The wave amplitude is given by Here is the amplitude

The wave amplitude is given by,Here, is the amplitude of the component wave of frequency ω at source point, is the propagation constant (=2πf/v), is the reflection co-efficient at water-reflector interface, is the reflection co-efficient at water-transducer interface, and x is the position measured from the transducer face. For simplicity, we take . In the present experiment, since a single transducer is used as the transmitter as well as the receiver, the response observed in the oscilloscope will be proportional to the wave displacement at x=0, i.e. .
The echo trains for n different sample lengths, l1, l2,… ,ln, are captured using digital oscilloscope DL 1640 and stored in PC for further analysis. From the average amc 7 Δt between successive echoes, ultrasound velocity v is determined (v=2ln/Δt) and k is calculated. From exponential fitting of the echo heights, effective attenuation αe is determined. To get the intrinsic attenuation constant α the following procedure is adopted. The Fast Fourier Transform (FFT) of the echo trains obtained for n lengths are determined. For each frequency component, n wave amplitudes are obtained for n different sample lengths. Oak Ridge and Oxford method for parameter fitting [8] is used to fit these n amplitudes according to relation (2) by adjusting the parameters k, r, α and . The computational method requires input guess values for k, r, α and . Input k is obtained from experimentally measured v, input α is the lowest value of αe obtained for large l, input r is calculated using the relation r=(ρ1v1−ρ2v2)/(ρ1v1+ρ2v2), ρ being the density, with suffixes 1 and 2 designating the reflector material and water respectively, and input is chosen arbitrarily. The experiment is repeated using steel, copper, brass, lead and glass reflectors.

Results and discussion
The velocities (v) and effective attenuation constants (αe) are determined for various lengths (l) of water columns in pulse-echo experiment with five different reflecting surfaces. The values for v are close to each other within experimental error. Average v is determined to be 1.4775×105cmsec−1. Fig. 1 shows the dependence of αe on l for glass reflector. We see that attenuation values are widely different particularly for small values of l. Similar variation in the attenuation constant has been reported by Martinez and co-workers [2]. Their work shows clearly that at short distances from the transmitting transducer, the attenuation values measured in pulse-echo method show wide variation. At long distances however, the results are consistent and it is possible to get an average value of 0.04417np/cm for the attenuation constant though it is not in agreement with other reported values [3–6]. Their measurement in through-transmission method uses two transducers aligned face-to-face, parallel to each other. By this the water medium in between behaves effectively like a bounded medium as in pulse-echo method and no better solution is obtained. The attenuation value obtained in this method is 0.04654np/cm.
Measurements with other four reflectors, viz., steel, copper, brass and lead show similar nature of variation of αe with l and this is illustrated in Fig. 2. No significant difference is noticed due to the differences in r because of the fact that fluctuation due to change in l is more prominent than the effect caused by the change in r. For longer l and smaller r the echo signals are weak and measurement of αe is more erroneous. The observed nature of Fig. 2 is also consistent with the numerical study presented in Ref. [1].
To determine the intrinsic attenuation constant α, FFT components are computed for pulse-echo signals captured for seven different column lengths (ln, n=1, 2,… ,7) under same exciting conditions. The exciting input is pulsed rf signal of carrier frequency 1MHz, peak-peak pulse height of ∼300V, pulse width of ∼3μs and pulse repetition time ∼2.45ms. Table 1 presents the liquid column lengths for different reflectors used to get pulse-echo signals. Fig. 3 shows representative echo signal captured with glass reflector for (a) l1=5.056cm and (b) l7=3.552cm and Fig. 4(a) and (b) shows their fft components respectively. In Fig. 4 unit frequency corresponds to 100MHz and relative amplitude is in arbitrary unit. The frequency range where the fft amplitudes are relatively high is shown in Fig. 4. The peak region is considered for computational parameter fitting. The error in experimental fft amplitudes is assumed to be within 10%. Best fit parameters with minimum χ2 error are determined using Oak Ridge and Oxford method [8]. In finding the best fit parameters, several factors have been taken into consideration. Firstly, Triassic Period is known that there is no dispersion for normal sound propagation in water [12] in the frequency domain of the present experiment. Usually, the transducer bandwidth is ∼10% of the central operating frequency, i.e. 0.95–1.05MHz for 1MHz transducer. So k remains constant for all frequencies in this range. Velocity measurement in pulse echo method gives reasonably accurate value and we take the experimental value of v to calculate k and vary k within experimental error. For input r, we depend on theoretical value and allow small variation around this theoretical value. αe determined from pulse-echo experiment is chosen as input α. We choose the particular set of values for the parameters for which χ2 error is minimum for maximum frequency components in the range considered. The best fit parameters along with their input values are presented in Table 2. The average k thus obtained is 42.74±0.03cm−1 giving v=(1.4693±0.0011)×105cmsec−1 and average α is 0.0435±0.0013np/cm. This value of α is more accurate and consistent with the measurement by Martinez et al. [2].