Investigation of hull pressure fluctuations generated by cavitating vortices

Transcription

1 First International Symposium on Marine Propulsors smp 9, Trondheim, Norway, June 9 Investigation of hull pressure fluctuations generated by cavitating vortices Johan Bosschers Maritime Research Institute Netherlands (MARIN), Wageningen, The Netherlands ABSTRACT The paper describes an ongoing research program at MARIN for the investigation of low frequency hull pressure fluctuations caused by cavitating vortices. Model scale measured hull pressure fluctuations generated by a cavitating vortex on a marine propeller have been analyzed with respect to variations between blade passages. It is shown that these variations have a significant influence on the amplitude spectrum from which it is concluded that the broadband energy content of the spectrum is just as important as the harmonics of blade passage frequency. The collapse and rebounds of the cavitating vortex show very distinct frequency components. These frequencies are reasonably well predicted by a theoretical formulation, obtained by applying the method of stationary phase to the dispersion relation for waves on cavitating vortices. Keywords Propellers, vortex cavitation, hull pressure fluctuations, Kelvin waves, dispersion relation. INTRODUCTION The dynamic behavior of cavities on or in the vicinity of ship propeller blades causes pressure fluctuations which excite the hull structure above the propeller. Since these pressure fluctuations are largely in phase across the aft body surface, cavitation is very effective in generating inboard noise and vibration. During the last decades, the cavitation-induced hull pressure forces have been reduced considerably, leading to strongly reduced inboard noise and vibration levels at harmonics of the blade passage frequency (BPF). With the reduction at the harmonics of BPF, the broadband spectral content of the pressure fluctuations has become important as well for ships with stringent noise and vibration requirements. The vortex type of cavitation is said to be the cause of the broadband hull pressure fluctuations but a detailed understanding is missing. The hull pressure signals are usually analyzed by the Fast Fourier Transform (FFT) from which an amplitude spectrum is generated. As the limitations of the FFT for non-stationary and non-linear signals are well known, the importance of analyzing the signal in the time domain has been stressed by e.g. Carlton and Fitzsimmons (6). For quantitative analyses of the data, however, it is still the amplitude spectrum by FFT that is being used. One of the limitations of FFT is that smearing of power of a tonal signal occurs over a range of frequencies due to stochastic variations of the single acoustic event, where an event is defined as the cavity dynamics during one blade passage through the ship wake peak. A procedure aimed to quantify the stochastic variations is discussed in Section., while the pressure signals for a single blade passage are discussed in Section.. The procedures are being explored in order to investigate the tip vortex as a source of hull pressure fluctuations and for the interpretation of the pressure fluctuations from a hull excitation point of view. The theoretical prediction of resonance frequencies of cavitating vortices is realized using the dispersion relation for perturbation waves on the cavitating core as first derived by Lord Kelvin, Thomson (88) and extended with compressibility effects by Morozov (974). In Section 3 an extension and further analysis of this dispersion relation is presented including a comparison with experimental data. ANALYSIS OF HULL PRESSURE SIGNAL. Stochastic aspects The hull pressure signals are usually analyzed with an FFT resulting in a spectrum. If the cavity dynamics of each blade passage would be perfectly repetitive one would only see (tonal) values in the spectrum at orders of BPF. In reality, however, significant variability occurs due to geometric variations between the propeller blades, temporal variations in the ship s wake field and variability of the cavitation dynamics itself. All variations lead to a broadband character of pressure pulses in the frequency domain. The variations in amplitude and phase (or frequency) of the signal between different blade passages can be interpreted as amplitude and phase modulation. The influence of these modulation effects on the power spectrum is shown in Figure using theoretical formulations of MacFarlane as presented by Baiter (98) and further discussed by Bark (988). A schematic

2 power spectrum power spectrum power spectrum frequency no modulation pulse train single pulse frequency amplitude modulation phase modulation modulated pulse train single pulse modulated pulse train single pulse frequency Figure : Influence of phase and amplitude modulation of a pulse train on the spectrum levels. spectrum is sketched of a single pulse together with the spectrum of a train of these pulses. The frequency is nondimensionalized with the pulse train frequency and the spectrum of the single pulse is corrected in magnitude for the number of pulses in the pulse train. Variations in amplitude between the pulses by random amplitude modulation do not affect the amplitudes of the tonals at orders of the pulse train frequency. However, the spectrum does now contain a broadband part of which the magnitude is proportional to the standard deviation of the amplitude variations. Random phase modulation decreases the amplitude of the tonals and distributes the power over a broadband region. The spreading is most pronounced at higher frequencies and may cause the disappearance of the tonals at higher harmonics. The stochastic variations of the pressure signal of a cavitating vortex are investigated for a two bladed skewed propeller operating in the wake field of a twin screw vessel investigated within an internal research project of MARIN. A two-bladed propeller variant was used instead of the original four-bladed propeller to avoid interference between hull pressure signals generated by different blades as much as possible. The measurements were made in the Depressurized Towing Tank and the presented pressure signal was measured directly above the propeller. The propeller blades were operating near design thrust value [Pa] P/ time [/rev] Figure : Definition of amplitude A and period P for the stochastic analysis (crest-trough value). coefficient per blade, corresponding to a thrust coefficient of K T =.7 for the four-bladed propeller. The cavitation number was σ n =.3. The cavitation pattern was characterized by a strong leading edge vortex of which images will be presented later. The stochastic variations of hull pressure fluctuations from blade to blade and from revolution to revolution are analyzed through a simple procedure illustrated in Figure. The time traces are first divided into ZN segments with Z the number of blades and N the number of shaft revolutions. Each segment or blade passage period is then searched for the maximum crest-trough value of neighboring extremes. The corresponding period P (twice the time between crest and trough), and amplitude A (half of the crest-trough value) are stored for each segment. A statistical analysis is then performed showing how the crest-trough amplitude and period (or frequency) varies with shaft revolution. The method has a disadvantage in that it is biased towards the high-frequency components (riding waves will reduce the crest-trough value of smaller frequencies). This can probably be solved by using a decomposed time signal through application of a band-pass filter or the Empirical Mode Decomposition proposed by Huang et al. (998). As a first step, it was however preferred to use the original measurement signal without any modification. The amplitude spectrum is presented in Figure 3 in nondimensional form. The hull pressure measurements are synchronized with the shaft rotation speed and an integer number of shaft revolutions was used for the length of the FFT such that blade passage frequencies would exactly match a FFT frequency value. The large components of the tonals at first and second BPF are mainly due to the thickness and loading of the blades. The spectrum is furtheharacterized by tonals at harmonics of BPF and a hump with maximum value close to the seventh BPF. For the original four bladed propeller this hump would have its maximum between the third and fourth BPF. The results of the stochastic analysis are presented in Figure 4 and Figure 5. Results are normalized such that the summation of the probability over all bins equals one. The probability thus represents the fraction of total blade passages for which the frequency or amplitude is within the bin width. The results were generated using 56 shaft revolutions. A

3 Kp, amplitude -3 probability (blade passage) frequency/blade passage frequency Kp, max amplitude Figure 3: Amplitude spectrum of the pressure fluctuations. probability (blade passage) frequency / blade passage frequency Figure 4: Probability distribution of the period P with a fitted log normal distribution. The frequency distribution shows large values near the seventh BPF, at the centre frequency of the hump in the amplitude spectrum. The results of the frequency and amplitude distribution both fit well with a log normal distribution. The mode or most frequent amplitude is near Kp=.4, which is more than a factor ten larger than the maximum amplitude in the spectrum for frequencies above the third BPF. No correlation could be found between the probability distribution of the frequency and the amplitude. The diagram with the probability of exceedance of the maximum amplitude shows that there is % chance that the maximum amplitude will exceed a value of Kp=.4. It is these large amplitudes that may cause noise and vibration hindrance on board. As the difference between the blades is very small, the stochastic effects are attributed to random variations of velocities in the wake field and cavitation susceptibility. At full scale tests these variations are expected to be smaller, but other mechanisms are usually present such as ship motions, rudder deflections and rpm changes. The probability of exceedance (blade passage) data fit Kp, max amplitude Figure 5: Probability distribution of the amplitude A with a fitted log normal distribution (top) and the corresponding probability of exceedance diagram (bottom). procedure presented is a first step towards more quantitative knowledge on these stochastic variations. The analysis shows that the phase (and frequency) modulation especially influences the higher harmonics of the blade passage frequency (tonals) in the spectrum causing a broadband hump. It can thus be concluded that a comparison between model scale and full scale for the higher harmonics should include the broadband part of the spectrum, for instance by integration to one-third-octave band levels.. Analysis of a single blade passage Having discussed the stochastic variations between blade passages, we now investigate the hull pressure signal for a single blade passage. A time trace is presented in Figure 6 in which it is seen that the maximum amplitude equals Kp=.37 which is larger than the mode presented in Figure 5. The presented time trace shows multiple peaks indicating the presence of multiple rebounds of the cavitating vortex.

4 4 Wavelet analysis frequency/blade passage frequency Kp.6 b d a c blade position [deg] (a), 54.5 deg blade position (b), 7 deg blade position.5. Kp, amplitude.5..5 (c), 85.7 deg blade position (d),.5 deg blade position Figure 7: High speed video images of the collapse of the cavitating vortex. Image (a) though (d) correspond to the annotations in the time trace in Figure frequency/blade passage frequency Figure 6: Analysis of a single blade passage. Top: timefrequency analysis using a Morlet wavelet with time trace. Bottom: FFT- spectrum. The results of a wavelet analysis presented in Figure 6 shows that the frequency of the pressure signal caused by the cavitating vortex increases with time. At the first collapse the frequency is approximately 5.5 times the blade passage frequency and it increases to approximately 8 times the blade passage frequency. The corresponding FFT spectrum is also presented in Figure 6. The spectrum is generated by zero-padding the time trace resulting in a non-dimensional frequency spacing of.44 and increasing the amplitude spectrum by the number of blade passage periods that fits in the extended time trace. It is observed that the amplitude in the spectrum of Figure 6 between four and eight times the blade passage frequency is approximately a factor ten larger than in the spectrum presented in Figure 3. The period that is selected for the maximum crest-trough value depends on which of the collapse and rebounds produces the largest amplitude. This may vary for each blade passage which also contributes to the variability of the period as presented in Figure 4. Four high speed video images showing the character of the cavitating vortex are presented in Figure 7. The cavity pattern is characterized by a strong cavitating leading edge vortex which sometimes develops as a small sheet with a re-entrant jet oriented parallel to the leading edge forming a cavitating vortex structure. During the first collapse, between image (a) and image (b), multiple cavitating vortices are generated coming from the tip and the re-entrant jet of the leading edge vortex. These vortices interact causing rebounds as observed in image (c) and image (d). 3 THEORETICAL ASPECTS OF THE DYNAMIC BEHAVIOR OF CAVITATING VORTICES 3. Derivation of a dispersion relation For the theoretical analysis of resonance frequencies use can be made of the dispersion relation for an inviscid cavitating vortex as originally derived by Lord Kelvin, Thomson (88). The incompressible formulation of Lord Kelvin has been extended to compressible flow for acoustic analysis by Morozov (974). A further extension with a constant axial free-stream velocity component is presented by Bosschers (8), which also includes a discussion on radiated noise aspects. A small review of the theory is presented here followed by an extension to correct for viscous effects and a discussion on its relevance. The starting point for the derivation of the dispersion relation is the convected Helmholtz equation for a disturbance velocity potential ϕ%. A cylindrical coordinate system ( r, θ, z) is adopted with a harmonic variation of the disturbance potential given by

5 π f / W Dispersion diagram for σ V =. n= + n= - n= + n= - n= + n= - Figure 8: Visualisation of the spatial deformation of a cavitating vortex for the modes n=, and at one time-step. ( ) i( k ) z z+ nθ ωt, % ϕ = φ r e () in which kz corresponds to the axial flexural wave number, n the azimuthal wave number which must be an integer and ω the angular frequency. The distortion of the cavitating vortex with average radius is described by a number of modes characterized by again k z, n, ω and an amplitude ˆr. The local cavity radius η is given by i ( k z + n θ ω t ) z η = r + rˆe. () c The mode n = corresponds to a breathing mode and involves volume variations. Mode n = corresponds to a serpentine mode, also called bending mode, helical mode or displacement mode as it is the only mode which leads to a displacement of the vortex centre line. The mode n = is the bell mode or double helix or fluted mode and leads to an elliptical shape of the vortex core. A visualization of the modes is given in Figure 8. The distortions are transverse propagating inertial waves and are often referred to as Kelvin waves, Saffman (995). The unperturbed velocities in the cylindrical coordinate system are given by: Γ U = ( U, V, W ) =,, W (3) π r in which the azimuthal velocity component due to a -D vortex filament is used. Combination of the linearized kinematic and dynamic boundary condition with the solution of the convected Helmholtz equation leads to a dispersion relation which involves Hankel functions. However, for small phase speeds or low frequencies as in the range of hull pressure fluctuations discussed here, the propagating wave in radial direction becomes an evanescent wave and the Hankel function reduces to the MacDonald function. Sound is then radiated due to the finite length of the vibrating cavity and it can be shown that the maximum pressure amplitudes of different modes n scale as ( k r ) n c, with k the acoustic wave number k = ω c. For the low frequencies and cavity diameters c g / W considered, always ( k ) <<, hence the higher order modes are not expected to contribute to the radiated noise and only the breathing mode n = is expected to be important. The potential flow outside the cavitating vortex allows the application of the Bernoulli equation and the azimuthal velocity at the cavity can be related to the difference between the free-stream pressure and vapour pressure. When introducing the cavitation number σ V, the following formulation is found for the mean nondimensional azimuthal velocity at the cavity radius Vc p pv = Γ = = σ. V (4) W π r W.5ρW c The non-dimensional form of the dispersion relation for low frequencies is then given by ω r κ K κ ω κ σ n ( ) ( κ ), c, = = + V n ±, W K n in which a non-dimensional wave number κ = kzrc has been introduced and K corresponds to the MacDonald n κ σ V = κ n= + n= - n= + n= - n= + n= - Figure 9: The dispersion diagram (top) for a cavitating vortex and the corresponding group speed (bottom) as a function of the non-dimensional wave number (5)

6 function for mode n. The dispersion relation gives only real values for the frequency and the distortions are therefore neutrally stable. Each mode contains two frequencies corresponding to the plus and minus sign in the right-hand-side. In the following, this sign is also used to identify the mode. The variation of the non-dimensional frequency ω with non-dimensional axial wave number at prescribed cavitation number is plotted in Figure 9. For further understanding of this dispersion diagram, the variation of the group speed is given as well. It is seen that the frequency variation at all modes becomes small for small wave numbers. Except for both breathing modes, the group speed becomes constant at small wave numbers, indicating asymptotic convergence to a constant frequency for decreasing wave number. The frequency of the two breathing modes becomes zero for vanishing wave number. The negative frequency and group speed observed for mode n = indicate that the distortions are propagating in opposite direction to the free-stream velocity. The distortions for mode n = always propagate at a speed close to the free-stream flow velocity. As the group speed also has a value close to the free-stream velocity, it is concluded that dispersion effects are very small for this mode. At smaller wave numbers, the frequency becomes zero and deformations for this mode become stationary and can easily be observed in cavitation tests. This mode is excited in vortex-vortex interaction, Saffman (995), and can also be identified in Figure 7d for example. It is seen that there is a coinciding frequency for the modes n = + and n = which may cause an interaction between them. The mode n = also has a zero frequency (and hence zero phase speed) which corresponds to a stationary deformation pattern. The significance of these points needs to be further investigated. Most interesting however, is that there is a wave number for which the group speed of the mode n = becomes zero. This condition corresponds to the dominant wave number in an asymptotic analysis for large time using the method of stationary phase, Whitham (974), and it shows the existence of a dominant frequency component. The corresponding frequency is negative indicating that the distortions are propagating upstream. Important to note is that this mode has a high acoustic efficiency. For larger cavitation numbers the group speeds of the modes n = and n = may become zero as well. 3. Correction for viscous effects The resonance frequency obtained by using the zero group speed condition of Eq. () is valid for potential flow only. Comparison of this frequency with experimental data already gives a reasonable correlation, Bosschers (8), but it was expected that further improvement could be obtained by inclusion of viscous effects. Instead of performing a full 3D stability analysis for viscous flow a simple correction method is used for the azimuthal velocity at the cavity radius given by Eq. (8). An velocity radius analytical formulation for a D axisymmetric cavitating vortex in viscous flow has been developed, Bosschers (9), which is an extension of the formulation of a (non-cavitating) Lamb-Oseen vortex. The azimuthal velocity at radial location r is given by ( ) v r ( r rc ) Γ r v = exp ζ, π r rv + ζ rc rv (6) where r v corresponds to the radius of the viscous core, to the radius of cavitating core and ζ is a constant such that the maximum azimuthal velocity for a non-cavitating vortex occurs on the radius of the viscous core, ζ =.564. This equation satisfies the viscous boundary condition of zero shear stress at the cavity surface. The equation can be used to derive an analytical expression for the pressure which is a function of the same parameters and which is given in Bosschers (9). The equations for the azimuthal velocity and pressure can be combined such that the non-dimensional azimuthal velocity becomes a function of cavitation number and the ratio of the parameters r v and which is used instead of Eq. (4). It is expected that this simple correction procedure for the resonance frequency is only valid for r r >. 3.3 Comparison with experiment The theoretical formulation for the resonance frequency using the zero group speed condition is compared with experimental data of Maines & Arndt (997). Resonance frequencies were found in the radiated noise of cavitating vortices generated by wings in cavitation tunnels. Different hydrofoils of elliptical planform were tested in two different cavitation tunnels, one in Obernach, Germany and one at St. Anthony Falls Laboratory (SAFHL), Minneapolis, USA. In both tunnels sound was radiated at a very distinct frequency component for a small range of cavitation numbers which could be related to the cavitating vortex through high speed video. This singing only occurred when the cavitating vortex was c v = =.5 =. =.5 =. inviscid Figure : Variation of the azimuthal velocity with radius for a D cavitating vortex

7 π f / W Exp c g = (p) c g = (v) / σ v Figure : Comparison of the non-dimensional resonance frequency between the experimental data of Maines & Arndt (997), symbols, and the zero group speed curves for potential flow (p) and with viscous correction (v). attached to the blade surface. The frequency was found to vary between 4 Hz and. khz, depending on tunnel velocity and lift coefficient. The viscous correction has been applied by selecting the vortex circulation such that the cavity diameter equals 4 mm for σ v = which are the mean values of the experimental data. The viscous core size was obtained by scaling data of Arndt (99) to the Reynolds number of the Obernach measurements using the formulation for a flat plate turbulent boundary layer. This results in a viscous core diameter of.8 mm. The cavity diameter was then varied between.5 and 7 mm and the corresponding relation between azimuthal velocity at the cavity radius and cavitation number was used instead of Eq. (4). The variation of the non-dimensional frequency with cavitation number is presented in Figure. It is seen that the experimental data falls between the zero group speed curves for potential flow and with viscous correction. The zero group speed condition may also be used to define a resonance frequency for the collapse and rebounds of the cavitating vortex described in Section. The wavelet analysis presented in Figure 6 provides the resonance frequency at the four time steps (a) through (d). The cavity diametean be obtained from the images presented in Figure 7 and the non-dimensional frequency can thus be computed. The results are presented in Table and show that the agreement with the zero group speed condition in Figure is only reasonable if the minimum diameter is used. Similar to the procedure used by Raestad (996), the dispersion relation can be given as a function of propeller parameters. The vortex strength can be written as a function of the propeller thrust coefficient K by τ KT nsd Γ =, Z T (7) Table : Review of estimated cavity diameters in figure 7 and corresponding non-dimensional frequencies. Cavitation number equals σ V =.. For image (b) and (c) the maximum and minimum diameters are given. Image diameter (mm) π f rc W (a) 9.4 (b), max 5. (b), min 4.6 (c), max 3. (c), min 3.5 (d) 5.9 in which n s corresponds to the shaft rotation rate, D denotes the propeller diameter, Z the number of blades and τ a proportionality constant which includes the ratio between vortex circulation and mean circulation on the blade. Unloading the tip leads to a decrease of τ. It is assumed that this constant depends only on the propeller geometry and not on the operating condition. By combining Eq. (7) with Eq. (4), a formulation for the radius of the cavitating core is found. For preliminary analysis a linear fit is used with a constant c for the ratio between the non-dimensional resonance frequency and the square root of the cavitation number as presented in Figure. The resonance frequency can then be written as σn f = c fb, τk T (8) where the blade passage frequency f b = Zn s is introduced and the cavitation number σ n is based on the shaft rotation rate and propeller diameter. The validity of Eq. (8) is analyzed for the two-bladed propeller operating at different thrust-coefficients and cavitation numbers at identical shaft rotation rate by analyzing the variability of the propelleonstant τ for the different operating conditions. The resonance frequencies are obtained from the mode of the probability distributions. Figure presents the variation of the measured frequency f and the parameter τ computed from Eq. (8). Both variables have been divided by their mean value. The results show a significant reduction in variability of τ compared to the frequency although there is still some variation left. Further investigation will be directed towards a more detailed analysis on how the resonance frequency should be selected. 4 CONCLUDING REMARKS An analysis has been made of the low frequency hull pressure signal generated by a two-bladed cavitating propeller operating in a wake field of a twin screw vessel. The cavity is a strong leading edge vortex which can also be described as a small sheet at the outer radii with a

8 normalized value tau frequency operating point Figure : Variability of the resonance frequency f and the proportionally constant τ computed from Eq. for various operating conditions. strong re-entrant jet vortex, oriented almost parallel to the blade leading edge. The following conclusions can be drawn: A stochastic analysis shows that the amplitude and frequency of the maximum crest-trough value varies significantly between blade passages. This leads to a reduction of the tonals at the higher harmonics of the blade passage frequencies in the amplitude spectrum and the formation of a broadband hump. Wavelet analysis of the pressure signal of one blade passage showed a narrowband signal of which the frequency increased with time during the collapse and rebounds of the cavitating vortex indicating the presence of a resonance frequency. An FFT spectrum of this signal shows a broadband hump in the frequency range. A theoretical dispersion relation for inertial waves on cavitating vortices has been derived for potential flow. A new criterion for the resonance frequency of cavitating vortices is defined by applying the condition of stationary phase given by the zero group speed. This results in a resonance frequency of the breathing mode n =, which is an effective noise source. This zero group speed condition gives a good prediction of the resonance frequency measured by Maines & Arndt (997) for a cavitating vortex generated by a wing if a simple viscous correction method is applied. Non-dimensionalizing the hull pressure data gives a reasonable agreement if the minimum cavity diameter is used. The same condition has also been used to provide a simple relation between the frequency of a collapsing vortex, cavitation number and propeller thrust coefficient. A preliminary analysis with measurement data for different operating conditions of the propelleonfirmed the overall trend but more detailed investigation is necessary. Future research will concentrate on further investigating stochastic effects in model and full scale hull pressure signals and on further analysis of the dispersion relation foavitating vortices. REFERENCES Arndt, R.E.A., Arakeri, V.H. and Higuchi, H. (99). Some observations of tip-vortex cavitation, J. Fluids Mech. Vol. 9, pp Baiter, H-J., Grüneis, F. and Tilmann, P. (98). An extended base for the statistical description of cavitation noise. Proc. Int. Symp. on Cavitation Noise, ASME Phoenix, AZ, USA, pp Bark, G. (988). On the mechanisms of propeller cavitation noise. Ph.D. thesis, Chalmers University of Technology, Göteborg, Sweden. Bosschers, J. (8). Analysis of inertial waves on inviscid cavitating vortices in relation to lowfrequency radiated noise. WIMRC Cavitation Forum, Warwick University, UK. Bosschers, J. (9). Modeling cavitating vortices in D viscous flow. To be submitted to the 7 th International Symposium on Cavitation CAV9, Ann Arbor, Michigan, USA, August 9. Carlton, J.S. and Fitzsimmons, P.A. (6), Full scale observations relating to propellers. Sixth International Symposium on Cavitation CAV6, Wageningen, the Netherlands. Huang, N.E., Shen, Z., Long, S.R., Wu, M.C., Shih, H.H., Zheng, Q., Yen, N.C., Tung, C.C. and Liu, H.H. (998), The empirical mode decomposition and the Hilbert transform for nonlinear and non-stationary time series analysis. Proc. R. Soc. A. Vol. 454, No. 97, pp Maines, B. and Arndt, R.E.A. (997), The case of the singing vortex. Journal of Fluids Engineering, Vol. 9, p Morozov, V.P. (974). Theoretical analysis of the acoustic emission from cavitating line vortices. Sov. Phys. Acoust., Vol. 9, No. 5, pp Raestad, A-E. (996). Tip vortex index an engineering approach to propeller noise prediction. The Naval Architect, July/August. Saffman, P.G. (995). Vortex dynamics, Cambridge University Press. Thomson, W. (Lord Kelvin) (88). Vibrations of a columnar vortex. Proceedings of the Royal Society of Edinburgh, Scotland, March, 88; Phil. Mag. X., pp Whitham G.B. (974), Linear and non-linear waves, John Wiley and Sons.