A simple and accurate formula for the sound velocity in water

doi:10.1016/S0301-5629(98)00091-X

Ultrasound in Medicine & Biology

Volume 24, Issue 7, September 1998, Pages 1065-1068

https://doi.org/10.1016/S0301-5629(98)00091-X Get rights and content

Abstract

The sound velocity in test objects and phantoms is often measured by performing a differential measurement with pure water. To promote standardization, a simple formula for the sound velocity in water is derived that renders true values within 0.20 m s⁻¹ over the temperature range 15–35 C. The formula is given by c = 1404.3 + 4.7 T − 0.04 T², with sound velocity c in m s⁻¹ and temperature T in C.

Introduction

Recently the AIUM Technical Standards Committee published the Stage 1 report “Methods for specifying acoustic properties of tissue mimicking phantoms and objects” (AIUM 1995). Herein, a simple formula for the sound velocity in pure water as a function of temperature was given, using rounded-off coefficients from a formula given by Wilson (1959). The AIUM report states that it is easy to measure sound velocity in phantoms with an accuracy of ± 3 m s⁻¹ with the substitution technique, where the sound velocity of water serves as the reference value.

Comparing the results of the simple formula with the data from Wilson (1959), we found discrepancies that already come close to the desired overall accuracy of 3 m s⁻¹. The deviations are caused by neglect of the higher order terms and rounding off in the AIUM report. We fear that a reference value with such a large error can lead to confusion in standardization. Therefore, we recommend that the reference value renders the true value with an error not larger than 10% of the desired measuring accuracy (i.e., 0.3 m s⁻¹). To obtain a simple formula with this accuracy, it is necessary to limit the range of temperatures for which it is valid. We derived a new formula for the range 15–35 C that is adequate for phantoms and test objects. To accomodate the inclusion of body temperature of humans, we also derived a formula for the range 10–40 C. Wilson (1959) estimates that the maximal error in his data is 0.12 m s⁻¹. Greenspan and Tschiegg (1959) indicate a maximal error of 0.05 m s⁻¹. More accurate results have been published by Del Grosso and Mader (1972), who state that systematic and random errors in their results lead to a maximal uncertainty of 0.015 m s⁻¹. To obtain an accurate formula, we therefore derived our approximations from the data of Del Grosso and Mader.

Section snippets

Materials and methods

The references mentioned above present the sound velocity c [m s⁻¹] as a power series of the temperature T [C]: $c=a_{0} +a_{1·} T+a_{2·} T^{2} +a_{3·} T^{3} +a_{4·} T^{4} +a_{5·} T^{5} .$ AIUM (1995) does not mention the temperature range. The other references state that their formulas are valid for the temperature range 0–100 C. The coefficients given by Del Grosso and Mader (1972) are given with nine significant figures. We found that we could round off the coefficients, as given in Table 1, without introducing an aberration from

Results

The coefficients for the least square fit lines are given in Table 2, they are marked by the label LS. The results of series with optimally rounded off coefficients are also given in Table 2 (formulas A1, B1–B3 and C1–C3). The calculated sound velocities from the coefficients given in Table 1, Table 2 are given in Table 3. Figure 2 represents the differences of various values with those calculated from the coefficients of Del Grosso and Mader. A linear fit (A1) was calculated for a small

Discussion and conclusion

Measurements of the sound velocity of phantom materials are often made with respect to water (substitution technique, AIUM 1995). To not compromise the accuracy of the data, it is desirable that all workers use a common reference. The formula proposed in AIUM (1995) introduces, in our opinion, too large an error.

One could argue that, now, with the help of computers, no approximate formulas are needed. Then, data taken from the publication of Del Grosso and Mader (1972) should be used. It should

References (6)

Methods for specifying acoustic properties of tissue mimicking phantoms and objects, Stage I
(1995)
V.A. Del Grosso et al.
Speed of sound in pure water
J Acoust Soc Am
(1972)
M. Greenspan et al.
Tables of the speed of sound in water
J Acoust Soc Am
(1959)

There are more references available in the full text version of this article.

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Technical NotesA simple and accurate formula for the sound velocity in water

Abstract

Introduction

Section snippets

Materials and methods

Results

Discussion and conclusion

Methods for specifying acoustic properties of tissue mimicking phantoms and objects, Stage I

Speed of sound in pure water

J Acoust Soc Am

Tables of the speed of sound in water

J Acoust Soc Am

Technical Notes
A simple and accurate formula for the sound velocity in water