JP2965245B2 - Mode eigenvalue measurement method - Google Patents

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION The present invention is, for example, with respect to wave propagation simulation method of calculating the sound propagation properties of the ocean, a environmental condition of the input, the normal mode eigenvalue necessary to estimate the speed of sound distribution in the seabed sedimentary lamination The method relates to a mode eigenvalue measuring method for measuring the characteristic value.

[0002]

2. Description of the Related Art Conventionally, techniques in such a field include:
For example, there is one described in the following literature. Literature; Acoustic. Soc. Am, 82 (3),
1987, Acoustical Society of America, (US),
SDRajan et al., `` Perturbativeinvesion methods for
obtaining bottom geoacoustic parameters in shallow
water ", p. 998-1017 The sound waves in the ocean repeat refraction in the seawater and reflections on the sea surface and the seabed, and propagate to the distant place. In particular, in the shallow ocean (hereinafter referred to as shallow sea area), sound waves that repeatedly reflect on the sea surface and the sea floor are dominant. To accurately grasp this sound field, the seafloor layer that affects the seafloor reflection loss is required. It is essential to know the speed of sound. Conventionally, the seafloor reflection loss has been estimated by emitting a pulse from a sound source and measuring the level of a seafloor reflected wave separated in time. However , in the shallow sea area, it is difficult to temporally separate the seafloor reflected waves, so another estimation method using continuous waves described in the above-mentioned document was proposed by Rajan et al. This is not a method of directly measuring the seafloor return loss, but measuring a normal mode eigenvalue (hereinafter, referred to as a mode eigenvalue) using the Hankel transform from the sound propagation characteristics in the horizontal direction, and assuming the mode eigenvalue as a sound velocity distribution (initial This is a method of estimating the speed of sound in the seabed sedimentary layer necessary for calculation of seafloor reflection loss by using the inverse problem solution method from the difference from the mode eigenvalue calculated from the value e).

A sound wave propagation characteristic in an environment where the medium characteristic such as water temperature and water depth does not change in the horizontal direction ( this is called a stratified medium) is given by the following Hankel transform pair.

(Equation 1) Here, r is the horizontal distance, k is the horizontal wave number, and J ₀ is the 0th-order Bessel function of the first kind. Now, when measuring the sound wave propagation characteristics p (r) in the horizontal direction, the wave number spectrum g (k) is calculated by equation (2), and the horizontal wave number k at which g (k) has a peak corresponds to the mode eigenvalue. The method of the said document is
This value is used to estimate the sound velocity distribution of the seabed sedimentary layer. As a method for obtaining the mode eigenvalue from the sound wave propagation characteristic p (r) , a synthetic aperture technique is used.

FIG. 2 is a functional block diagram of a mode eigenvalue measuring apparatus for implementing a conventional mode eigenvalue measuring method. The propagation characteristic measuring device 1 has a transmitter and a receiver underwater, and a sound wave having a constant angular frequency ω is output from the transmitter. The sound wave is received by the receiver, and the received signal p (r, t) is sent to the discrete Fourier transform processing unit 2. The distance r between the transmitter and the receiver is changed while the receiver is fixed and the transmitter is movable. The received signal p (r, t) received by the receiver is sent to the discrete Fourier transform processing unit 2, where the discrete Fourier transform represented by the following equation (3) is performed, and the angular frequency ω _p0 is obtained. wave propagation characteristic p is the distance characteristic of the sound field _{(r n, ω p0) (} n = 1,2, ..., n) are extracted.

(Equation 2) Here, the subscript ii of ω is a single subscript and represents the number of the sample angular frequency. Generally, omega _p0 is p (r _n, ω _ii) the maximum of the angular frequency omega _ii is used. This corresponds to the angular frequency of the received signal after the Doppler shift, and becomes a constant value if the transmitter moves in a uniform linear motion. The obtained p (r _n, ω _p0) is because the reception time _{different, p (r n, ω 0} ) = p (r n, ω p0) · exp (jω 0 r n / v) ·· (4) such that the received signal at the same time, i.e. simultaneously
The phase is corrected so that it can be regarded as the sound field observed at the time . Here, ω ₀ is the angular frequency of the sound wave output from the transmitter, v
Represents the moving speed of the transmitter.

[0005] Then, the phase-corrected sound wave Den 搬特 of p
(R _n , ω ₀ ) is sent to the Hankel transform processing unit 3 and
The Hankel transform described by equation (5) is performed and the wavenumber
The cutrum p (k, ω) is calculated.

(Equation 3) Wavenumber spectrum p (k, ω), the peak detection processing section 4
Is sent to the peak of the wave number k is detected, which is output as the mode eigenvalue.

[0006]

However, in the conventional mode eigenvalue measuring method , the wave number shifts due to the Doppler effect when the transmitter moves, especially when the moving speed is high. The peak of the wave number k corresponds to the received signal after the Doppler shift, and there is a problem that the peak deviates from the mode eigenvalue when stationary.

[0007]

According to a first aspect of the present invention, in a method for measuring a mode eigenvalue, a fixed frequency output from a transmitter having a linear motion at a constant velocity is provided. Of the sound waves received by the receiver fixed in the water, to each frequency from the frequency spectrum of multiple frequencies obtained by discrete Fourier transform .
There frequency spectrum is calculated each frequency spectrum of the frequency at which the maximum in the wave propagation characteristic calculation processing for calculating the sound propagation properties by performing phase correction so that each becomes a received signal at the same time, the discrete Fourier transform Interpolating the frequency intervals of the frequency spectra of a plurality of frequencies obtained as a result to obtain a frequency spectrum with a finer frequency interval, and a first peak detection for detecting a peak frequency of the interpolated frequency spectrum Processing, a phase compensation process for performing phase compensation of the calculated sound propagation characteristic by a phase difference caused by a difference between the detected peak frequency and the fixed frequency, and a Hankel transform of the phase-compensated sound propagation characteristic. To obtain the wave number spectrum and the peak wave number of the obtained wave number spectrum. And a second peak detection processing for outputting the peak wavenumber as a mode eigenvalues, running. According to a second aspect, in the mode eigenvalue measuring method,
Constant frequency output from a transmitter with constant linear motion
Based on a received signal of a sound wave of ω ₀ received by a receiver fixed in water, a frequency at each frequency is obtained from a frequency spectrum of a plurality of frequencies obtained by discrete Fourier transform.
Frequency spectrum of the frequency where the spectrum is maximum
, Each of which is subjected to phase correction so as to be a received signal at the same time to calculate a sound wave propagation characteristic, and a frequency spectrum of a plurality of frequencies obtained by the discrete Fourier transform. Interpolation processing to obtain a frequency spectrum of a denser frequency interval, a first peak detection processing to detect a peak frequency ω _p of the interpolated frequency spectrum, and the calculated sound wave propagation a Hankel transform processing for obtaining the wave number spectrum by the Hankel transform properties, the second peak detection process for estimating the peak wavenumber detects the peak wave number of the obtained wavenumber spectrum as the mode eigenvalue k _m ', the constant frequency - omega _0, the detected peak frequency
Number omega _p, and based on the mode eigenvalue k _m ', the correction value
To calculate the _{Δk m = k m '· (} ω 0 -ω p) / ω p, 該補
A mode eigenvalue correction process to output by correcting the peak wave number by positive .DELTA.k _m, running.

[0008]

According to the first aspect of the present invention, since the mode eigenvalue measuring method is configured as described above, the frequency interval of the frequency spectrum of a plurality of frequencies obtained by the sound wave propagation characteristic calculation processing is determined by the sinc interpolation method and the Bessel interpolation method. Interpolation is performed by a method or the like, and a frequency spectrum having a denser frequency interval is obtained. From the interpolated frequency spectrum, a peak frequency is detected in a first peak detection process. Since the sound wave propagation characteristic calculated in the sound wave propagation characteristic calculation process corresponds to the Doppler-shifted frequency due to the motion of the transmitter, the difference between the detected peak frequency and the frequency of the sound wave from the transmitter is calculated. After phase compensation of the sound wave propagation characteristic is performed by the phase difference, Hankel transform is performed. The wave number spectrum obtained by the Hankel transform corresponds to the sound wave output at a constant frequency while the transmitter is stationary, the peak wave number of the wave number spectrum is detected in the second peak detection processing, and the mode eigenvalue Is output. According to the second aspect, the frequency spectra of a plurality of frequencies obtained by the sound wave propagation characteristic calculation processing are interpolated, and the peak frequency ω _p of the interpolated frequency spectrum is detected in the first peak detection processing. Wave propagation characteristic calculated by the wave propagation characteristic calculation process, Hankel transform has been wavenumber spectrum is obtained, the peak wave number of the wave number spectrum is detected mode eigenvalue k _m 'is estimated by the second peak detection processing . By mode eigenvalue correction processing
The constant frequency ω ₀ , the peak frequency ω _p , and the
It is detected correction value .DELTA.k _m from over de fixed value k _m ', the complement
The peak wavenumber is output is corrected by the positive value .DELTA.k _m.

[0009]

EXAMPLES First Embodiment FIG. 1 is a functional block diagram of a mode eigenvalue measuring apparatus for carrying out the mode eigenvalue measuring method of the first embodiment of the present invention. The difference between the mode eigenvalue measuring method of the first embodiment and the conventional mode eigenvalue measuring method is that the wave number shifts due to the Doppler effect when the transmitter is moved, and thus the mode eigenvalue measurement method corresponds to a sound wave in a state where the transmitter is stationary. to the wave propagation characteristics, after compensating the phase for the wave propagation characteristic p corresponding to a frequency by Doppler shift (r _n, ω _0), is that which is adapted to the Hankel transform process. As shown in FIG. 1, the mode eigenvalue measuring apparatus of the first embodiment
Propagation characteristic measuring device 11, discrete Fourier transform processing unit 12,
It comprises an interpolation processing unit 13, a peak detection processing unit 14, a phase compensator 15, a Hankel conversion processing unit 16, and a peak detection processing unit 17. The propagation characteristic measuring device 11
It has a transmitter and receiver in water. The transmitter moves linearly at a constant speed, and the receiver is fixed. The output side of the propagation characteristic measuring device 11 is connected to a discrete Fourier transform processing unit 12 that performs a discrete Fourier transform on the received signal. The output side of the discrete Fourier transform processing unit 12 is connected to an interpolation processing unit 13 for interpolating a frequency spectrum. The output side of the interpolation processing unit 13 is connected to a peak detection processing unit 14 that detects a peak frequency from the interpolated frequency spectrum. Discrete Fourier transform processing unit 12
A phase compensator 15 for phase-compensating the sound wave propagation characteristic is connected to the output side of the peak detection processor 14.
The output side of the phase compensator 15, Hankel transform processing section 16 for calculating a wave number spectrum is connected to Hankel transform the phase compensated wave propagation characteristics, the more the output side of the Hankel transform processing unit 16, a wave number spectrum The peak detection processing unit 17 for calculating the wave number of the peak is connected. Hereinafter, with reference to FIG. 1, the processing content of the mode eigenvalue measuring method according to the first embodiment of the (A) ~ (F) Ru theory Aquiraz.

[0010] The (A) wave propagation property calculation processing channel measurement device transmitters that a uniform linear motion of 11, waves are output at a constant angular frequency omega _0. This sound wave is received by the receiver, and a time-series received signal p
(R, t) are sequentially sent to the discrete Fourier transform processing unit 12. Here, r is the horizontal distance between the transmitter and the receiver, and t is the time. In the discrete Fourier transform processing unit 12, by sampling received signals p (r, t) at a predetermined sampling period, have rows discrete Fourier transform shown in equation (3), the frequency of the plurality of constant angular frequency omega _ii spectrum p (r _n, ω
Newsletter _ii), and sends the frequency spectrum p (r _n, the omega _ii) to the interpolation processing unit 13. And the frequency spectrum p
(R _n, omega _ii) the frequency spectrum p (r _n, omega _p0) of the angular frequency omega _p0 becomes maximum after obtaining, as for formula (4), the frequency so that the received signal at the same time spectrum p (r _n, ω _p0) distance of the sound field by performing a phase compensation of
Wave propagation characteristic p (r _n, ω ₀₎ is the characteristic to obtain, and sends the wave propagation characteristics p (r _n, ω ₀₎ in position phase compensator 15. In this way, the difference between any reference time and the reception time
The phase change of the transmitted signal during the time corresponding to
be able to. (B) In the interpolation processing interpolation processing unit 13, for performing the measurement accuracy of the received signal frequency, the discrete Fourier transform processing unit 12 frequency spectrum p of the sent from _{(r n, ω ii) (} ii = 1,2 , ..., from II), sinc interpolation Houma other by using a Bessel interpolation method, the circumferential
Wavenumber spectrum p (r _n, ω _ii) is interpolated data of the angular frequency around the angular frequency omega _p0 having the maximum and sends it to the peak detection processing section 14. By performing such interpolation processing,
The accuracy of the peak detection process in the peak detection processing unit 14
Can be improved. Also, this interpolation method alone is not enough.
Input data input to the discrete Fourier transform processing unit 12
Data length by adding 0 data to the
Interpolation can be realized by Fourier transform
Noh. (C) first the peak detection processing peak detection processing section 14, the distance interval r _{n (n} = 1,2 performing Hankel transform processing the interpolated frequency spectrum,
, N) to calculate a frequency peak ω _p and output it to the phase compensator 15.

[0011](D)Phase compensation processingReason Angular frequency ω₀The sound wave output from the transmitter at
-Angular frequency ω due to the effect of_pIs received by mode
The eigenvalue changes in proportion to the angular frequency (mode eigenvalue
Is the angular frequency divided by the speed of sound).
Mode eigenvalue k when the wave is stationary_m When, Transmitter
Mode eigenvalue k when is moving_m'Between
Next approximationEquation (6) holds. Δk_m= K_m-K_m'= (Ω₀/ C₀) ・ (Ω₀−ω_p) / Ω₀ ・ ・ (6) where C₀Is the speed of sound at the depth of the transmitter. formula
The sound wave propagation characteristic p (r _n , Ω₀) Han
The peak wave number of the wave number spectrum after Kel transform is k_m''
Therefore, the mode eigenvalue k when stationary_mTo get
Has the wave number spectrum of equation (5) as Δk_mOnly translation
Any wave number spectrum that has moved may be used.This is a sonic transmission
The transfer characteristic p (r _n , Ω ₀ ) In advance _m
・ R _n It is sufficient to provide a phase rotation corresponding to. formula
From (6), Δk _m Is almost constant
Then, this is set as Δk.Therefore, the phase compensator 15
Is the sound propagation characteristic p (r_n, Ω₀) To exp (jΔk ・
r_n) And sends the result to the Hankel transform processing unit 16.(E) Hankel transformationReason In the Hankel transform processing unit 16, the phase-compensated sound wave propagation
The characteristics are transformed by Hankel transform in the same manner as in equation (5), and the
The spectrum is obtained and sent to the peak detection processing unit 17.(F) Second peak detection processingReason In the peak detection processing unit 17, the peak of the wave number spectrum isK
ToTo detect. The peak detected here is Doppler
Mode is phase-compensated, so the mode
Value k_mIs obtained. As described above, the first actual
According to the example,Discrete Fourier-transformed frequency spectrum
M p (r _n , Ω _ii ) To interpolate the frequency peak ω _p Detect
Out, transmitter From the frequency of the sound wave output from the
Calculate the amount of Doppler shift from
Horizontal sound wave propagation characteristic p (r _n , Ω ₀ )of
Perform Hankel transform after phase compensationdid
Removes Doppler shift due to transmitter movement
Mode eigenvaluek _m TodetectionCanYou.

Second Embodiment FIG. 3 is a functional block diagram of a mode eigenvalue measuring apparatus for performing a mode eigenvalue measuring method according to a second embodiment of the present invention. Elements common to the elements are denoted by common reference numerals. The mode eigenvalue measurement method of the second embodiment differs from the mode eigenvalue measurement method of the first embodiment in that the frequency of the received signal shifts due to the Doppler effect due to the movement of the transmitter.
Then , after the peak wave number corresponding to the received signal subjected to the Doppler shift is obtained, the deviation of the mode eigenvalue due to the Doppler shift is corrected. As shown in FIG. 3, the mode eigenvalue measuring device is a propagation characteristic measuring device 1
1, a discrete Fourier transform processing unit 12, an interpolation processing unit 13, a peak detection processing unit 14, a Hankel transform processing unit 16, a peak detection processing unit 17, and a mode eigenvalue correction unit 18. Discrete output side of the propagation characteristics measuring device 11
Fourier transform processing unit 12 is connected to the output side of the discrete Fourier transform processing unit 12, the interpolation processing unit 13 and the Hankel transform processing unit 16 is connected. A peak detection processing unit 14 is connected to an output side of the interpolation processing unit 13. A peak detection processing unit 17 is connected to an output side of the Hankel conversion processing unit 16. The output side of the peak detection processing units 14 and 17 is connected to a mode eigenvalue correction device 18 that corrects the mode eigenvalue by the Doppler shift. Hereinafter, with reference to FIG. 3, the processing content of the mode eigenvalue measuring method of the second embodiment of (a) ~ (f) Ru theory Aquiraz.

[0013] The (a) wave propagation property calculation processing channel measurement device transmitters that a uniform linear motion of 11, waves are output at a constant angular frequency omega _0. This sound wave is received by the receiver, and a time-series received signal p
(R, t) are sequentially sent to the discrete Fourier transform processing unit 12. In the discrete Fourier transform processing unit 12, the received signal p
(R, t) is sampled at a predetermined sampling period, have rows discrete Fourier transform shown in equation (3), a plurality of one
Frequency spectrum p (r _n of the constant angular frequency ω _ii,
to obtain omega _ii), the frequency spectrum p (r _n, ω _ii)
A letter to the interpolation processing unit 13. On the other hand, the frequency spectrum p
(R _n, omega _ii) the angular frequency becomes the maximum omega that put the _p0-frequency <br/> number spectrum p (r _n, omega _p0), since the reception time are different, like the equation (4) , performs phase compensation so that the received signal at the same time, sound wave propagation characteristics p (r, ω ₀₎
To the Hankel transform processing unit 16. In (b) the interpolation processing interpolation processing unit 13, for performing the measurement accuracy of the received signal frequency, the discrete Fourier transform processing unit 12 frequency spectrum p of the sent from _{(r n, ω ii) (} ii = 1,2 , ..., from II), sinc interpolation Houma other by using a Bessel interpolation method, the circumferential
Wavenumber spectrum p (r _n, ω _ii) is interpolated data of the angular frequency around the angular frequency omega _p0 having the maximum and sends it to the peak detection processing section 14. (C) In the first peak detection processing peak detection processing section 14, the distance interval r _{n (n} = 1,2 for performing Hankel transform processing the interpolated frequency spectrum,
.., N) to calculate the frequency peak ω _p and output it to the mode eigenvalue correction device 18.

The (d) The in Hankel transform processing Hankel transform processing unit 16, the sound wave propagation characteristic p (r _n,
ω ₀ ) is Hankel-transformed to calculate a spectrum in the wave number space, and sends it to the peak detection processing unit 17. Peak wave (e) the second peak detection processing peak detection processing section 17, a wave number spectrum p (k, ω ₀₎ sent from the Hankel transform processing unit 16
The number is detected and sent to the mode eigenvalue correction device 18. Here the peak wave number obtained is the same Doppler shifted mode eigenvalue k _m 'and conventional. (F) sound wave output from the wave transmitter mode eigenvalue correction processing angular frequency omega ₀ is received at the angular frequency omega _p due to the influence of the Doppler effect. Mode eigenvalue, since changes in proportion to the angular frequency, between the mode eigenvalue k _m in the case of wave transmitter and wave receiver are stationary, the Doppler shifted mode eigenvalue k _m ', the following < The relational expression (7) holds. _{Δk m = k m -k m '} = k m' · (ω 0 -ω p) / ω p ··· (7) Therefore, the mode eigenvalue correction device 18, calculates a correction value .delta.k _m from the equation (7) and, the correction value Δk in the mode eigenvalue k _m 'is the peak wave number detected by the peak detection processing section 17
by adding the _m, determine the mode eigenvalue k _m removal of the de Doppler shift amount. As described above, according to the second embodiment, the frequency spectrum subjected to the discrete Fourier transform is used.
Beam p (r _n, ω _ii) by interpolating the detected a peak ω _p of frequency
Out of the frequency of the sound wave output from the transmitter
Calculate the amount of Doppler shift from
Since to carry out the wavenumber of the peak corrected according to the amount, Ru can detect mode eigenvalue k _m removal of the Doppler shift caused by the movement of the transmitters.

[0015]

As described above in detail, according to the first aspect, the peak frequency is detected by interpolating the frequency spectrum subjected to the discrete Fourier transform, and the phase difference due to the difference between the peak frequency and the fixed frequency is detected. After performing phase compensation of sound wave propagation characteristics, the wave number spectrum is obtained by performing Hankel transform, the peak wave number is detected, and the mode eigenvalue is output, so the Doppler shift due to the movement of the transmitter is removed. Mode eigenvalue can be detected.
According to the second invention, it detects the peak frequency omega _p by interpolating the frequency spectrum obtained by discrete Fourier transform, the peak
Ω _p , constant frequency ω ₀ , and mode eigenvalue
It calculates a correction value .DELTA.k _m based on k _m ', the correction value .DELTA.k
_{By m} , the peak wave detected in the second peak detection processing
Since the number is corrected, the mode eigenvalue from which the Doppler shift due to the movement of the transmitter has been removed can be detected as in the first invention.

[Brief description of the drawings]

FIG. 1 is a functional block diagram of a mode eigenvalue measuring device according to a first embodiment of the present invention.

FIG. 2 is a functional block diagram of a conventional mode eigenvalue measuring device.

FIG. 3 is a functional block diagram of a mode eigenvalue measuring device according to a second embodiment of the present invention.

[Explanation of symbols]

11 channel measurement device 12 discrete Fourier transform processing section 13 interpolation processing section 14, 17 a peak detection processing section 15 phase compensator 16 Hankel transform processing unit 1 8 mode eigenvalue correction device

──────────────────────────────────────────────────続 き Continuing on the front page (72) Inventor Toru Morishita 1-7-12 Toranomon, Minato-ku, Tokyo Oki Electric Industry Co., Ltd. (72) Inventor Shunji Ozaki 1-7-112 Toranomon, Minato-ku, Tokyo Oki Electric Industry Co., Ltd. (58) Field surveyed (Int. Cl. ⁶ , DB name) G01H 3/00 G01S 3/82 G01S 15/88

Claims (2)

(57) [Claims] 1. A discrete Fourier transform is performed on a sound wave of a constant frequency output from a transmitter having a constant linear motion based on a received signal received by a receiver fixed in water. All from the frequency spectrum of a plurality of frequencies to each frequency
There frequency spectrum is calculated each frequency spectrum of the frequency at which the maximum in the wave propagation characteristic calculation processing for calculating the sound propagation properties by performing phase correction so that each becomes a received signal at the same time, the discrete Fourier transform Interpolation processing for interpolating the frequency intervals of the frequency spectrums of a plurality of frequencies obtained as described above to obtain a frequency spectrum of denser frequency intervals, and a first peak detection for detecting a peak frequency of the interpolated frequency spectrum Processing, phase compensation processing for performing phase compensation of the calculated sound wave propagation characteristic by a phase difference due to a difference between the detected peak frequency and the fixed frequency, and Hankel transform of the phase-compensated sound wave propagation characteristic. To obtain a wave number spectrum, and a peak wave number of the obtained wave number spectrum. Second and peak detection processing mode eigenvalue measuring method characterized by performing the detecting and outputting the peak wavenumber as a normal mode eigenvalue. 2. A discrete Fourier transform is performed on a sound wave having a constant frequency ω ₀ output from a transmitter having a constant linear motion based on a reception signal received by a receiver fixed in water. Each frequency is obtained from the obtained frequency spectrum of multiple frequencies.
Frequency spectrum calculated respectively frequency <br/> number spectrum of frequencies with a maximum at a wave propagation characteristic calculation processing for calculating the sound propagation properties by performing phase correction so that each becomes a received signal at the same time Interpolating the frequency intervals of the frequency spectrum of a plurality of frequencies obtained by the discrete Fourier transform to obtain a frequency spectrum of a denser frequency interval; and a peak frequency ω _{p of the} interpolated frequency spectrum.
A first peak detection process of detecting the calculated wave number spectrum, a Hankel conversion process of obtaining the wave number spectrum by performing the Hankel conversion of the calculated sound wave propagation characteristic, and detecting the peak wave number of the calculated wave number spectrum to calculate the peak wave number. a second peak detection process for estimating the normal mode eigenvalue k _m ', the fixed frequency omega _0, the peak frequency omega of the detected
_p, and the basis of the normal mode eigenvalue k _m ', complement
To calculate the positive value _{Δk m = k m '· (} ω 0 -ω p) / ω p,
Mode eigenvalue measuring method characterized by a mode eigenvalue correction process to output by correcting the peak wave number by the correction value .DELTA.k _m, executes.