XXV. Remarks on a Paper by Dr. Sondhauss.

By the Hon. J. W. STRUTT, Fellow of Trinity College, Cambridge*. IN N Nos. 5 and 6 of Poggendorff's Annalen for this year there is a paper by Dr. Sondhauss "On the Tones of Heated Tubes and Aerial Vibrations in Pipes of various forms," in which are given formulæ of considerable generality embodying the results of original and other experiments. Many years ago† Dr. Sondhauss had investigated the influence of the size and form of flask- or bottle-shaped vessels on the pitch of the sounds produced when a stream of air is blown across their mouth, and had obtained as an empirical formula for flasks with rather long cylindrical necks,

n= C

σ

(F.)

where n is the number of vibrations per second, a the area of the section of the neck whose length is L, and S the volume of the body of the flask. C is a constant determined by the experiments. On the other hand, when L is very small compared with the diameter of the neck, which then becomes a mere hole,

In the paper now under discussion it is sought to fill up the gap, as it were, and the following formula is arrived at as applicable for all proportions of L and σ,

in which a= velocity of sound.

(VII.)

c is a constant, of which Dr. Sondhauss says that it relates to the change in the velocity of sound in closed spaces from which the sound-waves have only a restricted exit; and its value, as found from the experiments, is approximately 2.3247.

a result which Dr. Sondhauss applies to cylindrical tubes closed at one end. This being admitted, it readily follows that for a pipe open at both ends,

An extension is next made to the case of more than one neck, but it will not be necessary for my purpose to repeat the formulæ. A few days before I saw Dr. Sondhauss's work I had myself completed a paper on a similar subject, which has since been sent to the Royal Society. The formula there given were in the first instance obtained theoretically, though some of them were afterwards verified by a rather laborious series of experiments. But on the present occasion I shall leave the theory on one side, and wish only to discuss some differences between the results of Dr. Sondhauss and my own, regarded from an experimental point of view. The rational formula corresponding to (VII.) is

where, however, S has not quite the same meaning as with Dr. Sondhauss, but includes the volume of about half the neck, and

is therefore nearly identical with the (S+L) of (VII.). On

this understanding (VII.) may be written

The rational formula (C) was first given by Helmholtz in his admirable paper in Crelle, on Vibrations in open Pipes; it is only strictly applicable to openings of circular form. The difference between (A) and (VII.) is never very great, being on one side when L is small, and on the other when L is large, and accordingly vanishing for some intermediate value. The greatest difference is shown in (VIII.) and (B) when L is very large. I therefore consider Dr. Sondhauss's opinion and anticipation to be in the main justified by my investigation, when he says, "I remark that I regard the formula (VII.) . . . not merely as an empirical formula useful for interpolation, but am convinced that it forms the theoretical expression of a natural law. From the zeal with which the field of mathematical physics is now cultivated, we may expect that the laws which I have discovered experimentally will soon be proved by analysis." But I must observe that (A) is only true subject to a series of limitations, which Dr. Sondhauss seems scarcely, if at all, to have contemplated. All the dimensions of the vessel (with a partial exception of the length of the neck) must be small compared with the quarter wave-length, and the diameter of the neck must be small against the linear dimension of the body of the vessel. The latter condition excludes the case of S small or nothing, to which Dr. Sondhauss pushes the application of his formula. But there is a rational formula proper for closed cylindrical tubes, as has been proved by Helmholtz in his paper on open pipes, to which Dr. Sondhauss refers, but apparently without availing himself of the results. It runs,

a

n=

L+
4

στ

(D)

but is only strictly true when σ is small against L. Although I am of opinion that (A) and (D) and the transition between them cannot be comprehended in the same theoretical investigation, yet it is easy to adjust (A) so as algebraically to include (D).

Thus

becomes, when S (which now refers to the volume of the body

only) is put equal to zero,

supposing the condition fulfilled as to the relative magnitudes of L and o. Now this form of (A) is perfectly legitimate, the 4 value of being 405. In the formula (VII.) ·405 is replaced by or 430.

1

But Dr. Sondhauss will naturally point to the comparison of (X.) with the experiments of Wertheim, which he justly regards as very satisfactory. I have examined the series of twenty-two experiments with cylindrical tubes of circular form closed at one end, and have calculated for comparison the results of (D): It will be seen from the annexed Table that, good as is the agreement of the observations with (X.), it is still better (on the whole) with (D). Indeed I must confess that the differences in some cases, where is by no means small compared with L, are much less than I should have expected.

The foregoing Table shows that although (X.) represents Wertheim's observations with considerable accuracy, yet Helmholtz's rational formula (D) is on all grounds to be preferred.

Dr. Sondhauss expresses himself strongly as to the difficulty which exists in determining accurately the pitch of the very uncertain sound produced by tubes whose diameter is not small compared with their length, an opinion which I entirely share. It is indeed difficult to understand how Wertheim obtained results of such precision. But I cannot agree with Dr. Sondhauss when he goes on to say that resonance is not a sure guide in determining accurately the pitch of a pipe; for it was by this method exclusively that the determinations recorded in my paper were made. I have there given at length my reasons for adopting it, and for doubting the results of the method of blowing, although such experiments as those of Wertheim go to show à posteriori that in his hands at least it was not unworthy of dependence. Other experiments of Wertheim are calculated from formula (IX.) and show a tolerable agreement. The difference between (IX.) and Helmholtz's theoretical formula (C) relates only to a constant multiplier, and corresponds to a difference of pitch of about a quarter of a semitone. The discordances are attributed (no doubt correctly) to the unsuitable form of some of the vessels, and consequent imperfect fulfilment of the theoretical condition to which (C) is subject.

We come next to vessels in the form of flasks with a cylindrical neck of sensible length. Dr. Sondhauss gives a Table containa ing the results of a comparison of (VII.) with some experiments of his own. The average discordance amounts to about a semitone. Although it was evident beforehand that in most cases the limitations on formula (A) were grossly violated, I thought it worth while to calculate in accordance with (A) the theoretical pitch, and have given the results in the form of a Table:

The result of the formula (VII.) ought evidently here to be greater than that of (A). On a recalculation I find 85.8 instead of 83.2.