ESTIMATION OF INCIDENT AND REFLECTED WAVES IN IRREGULAR WAVE EXPERIMENTS

  J. Indones. Math. Soc. (MIHMI) Vol. 12, No. 2 (2006), pp. 141–153.

  

ESTIMATION OF INCIDENT

AND REFLECTED WAVES

  

IN IRREGULAR WAVE EXPERIMENTS

_Abstract.

A. Suryanto

  We discuss a method to decompose the measured wave elevations obtained

from irregular wave experiments. The decomposition is performed by comparing the

general solution of the linear wave equation and the wave elevation measured at some fixed

positions. To minimize the error in matching the Fourier coefficient at each frequency of

the solution, a least square method is applied. Error analysis as well as the validation of

the method will also be given.

  1. INTRODUCTION In model testing of ships, it is important to simulate open ocean conditions as closely as possible. Such simulations are often performed in a wave tank at a hydrodynamic laboratory. At one side of the tank, waves are generated by a wave maker over the whole width of the tank; and at the other side of the thank, an efficient ’beach’ is installed to prevent the reflection of the incoming waves. However, even the beach is installed to minimize the wave reflection, in general the incident waves cannot be absorbed completely and part of it will be reflected by the beach. The reflection of waves will of course disturb the incident waves generated by the wave maker. Besides on the type of incident waves, the reflected waves often depend on the type or structure of the beach. To test the properties of the beach, one wants to know how much the incident waves are reflected by the beach.

  In a hydrodynamic laboratory, the measurements of wave elevations are con- ducted using wave probes that are often located at fixed positions. The measured

  Received 12 March 2005, Revised 8 March 2006, Accepted 17 March 2006. _{2000 Mathematics Subject Classification Key words and Phrases} : 76B15.

  : hydrodynamics laboratory, incident and reflected waves (spectra), least square method.

A. Suryanto

  wave elevations consist of the incident waves, the reflected waves and their inter- actions. To investigate the properties of the beach, we need to decompose the measured wave profiles into its components. A number of previous studies have considered the measurement of wave reflection using fixed wave probes. For exam- ple, a method involving two fixed wave probes was introduced by Thornton and Calhoun [7] and Goda and Suzuki [2]. In this method, the measured wave (eleva- tions) signals are decomposed into the incident and reflected wave. The reflection coefficient is found by calculating the ratio of reflected wave height to the incident wave height. However, as indicated by Goda and Suzuki [2], this method fails when the spacing between the two wave probes is equal to an integer number of half wavelengths. In order to reduce the probes spacing problem and the sensitivity to noise and deviations from the linear theory, Mansard and Funke [5] proposed a method involving three probes which is based on a least squares technique.

  Recently, Brossard et al. [1] proposed a method to measure the reflection coefficient using one or two moving probes. However, as pointed out by Isaacson [4] and Nallayarasu et al. [6], transversing the wave probe for a wide range of experiment seems to be pratically difficult and if proper arrangement is not made to move the probe, the data obtained may not be reliable. Hence, we will concentrate only on the method which uses fixed probes. By following Zelt and Skjelbreia [8], we generalize the method of Mansard and Funke [5] by including arbitrary number of probes. However, the reflected waves in this paper will not only be characterized by the reflection coefficients but also by their phase shifts.

  2. REFLECTION MODEL To derive a reflection model, we assume that the wave in the basin is one- dimension and linear. Under this assumption, the wave satisfies the following partial differential equation, see [3]

  2

  2

  2

  ∂ η = ∂ R η. (1) _{t x} This equation is a linear dispersive wave equation which has monochromatic solu- tions _{i x−ω t i x−ω t}

  

(k ) (−k )

  a e + b e , where the wave number k and the frequency ω are related by the exact dispersion relation, for any amplitude a and b. Using the fact that a superposition of any two solutions is again a solution of a linear differential equation, the general solution of (1) can be written as _∞

  Z n o _{i i}

  (K(ω)x−ωt) (−K(ω)x−ωt)

  η(x, t) = A(ω)e + B(ω)e dω. (2) _−∞ This solution represents two wave trains that run in opposite direction; every component of the first train runs with the same frequency and velocity as the

  Estimation of incident and reflected waves

  associate component of the second train. Notice that we express the wave number K as a function of the wave frequency ω. Indeed, for a given frequency ω, the wave number K is determined uniquely by the exact linear dispersion relation.

  Now consider an incident wave which is generated by the wave maker and propagates to the beach. Assume that the beach has a complex-valued reflection coefficient R, which can be written in exponential form as _iβ

  R = r e , (3) where r = |R| is the ratio of reflected wave height to the incident wave height; √

  β is the phase shift during reflection; and i = −1. For later discussion, we refer to r as the reflection coefficient. For a solid wall, the reflection coefficient is 1; i.e., the magnitude of the reflected wave is equal to that of the incident wave.

  When the incident wave is completely absorbed by the beach, the magnitude of the reflected wave is zero. As a result, the reflection coefficient is zero. As we see below, the reflection coefficient and the phase shift will depend on the frequency. We assume that the reflected wave due to the beach travels back to the wave generator with the same frequency and the same velocity as the incident wave but in opposite direction. And thus, using these assumption, we can consider (2) as incoming waves and reflected waves. Without loss of generality, we assume that _∞ R _i

  (K(ω)x−ωt) _−∞ A(ω)e dω is the incoming wave which propagates from the left and is reflected when it hits the ’beach’ with reflection coefficient r(ω) and phase shift

  β(ω), from which we have _iβ

  (ω)

  B (ω) = r (ω) A (ω) e . (4) Referring to the words ’incoming’ and ’reflected’ waves, it is convenient to write the wave height A (ω) and B (ω) as A inc (ω) and A ref (ω), respectively, such that (2) becomes _∞

  Z n o _{i i}

  (K(ω)x−ωt) (−K(ω)x−ωt)

  η(x, t) = A inc (ω)e + A ref (ω)e dω, (5) _{−∞ iβ (ω)} where A ref (ω) = r (ω) A inc (ω) e , r (ω) ∈ [0, 1] and β (ω) ∈ [−π, π]. The formula (5) gives a general solution of equation (1) in terms of the inci- dent and the reflected spectral functions A inc (ω) and A ref (ω). To get the complete wave field, the spectral functions A inc (ω) and A ref (ω) should be determined from experiments. By knowing the incident and reflected spectra, the reflection coef- ficient and the phase shift can be determined at once, i.e. by applying formula (4).

  It is noted that we have assumed the position of the ’beach’ to be the origin of the horizontal coordinate x and the reflected wave is never reflected by the wave maker (or the reflected wave never reaches the wave maker).

A. Suryanto

  3. ESTIMATION OF THE REFLECTION COEFFICIENT AND THE PHASE SHIFT

  In this section we present a method to measure the wave reflection from wave experiments. This method uses a number of wave probes that are located at fixed points. Each of these probes gives a signal that is the wave elevation as a function of time at its position: _∞

  Z _{− iωt} s(x n , t) = s n (ω)e dω, n = 1, 2, ..., p (6) _−∞ b where x n n (ω) is the spectrum of the signal is the position of the n−th probe; bs and p is the number of probes. The signal spectrum can be obtained by taking the

  Fourier transform of the signal: _∞ Z

  1 _iωt s n (ω) = s (x n , t) e dt. (7) b

  2π _−∞ Then, by comparing the coefficients in equation (6) and (5): s(x n , t) = η(x n , t) for all n we obtain the following equations: _{iKx n iKx n −}

  A inc (ω) e + A ref (ω) e n (ω), n = 1, 2, ..., p. (8) = bs

  For p = 2, the expressions for A inc (ω) and A ref (ω) have been derived exactly by Goda and Suzuki [2]. In 1980, Mansard and Funke [5] solved the case p = 3 using a least square method. We will follow their approach for arbitrary p. To do this, we introduce ε n

  η n (ω) at (ω) in (8), i.e. ’errors’ in matching the Fourier coefficient b frequency ω at probe n _{iKx iKx n n −} ε n (ω) = A inc (ω) e + A ref (ω) e n (ω).

  − bs These errors may arise from difficulties in precise measurements, and may also caused by non-linear wave interactions or other spurious wave effects that give rise to measured wave heights and phases that differ from the linear theory as assumed. By applying a least square method, one may find A inc (ω) and A ref (ω) such that _p

  P

  2 _{n =1} (ε n (ω)) is minimized. This minimization problem can be solved by setting _p

  P

  2

  the partial derivatives of (ε n (ω)) with respect to A inc (ω) and A ref (ω) to be _n

  =1

  zero. This gives _p

  X ¡ − ¢ _{iKx n iKx n iKx n}

  A inc (ω) e + A ref (ω) e n e = 0, (9a) _n − bs _p =1

  X ¡ − ¢ − _{iKx iKx iKx n n n}

  A inc (ω) e + A ref (ω) e n e = 0, (9b) _n − bs

  =1

  Estimation of incident and reflected waves

  or _{p p}

  X _{n iKx n}

  X

  2iKx

  A inc (ω) e + pA ref (ω) = s n (ω) e , (10a) _{n n} b

  =1 =1 _{p p}

  X _{− n − n}

  X

  2iKx iKx

  pA inc (ω) + A ref (ω) e = s n (ω) e . (10b) _{n n} b

  =1 =1

  The solution of these two equations is given by _{p p p} P P − P − _{iKx n n iKx n}

  2iKx

  s n (ω) e e s n (ω) e _{n =1 n =1 n =1} b − p b A inc (ω) =

  , (11a) _{p p p} D P − n P n P n _{iKx 2iKx iKx} s n (ω) e e s n (ω) e _{n n n} b − p b

  

=1 =1 =1

  A ref (ω) = , (11b)

  D µ ¶ µ ¶ _{p p}

  P P −

  2iKx n 2iKx n

  2 where D = e e . _{n n} − p =1 =1

  By writing _iθ

  (ω)

  A inc inc , (12) (ω) = |A (ω)| e we have _ref

  |A (ω)| r (ω) = (13) _inc |A (ω)| and

  β (ω) = arg (A (ω)) , (14) _{ref (ω)) − arg (A inc} where arg(z) ∈ [−π, π) is the argument of complex value z. From (11a) and (11b), it can be seen that the least square method fails when

  D = 0. To evaluate this denominator, it is more convenient to write it as _{p p}

  X X

  2 D = 4 sin K(x m n ). (15) _{m n<m} − x =1

  Then it is clear that the denominator D can be zero only if the position of probes are such that sin (K(ω)(x m n − x )) ≡ 0 for all m and n. Furthermore, the algebraic manipulation of (11a) and (11b) leads to the fol- lowing formula for A inc (ω) and A ref (ω) _p

  X A inc (ω) = C n s n (ω) , (16a) _n b

  =1 _p

  X A ref (ω) = C n s n (ω) , (16b) _n b

  =1

A. Suryanto

  _{− P −} ·µ ¶ ¸ _p

  1 iKx n n x m 2iK(x )

  where C n = e e and C n represents the complex _{D m} − p

  =1

  conjugate of C n . It can be checked that for p = 2, the least square method repro- duces the results of Goda and Suzuki [2]; while for p = 3, formulae (16) are exactly the same as those of Mansard and Funke [5].

  4. SENSITIVITY ANALYSIS In this section we will show that the accuracy of the decomposition method

  (16) depends very much on the probe’s spacing. To see this, by following Zelt and Skjelbreia [8], we suppose that the wave signal at x = x n obtained from experiments includes the residual signal E n (t): s(x n , t) = ξ(x n , t) + E n (t) , (17) where ξ(x n , t) is the exact signal at x = x n . These errors may be caused by the non-linear wave interactions, signal noise, measurement errors, etc. _{n (ω) = b ξ n (ω) + ε n (ω) , where}

  From equations (16) and using the fact that bs b ξ n (ω) and ε n (ω) are the Fourier transform of ξ(x n , t) and E n (t) respectively, the exact incident wave spectrum and the exact reflected wave spectrum can be written as _p

  X b e

  A inc (ω) = C n ξ n (ω) _n

  =1 _p

  X = A inc C n ε n (ω) , (18a)

  (ω) − _{n p} =1 X b e

  A ref (ω) = C n ξ n (ω) _n

  =1 _p

  X = A ref C n ε n (ω) . (18b)

  (ω) − _{n =1} If the residual signal is zero then we have exact incident and reflected wave spectra, otherwise the error at probe n, i.e., ε n (ω) will be amplified by the factor

  C n . The amplification of errors associated with probe n can be expressed by ¯ Ã ! ¯ _p ¯ ¯

  X 1 − _{n x m}

  2iK(x ) _{n e . (19)} ¯ ¯

  |C | = − p ¯ ¯

  D ¯ ¯ _m

  =1

  This amplification factor has maximum value if the denominator D is zero. For p = 2 (see Goda and Suzuki [2]), this condition satisfied by |K (x

  1 − x 2 )| = nπ, for Estimation of incident and reflected waves

  any integer n. This means that the method fails when the distance between the two probes appraches an integer number of half wavelength. The problem related to this probe spacing can be reduced by adding the number of probes with unequal spacing. Indeed, for arbitrary number of probes p, the denominator is zero only if the probes are placed such that sin (K(ω)(x m n

  − x )) ≡ 0 for all combination of m and n. Further, if the residual signal is non zero then the absolute errors caused by the estimation of the reflection coefficient and the phase shift are given by

  E r exact (20a) = |r − r|

  E β exact (20b) = |β − β| where

  ¯ ¯ ¯ ¯ e

  A ref (ω) ¯ ¯ r exact = ¯ ¯ ¯ e ¯

  A inc (ω) β exact = arg( e A ref A inc (ω)).

  (ω)) − arg( e From equations (18) and (20), it can be seen that if the measured wave spectrum is much larger than the residual spectrum then the absolute error of the reflection coefficient and the phase are relatively small. On the other hand, the ’small’ measured wave spectrum will produce large errors in the estimation of the reflection coefficient and the phase shift.

  5. METHOD VALIDATION To assess the validity, the method to estimate the reflection coefficient and the phase shift is tested against ”artificial linear waves” in 5 m water depth:

64 X

  η (x, t) = A j cos(k j j t) + 0.1 A j cos(k j x + ω j t), (21) _j x − ω

  =1

  where A j

  = 0.5 − 0.5 cos(2πj/64), ω j = ω + j∆ω. The wave numbers k j and the frequencies ω j are related by the exact linear dis- persion relation. Observe that we have set the reflection coefficient r = 0.1 and the phase shift β = 0 for all frequencies. Three time series of wave elevation consisting of 4096 points per wave probe with a time step ∆ = 0.04723 seconds were created at locations x = 1.94874 and

  1 = −15.12, −17.36, −19.04 m using ω 2π

  ∆ω = . Hence, we generate ”artificial wave” signals with frequencies in the

  4096∆

A. Suryanto

  range I = (1.981, 4.027). In addition, an uncorrelated random noise signal was added at each of the wave signals to simulate the errors due to the measurements or other effects that cannot be measured directly. Each of the generated noise sig- nals has a normal distribution function with mean µ = 0 and standard deviation σ = 1 mm. The components of the generated wave (elevation) signal are presented in Figure 1.

  (b) Incident wave signal as in (a) (a) Incident wave signal but viewed in small interval

(c) Reflected wave signal (d) Noise signal

  Recalculated signal Input signal using least square method

  

Figure 1: Component of the artificial wave (elevation) signal at x = −15.12m. The

components of the input signal are free of noise. The components of the recomputed

signal are obtained from the ’measured signal’ that containts the noise signal.

  It is noted that the transformation from time domain to frequency domain

is done using the Fast Fourier Transform (FFT) available in Matlab 6.5. In the

frequency domain, the wave train is decomposed into incident and reflected waves

using the method presented in Section 3 with p = 3. A computer program has been

made in Matlab 6.5. The amplitude spectrum and the phase spectrum of incident

wave are plotted in Figure 2. In this figure, the amplitude and the initial phase

of the incident wave from input data are compared with that of the incident wave

estimated by the least square method. The difference in the amplitude spectrum

between them can not be visually detected. However, in the range of frequencies

where the energy-spectrum is very low, there is a large difference between the phase

of the input data and that from least square method. This large difference can be

explained by the fact that on the interval of these frequencies the power spectrum

  _{0.8 1.0 1.2} Estimation of incident and reflected waves _{Amplitude 0.0 0.2 0.4}

  0.6 ₁

₂

_{3 4 5} frequency (rad/sec) _{2 3 4} (a) The amplitude of the incident wave spectrum _{phase (rad) -4 -3 -2 -1}

  1

  1 _{frequency (rad/sec)} ^{2 3 4 5} (b) The phase of the incident wave spectrum

Input

  Recalculated using least square method

Figure 2: The amplitude and the phase of the incident wave spectrum at x =

−15.12m. Very good agreement between the input and the recalculated incident

wave spectrum using the least square method is obtained in the frequency interval

where the power spectrum is large.

  intervals, i.e. the noise-to-signal ratio is quite large. Indeed, we actually simulate waves in the interval frequencies I = (1.981, 4.027). However, the power spectrum oustide the interval I is not zero. It is clear that the error is mainly caused by the noise signal and/or the numerical error due to the FFT. Another consequence is that the estimation of the reflection coefficient and phase shift outside the ω−interval [2, 4] is subject to errors as shown in Figure 3. As mentioned above, the reflection coefficient and the phase shift should be 0.1 and 0.0, respectively. Figure 3 shows that a good agreement is only in the region where the incident spectrum is ’large’ enough. Due to this fact we neglect the analysis for the frequencies where the power spectrum is less than α % of the maximum power spectrum. As a result, the least square method only gives the reflection coefficient and the phase shift on the frequency interval (2.241, 3.735), (2.306, 3.670) , (2.339, 3.637) and (2.371, 3.605) for

A. Suryanto

  (a) Reflection coefficient (a) Reflection coefficient (b) Reflection coefficient as in (a) (b) Reflection coefficient as in (a) but viewed in small interval but viewed in small interval

  (c) phase shift (c) phase shift (d) Phase shift as in (c) (d) Phase shift as in (c) but viewed in small interval but viewed in small interval

Figure 3: The reflection coefficient and the phase shift estimated by the least square

method. In the region where the power spectrum is very low, i.e., outside ω-interval

[2, 4], the estimation of reflection coefficient and phase shift are subject to errors.

  α = 2.5, 5, 7.5 and 10, respectively. The averaged relative errors in the estimation of the reflection coefficient (ε r ) and the averaged absolute errors in the estimation of the phase shift (ε β ) on these frequency intervals are presented in Table 1 and Figure 4. We did not present the error of the estimation of the phase shift in terms of the relative error because the exact value of the phase shift is zero. We also present the averaged errors for α = 0. The averaged errors for α = 0 are only calculated within the frequency interval I. Notice that the theoretical errors are calculated using formulae presented in Section 4. From Table 1 and Figure 4 we can see that the computed errors and the theoretical errors are in good agreement. Another important observation is that the error decreases whenever we increase the value of α. This indicates that the lower power spectrum gives the higher error.

  Table 1. Averaged relative errors of r and averaged absolute error of β ε r (%) ε β

  α % theory computed theory computed 0.0 44.145 48.040 0.21443 0.21786 2.5 2.005 2.006 0.02193 0.02186 5.0 1.557 1.557 0.01909 0.01903 7.5 1.548 1.548 0.01728 0.01725

  10.0 1.242 1.242 0.01600 0.01660

  Estimation of incident and reflected waves

  Based on the estimated incident and reflected spectrum, we recomputed the incident and the reflected wave (elevation) signals as shown in Figure 1. Notice that the input incident and the reflected wave signal at this picture are plotted before the noise signal is added. On the other hand, the recomputed incident and reflected wave signals are obtained by decomposing the ”measured wave signal” which contains the noise signal. From this picture, we see that at the interval where the elevation is ’small’ there is a large difference between the input signals and the recomputed signals. It should be noted that the main source of the inaccuracy of the decomposition is due to the presence of the noise. The resultant of the wave components of the ’measured’ signals and that of the recomputed signals are plotted in Figure 5. It can be seen that the difference between the input and the recomputed signals is very small. The satisfactory agreement between the input and the estimated incident spectrum, reflection coefficient, phase shift and wave elevation shows that the least square method is a useful method to decompose the measured wave.

  (a) Relative error of (a) Relative error of r r (b) Absolute error of β (b) Absolute error of

  

Figure 4: Averaged relative errors of r and averaged absolute error of β. The

relative error of β is not calculated because the value of β is zero.

  Recalculated using Recalculated using Input Input least square method least square method

  

Figure 5: The ’measured’ wave signal and the recomputed wave signal at x =

−15.12m.

A. Suryanto

  6. CONCLUSIONS A method to estimate the incident and the reflected waves has been discussed. In this method, the beach is modelled in such a way that the reflection properties of the beach are characterized by a reflection coefficient and a phase shift. The reflection coefficient and the phase shift must be determined from measurements, i.e., by decomposing the measured wave elevations into its components. Considering a linear wave equation as the model for the free surface wave, the decomposition of the wave elevations which are obtained from some point measurements can be done by finding an approximation of the general solution of the wave equation, i.e. by implementing a least square technique.

  From the analysis, it is shown that the spacing of the wave probes has great influence on the accuracy of the results. However, the problem that corresponds to this probe spacing can be reduced by incorporating more number of probes with unequal spacing.

  The sensitivity of the methods to estimate the reflection coefficient and the phase shift with respect to the error (or noise) in the measurements has been presented. In the significant frequency interval, i.e. at frequencies where the power spectrum is large enough, a small error in the estimation of the reflection coefficient and the phase shift is obtained by the least square method.

  Acknowledgement. This work was funded by EU Indonesia - Small Projects Facility No.2004/079−057: Building Academia - Industry Partnership in the sectors of Marine and Telecommunication Technology. The author would like to thank Dr.

  Andonowati and Prof. E. Van Groesen for inviting him to joint this project. Part of this research was conducted at the Modelling and Simulation Co-Laboratory of KPPMIT, Institut Teknology Bandung, Indonesia.

  

REFERENCES

J. Brossard, A. Hemon, W. Rivoalen

1. , “Improved analysis of regular gravity waves

and coefficient of reflection using one or two moving probes”, Coastal Engrg. 39 (2000), 193–212.

  

2. Y. Goda and Y. Suzuki , “Estimation of incident and reflected waves in random wave

experiments”, Proc. 15th Coastal Engrg. Conf. 1 (1976), 828–845.

  3. E. van Groesen , Advanced modeling in science, Lecture Notes MSc. No 150910, University of Twente, The Netherlands, 1998.

  M. Isaacson

4. , “Measurement of regular wave reflection”, J. Waterway, Port, Coastal

& Ocean Engrg. , 117(6) (1991), 553–569.

  E.P.D. Mansard and E.R.Funke

5. , “The measurement of incident and reflected spec-

tra using a least squares method”, Proc. 17th Coastal Engrg. Conf. 1 (1980), 154–172.

  Estimation of incident and reflected waves

  

6. S. Nallayarasu, C. Hin Fatt, N.J. Shankar , “Estimation of incident and reflected

waves in regular wave experiments”, Ocean Engrg. 22(1) (1995), 77–86.

  

7. E.B. Thornton and R.J. Calhoun , “Spectral resolution of breakwater reflected

waves”, J. Waterway, Port, Coastal & Ocean Engrg., 98(4) (1972), 443–460.

  

8. J.A. Zelt and J.E. Skjelbreia , “Estimating incident and reflected wave fields using

an arbitrary number of wave gauges”, Proc. 23rd Coastal Engrg. Conf. 1 (1992), 777–789.

A. Suryanto : Applied Mathematical Modeling and Computation Laboratory, Jurusan Matematika, Universitas Brawijaya, Jl. Veteran Malang 65145 Indonesia.

  E-mail: suryanto@brawijaya.ac.id