), v(z), c(z) would be obtained as polynomials of degree N - - 2 . Thus, two grenades would determine constant values for the wind and speed of sound, three grenades would determine a linear variation with height, while four grenades would determine a quadratic variation with height, etc. Certain points m a y be noted in connection with the above results: (a) I t is not necessary to assume that the wavefront received at the microphone array is plane. In practice, a and b would be determined by spacing microphones along the Ox-, Oy-directions, and taking the derivatives of the distance v. time relations for the passage of the wavefront along each of these lines at O. I f the wavefront were curved, these relations would show a finite acceleration, but a and b would still be the derivatives at O. In practice, extending the microphones much beyond the region over which the wavefront is sensibly plane would improve the accuracy of a and b only very slightly. (b) I t is not necessary to know the surface air temperature at the microphones. In SCEL's experiments, this quantity was accurately measured (to 0.2°C) in order to obtain the surface speed of sound, which is required for calculating the directions of arrival of the sound waves. Since surface-air temperature is a superfluous measurement, the particular value used for it would hardly be expected to influence the final results. This statement is supported by SCEL's investigation into the effect of errors, which showed that a I°C error in surface-air temperature would introduce an error of only 0"02°C into the computed upper-air temperatures. 350
l~esearch notes (c) The small-order terms in (I) comprise second- and higher-order contributions from the variations with height in the wind velocity and speed of sound. Expressions for the second-order contributions have been derived (G~ov~s, 1956a), and an estimate has bean made of their magnitude for certain typical atmospheric and experimental data. In the case of nearly vertical rays, the contributions are not expected to exceed 0.8 m/see; and so these terms could be neglected, if the errors from experimental sources were thought likely to be greater than this amount. (d) A further source of error affecting the final results would be an insufficient number of grenades. The more oscillatory the nature of u(z), v(z) and c(z), the closer would need to be the spacing of the bursts in order to detect these variations. (e) In (1), the effect of vertical air motion has been neglected. This has been necessary because observations on a single ray from each grenade burst suffice to determine only three mlknowns, and these have been chosen to be u(z), v(z) and c(z). The effect of a vertical component of wind velocity w(z) would be to increase U(z), V(z), and C(z) by amounts
--W(z)z/ar, --W(z)z/br and [1 @ (z/kr)2]l/2W(z) respectively, where W(z) =
f0'
w(~) d~.
Returning to the results of SCEL's experiments (ST~oUD et al., 1956), temperatures in the region of 50 km were found to be distributed about a mean, which was 15°C lower than standard-atmosphere values based on all recent experimental results. This difference was also found in the T-day experiments, when various methods were used within the same 24-hour period. On the other hand, comparisons with balloon measurements in the region of 30 km have been much closer. Since wind speeds were always much greater at 50 km t h a n at 30 km, it m a y be t h a t the analysis has not been completely effective in correcting temperatures for the effect of wind. A recalculation of the results of these firings on the basis of equations (1) would seem to be well worth while. SCEL's work on the rocket-grenade experiment has demonstrated the feasibility of using sound propagation for obtaining upper air winds and temperatures at least up to 80 kin. Their success has encouraged this department to undertake this type of experiment in the forthcoming British programme of rocket experiments (Jo~ES and MASSEY, 1956). These grenade experiments will differ from SCEL's method, chiefly in the use of a more extensive distribution of sound-detecting equipment on the ground. I t is hoped by this means to be able: (i) to improve the accuracy of the temperature and wind determinations, as these will not be so critically dependent on the measurement of small time differences between the various microphones; (ii) to determine vertical air motion if this is at all appreciable; (iii) to be able to derive pressures from the hydrostatic equation with good accuracy; (iv) to cheek the consistency of the observations by the method of least squares; (v) to average out the effect of any horizontal inhomogeneities in the wind and temperature structure of the atmosphere. A rigorous theory for analysing the data of this form of the rocket-grenade experiment has been developed and is in publication (GRoves, 1956b).
Department of Physics University College, London FEI~E~eE M. et al. GROVES G. V. Jo~Es F. E. and ]~ASSEY H. S. W. STROUD W. G. et al. WEIS~ER A. G.
G. V. G~ovEs t~EFERENCES 1956 J. Met. 18, 5. 1956a J. Atmosph. Terr. Phys. 8, 24. 1956b J. Atmosph. Terr. Phys. 8, 189. 1956 Nature 177, 643. 1956 J. Geophys. Res. 61, 45. 1956 J. Met. 13, 30. 351
Despite severe interference of the type mentioned good records have been obtained (apart from zero drift) even when the receiver was left unmonitored for three weeks. The rotation of the sky could be followed, the smooth maximum corresponding to the euhnination of the Sagittarius region. A representative portion of the recordings is illustrated in Fig. 2. Continuous records were taken between 18 November 1952 and 30 January 1953 giving a total of 1700 hours. Although this work has had to be postponed it was felt that the instrumentation described here might be of assistance to other experimenters. M. LA~TINEUR
Institut D'Astrophysique, Paris Cavendish Laboratory, Cambridge
J. ]). WHITEHEAD I~EFERENCE
LOVELLA. C.B.
1955 Joddrell Bank Radio Astronomy Symposium
A rigorous method of analysing data of the rocket-grenade experiment (Received 27 June 1956)
BETWEEN July 1950 and September 1953, the Signal Corps Engineering Laboratories' (SCEL) rocket-grenade experiment was carried out successfully on twelve occasions with Aerobee rockets at White Sands. The data from these firings have now been analysed, and results of temperatures and wind velocities between 30 and 80 km altitude have recently been published (STRoVD et al., 1956). Any attempt at determining the speed of sound in the open atmosphere has to take account of the possibility of the results being seriously influenced by the motion of the atmosphere. In the early determinations of the speed of sound in air, made by timing a sound wave between two points some miles apart, the effect of wind was mainly eliminated by arranging simultaneous transmissions in both directions. The procedure has been similar in the abnormal sound propagation experiments of C~ARY and other workers, the contributions from wind and temperature being separated out by timing sound waves travelling in various different azimuthal directions. Another way, by which the presence of wind may be detected, is its effect on the direction of arrival of the sound waves at the reception point. This method has been adopted in SCEL's experiments, using an L-shaped array of five microphones spaced 1000 feet apart on the ground roughly vertically beneath the grenade tmrsts. The theoretical problem of separating out the effects of wind and temperature in the rocket-grenade experiment has been treated by SCEL by a method of successive approximation (FERENCEet al., 1956). Firstly, the wind is found using temperatures that are uncorrected for wind, and then the speed of sound (and hence temperature) is found using these wind values, etc. The longest part of the calculation appears to be finding the wind velocity, as this requires the solution of 19 independent equations by successive approximation in order to obtain the two unknown components of wind velocity (WEISNER, 1956). The object of the present note is to point out that a theory is now available (GROVES, 1956a), which enables the separate contributions from wind and temperature to be calculated directly from the experimental data. Besides the usual assumption of a horizontally homogeneous atmosphere, the theory is based on the assumption that the s
349
Research n o t e s
atmosphere departs only slightly from being a uniform medium of propagati(m, i.e. that the variations with height in the speed of sound and in the wind velocity are small compared with the speed of sound. On presently available evidence this assumption appears to be justified up to 90 km altitude and possibly even higher, as variations in wind- and soundvelocities with height are of the order of about 50 m/see, and so are small compared with the speed of sound ( ~ 3 2 0 m/see). This assumption replaces the assumption made by SCEL t h a t winds and temperature are constant in any horizontal layer defined by two eonseeutive grenade bursts, and Jeads to a more refined mathematical treatment of the problem. Let (x, y, z) be the co-ordinates of a grenade burst with respect to Oxyz-axes, where O is the reception point on the ground at which the microphone array is situated, and let t be the travel time of the sound wave to O. Let a and b be the velocities with which the ineoming sound wave passes 0 in the directions Ox and Oy respectively. In terms of this single set of observations, it can be shown (GRoves, 1956a) t h a t
v(~) ~
f0
v(~) d~ = - - z { ( y + z2/b'r)/t + small-order terms}
V(z) ~ C(z) ~-
~,(~) d~ = --_,{(x + ~/~.)/t + small-order terms}
f:
(1)
c(~) d~ ~ z{[1 + (z/k~-)2]l/°'z/t + small-order terms}
where u(~), v(~), are the Ox, Oy-eomponents of wind velocity and c(~) is the speed of sound at height ~, and - r = t + x/a + y/b l / k ~ = 1/a 2 + 1/b 2.
From observations on Y grenades, bursting at heights z = z 1 . .. z x, values can therefore be calculated for the functions U(z), V(z) and C(z) at z = z1. . . . zx. B y either graphical or numerical differentiation of these values, empirical determinations of u(z), v(z), c(z) are obtained. With N grenades, U(z), V(z), C(z) m a y be expressed as polynomials of degree N - - 1 in z, and then u(z), v(z), c(z) would be obtained as polynomials of degree N - - 2 . Thus, two grenades would determine constant values for the wind and speed of sound, three grenades would determine a linear variation with height, while four grenades would determine a quadratic variation with height, etc. Certain points m a y be noted in connection with the above results: (a) I t is not necessary to assume that the wavefront received at the microphone array is plane. In practice, a and b would be determined by spacing microphones along the Ox-, Oy-directions, and taking the derivatives of the distance v. time relations for the passage of the wavefront along each of these lines at O. I f the wavefront were curved, these relations would show a finite acceleration, but a and b would still be the derivatives at O. In practice, extending the microphones much beyond the region over which the wavefront is sensibly plane would improve the accuracy of a and b only very slightly. (b) I t is not necessary to know the surface air temperature at the microphones. In SCEL's experiments, this quantity was accurately measured (to 0.2°C) in order to obtain the surface speed of sound, which is required for calculating the directions of arrival of the sound waves. Since surface-air temperature is a superfluous measurement, the particular value used for it would hardly be expected to influence the final results. This statement is supported by SCEL's investigation into the effect of errors, which showed that a I°C error in surface-air temperature would introduce an error of only 0"02°C into the computed upper-air temperatures. 350
l~esearch notes (c) The small-order terms in (I) comprise second- and higher-order contributions from the variations with height in the wind velocity and speed of sound. Expressions for the second-order contributions have been derived (G~ov~s, 1956a), and an estimate has bean made of their magnitude for certain typical atmospheric and experimental data. In the case of nearly vertical rays, the contributions are not expected to exceed 0.8 m/see; and so these terms could be neglected, if the errors from experimental sources were thought likely to be greater than this amount. (d) A further source of error affecting the final results would be an insufficient number of grenades. The more oscillatory the nature of u(z), v(z) and c(z), the closer would need to be the spacing of the bursts in order to detect these variations. (e) In (1), the effect of vertical air motion has been neglected. This has been necessary because observations on a single ray from each grenade burst suffice to determine only three mlknowns, and these have been chosen to be u(z), v(z) and c(z). The effect of a vertical component of wind velocity w(z) would be to increase U(z), V(z), and C(z) by amounts
--W(z)z/ar, --W(z)z/br and [1 @ (z/kr)2]l/2W(z) respectively, where W(z) =
f0'
w(~) d~.
Returning to the results of SCEL's experiments (ST~oUD et al., 1956), temperatures in the region of 50 km were found to be distributed about a mean, which was 15°C lower than standard-atmosphere values based on all recent experimental results. This difference was also found in the T-day experiments, when various methods were used within the same 24-hour period. On the other hand, comparisons with balloon measurements in the region of 30 km have been much closer. Since wind speeds were always much greater at 50 km t h a n at 30 km, it m a y be t h a t the analysis has not been completely effective in correcting temperatures for the effect of wind. A recalculation of the results of these firings on the basis of equations (1) would seem to be well worth while. SCEL's work on the rocket-grenade experiment has demonstrated the feasibility of using sound propagation for obtaining upper air winds and temperatures at least up to 80 kin. Their success has encouraged this department to undertake this type of experiment in the forthcoming British programme of rocket experiments (Jo~ES and MASSEY, 1956). These grenade experiments will differ from SCEL's method, chiefly in the use of a more extensive distribution of sound-detecting equipment on the ground. I t is hoped by this means to be able: (i) to improve the accuracy of the temperature and wind determinations, as these will not be so critically dependent on the measurement of small time differences between the various microphones; (ii) to determine vertical air motion if this is at all appreciable; (iii) to be able to derive pressures from the hydrostatic equation with good accuracy; (iv) to cheek the consistency of the observations by the method of least squares; (v) to average out the effect of any horizontal inhomogeneities in the wind and temperature structure of the atmosphere. A rigorous theory for analysing the data of this form of the rocket-grenade experiment has been developed and is in publication (GRoves, 1956b).
Department of Physics University College, London FEI~E~eE M. et al. GROVES G. V. Jo~Es F. E. and ]~ASSEY H. S. W. STROUD W. G. et al. WEIS~ER A. G.
G. V. G~ovEs t~EFERENCES 1956 J. Met. 18, 5. 1956a J. Atmosph. Terr. Phys. 8, 24. 1956b J. Atmosph. Terr. Phys. 8, 189. 1956 Nature 177, 643. 1956 J. Geophys. Res. 61, 45. 1956 J. Met. 13, 30. 351