Page 21 - January 2009

Walker makes a further refinement of the echo method. He discovered that it is quite natural to develop an alternat- ing rhythm: clap-echo-clap-echo, with equal intervals between echo and clap. Instead of attempting to match a sin- gle clap-echo against a single swing of the pendulum, he establishes this rhythmic clapping and compares it to multi- ple swings of the pendulum.
This rhythmic clapping is brilliant. Human ability to establish such a rhythm is remarkable—a series of claps can be evenly interspersed between the series of echoes with a remarkable uniformity of interval. [It is, however, all too easy to get bad results. Care must be taken to select a wall without overhangs, steps, or inside corners. All other reflectors must be sufficiently distant to have no influence. And changes in ambient noise can distract sufficiently to upset the rhythm.]
Had this technique been pursued more seriously, it may have been capable of the precision required to determine the temperature dependence of sound speed 50 years before it was actually measured. Matching the periodicity of the clap- echo pattern with the periodicity of the pendulum is an early variety of synchronous averaging. Over a long period of observation, the clapping rate and the pendulum rate could have been matched closely by either increasing or decreasing the distance to the wall. Walker used a simple pendulum—a lead bullet on a wire—and this limited the period of observa- tion as the pendulum’s oscillation decayed. [In addition try- ing to keep the required rhythm while both listening for echoes and watching the pendulum would very likely have created problems. It would have been difficult to avoid unin- tentional phase locking between the clapping rate and the pendulum unless a second observer was used. If Walker had concentrated only on clapping the boards and a second observer watched the pendulum, this could be avoided.]
Walker suggested the use of an “automatic” clock but apparently did not try it himself. If Walker had Derham’s automatic clock and if Walker had chosen to perform the echo experiment on the coldest and hottest days available to them in England, he may have uncovered the temperature dependence of sound speed.
What would have been required to see the temperature dependence of sound speed and could such an experiment have succeeded in 1700? In air, the speed of sound (in m/s) is equal to 20.05 times the square-root of the absolute temper- ature in kelvin (the temperature in celsius plus 273.1). Consequently, the variation in sound speed with temperature in the vicinity of 15 °C is 0.59 m/s per degree celsius. From winter to summer in England, it would not have been diffi- cult to find temperatures from at least 0 to 25 °C. This tem- perature range would produce a seasonal sound speed differ- ence of almost 15 m/s—slightly more than 4 percent. Derham claimed a timing resolution of 0.25 seconds. With this timing resolution a total travel time of at least 6.25 sec- onds would be just sufficient to resolve a 4 percent difference in sound speed. A measurement of ten times that duration would resolve easily the 4 percent variation in sound speed. So the question reduces to this: can we design an experiment to give a one-minute period of coherent observation? Derham’s timing measurements from Blackheath to Upminster had a travel time of about 60 seconds but the
dependence on wind is so strong that it would have been challenging to uncover the temperature dependence. [Furthermore, refraction—unknown to anyone at that time—would have confounded the interpretation. It would have been natural to select the coldest conditions and com- pare those results to the hottest conditions but these two extremes would most typically be accompanied by signifi- cantly different actual path lengths because of differences in refraction.]
If, however, rhythmic clapping in the echo method could be maintained for 60 seconds, there would be a chance of success. In the echo method, the effects of wind are second order and a short baseline would avoid the effects of refrac- tion. A careful, rhythmic echo measurement may have uncovered the temperature dependence of sound speed using the clocks available in 1700.
Of course, echo methods are not suitable for assessing the effects of wind. A round-trip measurement has a much weak- er dependence on wind speed than the one-way measurement. [It is sometimes naively stated that the echo method cancels any effects of wind but that is not true. The round-trip meas- urement cancels the first-order influence of the wind but the measured speed still depends on the square of the ratio of the wind speed to the sound speed. In addition, the error intro- duced by the wind biases the round-trip measurement—the measured speed is always less than the thermodynamic sound speed regardless of the direction of the wind.]
In retrospect, what is surprising is that the echo methods did not produce particularly good results in the late 1600’s and early 1700’s. Even Walker who seemed to have an excep- tional grasp of the structure of a good measurement did not produce a particularly good value for sound speed. For the eleven measurements he reports, the average (with one stan- dard deviation) is 1305 ± 120 ft/s. By and large, the credible measurements were long-baseline, one-way measurements by flash-and-report timing of guns or cannons. Furthermore, long one-way measurements were necessary to identify the effects of wind.
Like Walker, Derham believed that an “automatic” clock was a far better measurement tool for sound speed measure- ment than a simple pendulum. Walker describes the typical timing pendulum as a length of cord or wire with a lead bul- let crimped onto the end. This is, in fact, the apparatus described by Newton in his measurement of sound speed. Newton’s approach was a modification of the simple echo method but not as elegant as Walker’s. Newton located him- self 208 feet from a reflecting wall and compared the echo transit time to two pendulums. [This is the experiment described in the first edition of Principia.]
He noted that the echo returned before one half-cycle of an 8-inch pendulum but after one-half cycle of a 5.5-inch pendulum. From these observations, Newton concluded that the speed of sound must be greater than 920 ft/s but less than 1085 ft/s. These values bracketed his flawed prediction of 968 ft/s so he had little incentive to refine the measurement. [Convincing experimental evidence that the value of 968 ft/s was far too low came after the first edition of Principia. Newton did try again later with revised results published in the second and third editions.]
20 Acoustics Today, January 2009