Application Note:
Time of Flight and Material Properties

Current trends in ultrasonic instrumentation and signal processing make it feasible to measure variations in ultrasonic time-of-flight (ToF) with precision in the picosecond range.  This capability opens up new possibilities for material characterization and non-intrusive temperature measurements.  For a propagation path of length, L, the ToF variations are related to a material’s thermal expansion coefficient, a, and its elastic modulus temperature coefficient, g (see appendix A).  The variation in temperature, dQ, is related to the fractional variation in the ToF and is given by the following expression:

Eq. 1Eq. 1

Where a is the thermal expansion coefficient and g is the elastic modulus temperature coefficient.  By making precise TOF measurements, one can estimate the variation temperature.  Conversely if the temperature change is measured independently and is uniform across the propagation path, the fractional change in TOF can be used to measure the modulus temperature coefficient, g.

The materials parameters can be either acquired from the literature or measured directly.  Methods for measuring the thermal expansion coefficient are well known and data is readily available in the literature.  Measurements of modulus over the relevant temperature range are less common but are generally available for many metals, composites and ceramics. 

Figure 1 shows a comparison of measured fractional ToF variation for alumina with that calculated using literature data for the thermal expansion coefficient, a, and the modulus temperature coefficient, g.  Literature data was only available up to a temperature of 1500C.  For the measured data, a thermocouple was used to measure the temperature up to 1000C and an optical pyrometer used in the range from 1000C to 2100C.  Overall, the agreement between the literature derived results and the measured data is excellent.  Figure 2a shows the literature data for alumina the thermal expansion coefficient and Figure 2b shows the elastic modulus temperature coefficient data.

ToF in Alumina

 Alumina Rod Torched

 Alumina Rod Melted
Figure 1 a) Comparison of the measured and calculated ToF for alumina from ambient to 2100C; b) torch-heated 250 mm long alumina rod with annular step at heated end and ultrasonic sensor attached at the opposite end;  Lower micrograph shows melted tip as experiment exceeded 2100C.

Thermal Expansion Coefficient

Modulus Data
Figure 2 a) Thermal expansion coefficient data for alumina b) Modulus data for alumina


ToF Property Relationships

For a propagation path of L and an ultrasonic velocity of C the ToF for a pulse-echo propagation path is given by:

Eq. 2 Eq. 2

The fractional variation in the ToF with temperature, Q, is given by:

Eq. 3 Eq. 3

There are 2 contributions to the ToF change the expansion of the material and the variation in the velocity with temperature.  For most solids the thermal expansion coefficient is positive and the ultrasonic temperature coefficient is negative.  Thus in equation 2 both effects increase the ToF as temperature increases.

The ultrasonic velocity can be related to the density, r, and the Young’s modulus, E, of the material given by:

Eq. 4 Eq. 4

By substituting this expression into Eq. 3 and defining the modulus temperature coefficient, g in Equation 5.  It can be shown that the velocity coefficient, b is related to the modulus temperature coefficient  and the thermal expansion coefficient as shown in Equation 6. 

Eq. 5 Eq. 5

Eq. 6 Eq. 6

The fractional change in ToF can be then directly related to the thermal expansion coefficient and the temperature variation in the modulus which is given by Equation 7.

Eq. 7 Eq. 7