The Dispersion Relation of Short Wind Waves from Space–Time Wave Measurements*

1936 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 21

The Dispersion Relation of Short Wind Waves from Space±Time Wave Measurements*

DAVID W. W ANG AND PAUL A. HWANG Oceanography Division, Naval Research Laboratory, Stennis Space Center, Mississippi

(Manuscript received 17 November 2003, in ®nal form 13 March 2004)

ABSTRACT To study the dispersion relation of short wind waves, a linear wave gauge array (WGA) is con®gured and mounted on a wave-following buoy to conduct in situ space±time measurements of short gravity waves. Results from two ®eld deployments of the WGA buoy in growing seas are presented. The two-dimensional (2D) wavenumber±frequency spectra derived from the space±time measurements provide a direct examination on the relation of wave frequency and wavenumber of short waves in the along-wind direction. Both wavenumber-based and frequency-based phase velocities are extracted from the 2D spectra. The effect of higher harmonics resulting from the Fourier decomposition of nonlinear wave pro®les is more prominent to the frequency-based phase velocity than the wavenumber-based phase velocity. The wavenumber-based phase velocity is consistent with that according to the linear dispersion relation, while the frequency-based phase velocity becomes larger due to the higher harmonics.

1. Introduction g 0.5 CL(k) ϭ or (2) The wave dispersion relation serves as a key discrim- ΂΃k inator of the dynamical properties of the surface wave g ®eld. For deep water gravity waves, the linear dispersion CL(␻) ϭ . (3) relation between the magnitude of wavenumber k and ␻ frequency ␻ can be expressed as Studies based on wave measurements from laboratory and ®eld experiments have shown that the short wave ␻ 2 ϭ gk, (1) components do not follow the linear dispersion relation, and the phase velocity C(␻) is signi®cantly larger than where g is the gravitational acceleration. This relation that according to the linear dispersion relation (e.g., provides an avenue to connect wave characteristics ob- Ye®mov et al. 1972; Mitsuyasu et al. 1979; Ramamon- served between spatial and temporal domains. The dis- jiarisoa and Mollo-Christensen 1979; Rebuffat and Ra- persion relation is crucial to remote sensing applica- mamonjiarisoa 1981). Effects of nonlinear steep dom- tions, such as scattering of radar waves by ocean surface inant waves on the dispersion relation and phase ve- roughness. One quanti®able variable that can be ex- locity of short waves have been studied (Longuet-Hig- tracted from the dispersion relation between the wave- gins and Phillips 1962; Huang and Tung 1976; Masuda number and frequency is the phase velocity C ϭ ␻/k. et al. 1979; Laing 1986). It is generally believed that In practice, the phase velocity can be derived either as waves in open oceans are less steep and do not affect a wavenumber-based variable C(k) from spatial mea- the dispersion relation of short waves. Due to the dif- surements or a frequency-based variable C(␻) from tem- ®culties of conducting reliable in situ measurements of poral measurements. According to the linear dispersion short waves in open oceans, there is a lack of solid relation (1), we can have evidence to validate this assumption. Most ®eld studies on the dispersion relation focus on either long gravity waves [up to 0.5 Hz in Donelan et al. (1985), up to 1 Hz in Hara and Karachintsev (2003), and up to 12.6 rad * U.S. Naval Research Laboratory Contribution Number NRL/JA/ Ϫ1 7330-03-21. m in Gotwols and Irani (1980)] or gravity-capillary waves (Hara et al. 1998; wavenumbers ranging from 50 to 800 rad mϪ1). It is of great interest and importance Corresponding author address: Dr. David W. Wang, Meso- and to have reliable ®eld measurements for short gravity Finescale Ocean Physics Section, Department of the Navy, Naval Research Laboratory, Code 7330, Stennis Space Center, MS 29529- waves to bridge the data gap and improve our under- 5004. standing. E-mail: [email protected] Methods used to study the dispersion relation are

Unauthenticated | Downloaded 09/30/21 02:46 AM UTC DECEMBER 2004 WANG AND HWANG 1937 based on the assumption that the wave ®eld can be range of 1 m, based on the design by Chapman and treated by a linear superposition of sinusoidal wave Monaldo (1991). The wires are installed at a spacing of components. One commonly used method is the cross- 0.0508 m, which provides a spatial measurement for spectral method, which estimates frequency-based wavenumbers ranging from 6.18 to 61.8 rad mϪ1. The wavenumber k(␻) and phase velocity C(␻) from the temporal sampling rates for the deployment of St. An- cross spectra of temporal wave measurements at two drew Bay and the Gulf of Mexico are 25 and 50 Hz, adjacent locations. The method has its limitations and respectively. The mean measurement resolution of the bias (Huang 1981; Dudis 1981) and is only considered 20 wires is 0.23 mm with a standard deviation of 0.0028 to be valid for phase velocities in the frequency region mm for the St. Andrew Bay deployment and 0.19 mm around the dominant wave component. Nonetheless, it with a standard deviation of 0.0037 mm for the Gulf of has been widely adopted in many studies because of its Mexico deployment. The measurement resolutions are simplicity and availability (e.g., Ye®mov et al. 1972; obtained from static calibrations. The potential impacts Rikiishi 1978; Ramamonjiarisoa and Mollo-Christensen to the resolutions from the capillary effect and self- 1979; Mitsuyasu et al. 1979; Thornton and Guza 1982; disturbance of the capacitance probe are not considered Elgar and Guza 1985; Laing 1986). The relation be- and require further study. This wave gauge array pro- tween wavenumber and frequency of a random wave vides a direct space±time measurement of surface wave ®eld in open oceans should be best studied via the use elevation ␩(y, t) along a transect direction over the of a wavenumber±frequency spectrum, which maps the gauge array length (1 m). energy distribution in both space and temporal domain. During the two ®eld deployments, the array is mount- Following the energy distribution on the 2D plane, we ed on a surface-following buoy and is oriented along can directly examine the relation between wavenumber the direction of the buoy's bow. The buoy is deployed and frequency and extract the phase velocity. in free drifting mode to reduce the relative velocity be- Using the wavenumber±frequency spectra estimated tween the surface current and the structural members from directional wave measurements, Donelan et al. piercing through the water surface. This operation can (1985) and Hara and Karachintsev (2003) examined the signi®cantly reduce local wave generation by the buoy dispersion relation and phase velocity. Their results are members and minimize ¯ow distortion to the sensing limited by the directional resolutions and the data-adap- region (Hwang et al. 1996). This free-drifting operation tive techniques (Irani et al. 1986). Gotwols and Irani also reduces the potential effect of Doppler frequency (1980) studied the phase velocity of short waves using shifts (see the appendix). The buoy's bow is aligned into 2D spectra derived directly from wave images recorded the local wind direction by a ®n (about 0.4 m2 in area) by a video camera. Their results are limited by the ef- attached on the mast. Under steady wind conditions, the fects of sky radiance and re¯ection in the image (Irani orientation of the wave gauge array (referred to as the et al. 1986). The above studies are all subject to un- along-wind array) is approximately in the direction of certainties and limitations imposed by the measurement dominant wind wave. The locations of the free-drifting systems and data processing schemes. To have an ac- buoy are registered by the GPS recording onboard the curate examination of the dispersion relation, it is highly buoy. The recordings are then used to estimate drifting desirable to use wavenumber±frequency spectra derived speed and direction. Effects from the buoy's transla- from in situ high-resolution space±time measurements tional and angular motions on wave measurements from of short waves. However, the dif®culties of conducting the arrays are corrected using the outputs from motion such space±time data acquisition make this rarely prac- sensors mounted on the buoy following the approach ticed in the ®eld experiments. by Hanson et al. (1997). More details about the buoy Recently, we con®gured a linear wave gauge array designs and its operations can be found in Hwang et al. (WGA) mounted on a wave-following buoy to carry out (1996). An ultrasonic anemometer (Handar 425A) in situ space±time measurements for short wind waves. mounted on the buoy mast at about 1.3 m about mean In this study, we present results from two ®eld deploy- water line provides wind speed and direction measurements under active wave generation conditions: one in ment at 1-Hz sampling rate. Average wind speed at 10- a well-sheltered estuary of St. Andrew Bay, Florida, and m elevation U is then estimated from the measured wind another in nearshore waters of the Gulf of Mexico. Sec- speed combined with humidity and air and water tem- tion 2 provides a description of the wave gauge array peratures using the method by Liu et al. (1979). system and environmental conditions of the two deployments. Analysis results are presented in section 3. b. St. Andrew Bay deployment Concluding remarks are given in section 4. The ®rst ®eld deployment of the WGA buoy was 2. Measurement systems and ®eld deployments carried out in St. Andrew Bay (12 October 2001), lo- cated on the northwest coast of Florida (Fig. 1a). This a. Measurement systems bay is a part of an estuary system and is well sheltered A linear WGA is con®gured from a set of 20 capac- from the long waves of the Gulf of Mexico by the itance wires of 1-mm diameter with a measurement barrier islands. During a steady southeasterly wind

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FIG. 1. Location maps and free-drifting trajectory of the WGA buoy for the ®eld deployments in (a) St. Andrew Bay and (b) the Gulf of Mexico. The starting locations are marked with ``x.'' event, the WGA buoy was deployed in free drift near datasets are shown in Fig. 2a. The inverse wave age the southeast corner of the bay for about 3 h (1500± of the four selected datasets, U/Cp , decreases from 4.65 1700 UTC). Details of this deployment can be found to 2.32. in Hwang and Wang (2004). In the sheltered waters of the bay, the buoy is very stable. The root-mean-squares c. Gulf of Mexico deployment of the pitch and roll motions are less than 2Њ and the yaw motions are about 4Њ. Four datasets from the be- The second ®eld deployment of the WGA buoy was ginning, middle, and end of the deployment are se- carried out in the coastal waters south of the Grand Isle lected for the analysis. Details of wind and wave con- in the Gulf of Mexico (16 February 2003). After a cold ditions of the datasets are listed in Table 1. Each dataset front passed through the area, the buoy was deployed spans over 2.7 min, which contains about 70±140 wind in free drift under a steady northwesterly wind for about waves, depending on the peak wind wave frequencies 7 h (1700±2300 UTC). The buoy gradually drifts south- (0.44±0.88 Hz; see Table 1). The average drifting eastward with speeds between 0.4 and 0.5 m sϪ1. The speeds of the selected datasets are 0.22, 0.25, 0.35, and drifting track starts from a fetch of about 0.5 km and 0.63 m sϪ1 . The wave frequency spectra for the four ends at a fetch of about 15 km (Fig. 1b). In the study,

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TABLE 1. Wind and wave conditions of the selected datasets from the St. Andrew Bay deployment (12 Oct 2001). The steepness ka is 2 computed based on wavelength and wave amplitude represented by fp and Hs, which has a ϭ Hs/2 and k ϭ (2␲ fp) /g; g is the gravitational acceleration; and Cp is the phase velocity at the peak frequency fp according to the linear dispersion relation (3).

Time U ␪u Hs (wind sea) fp Air temp Water temp Ϫ1 (UTC) (m s ) (Њ) (m) (Hz) U/Cp ka (ЊC) (ЊC) 1500 8.4 162 0.10 0.88 4.65 0.156 27.5 24.2 1530 8.6 157 0.15 0.66 3.57 0.132 27.4 24.2 1600 8.6 151 0.23 0.51 2.74 0.121 27.1 24.2 1700 8.3 147 0.32 0.44 2.32 0.125 26.6 24.2 three datasets are selected from measurements at the deployment. Time series of wave elevation measurements beginning (1700 UTC), middle (2000 UTC), and end from gauges at the upwind end, middle, and downwind (2300 UTC) of the deployment. Each dataset lasts 5.4 end of the array are displayed in Fig. 3a. Also shown is min, which contains about 100±130 wind waves with the 2D image of the space±time wave elevations (Fig. peak frequencies from 0.32 to 0.42 Hz (Table 2). The 3b). The positive distance in the along-wind direction (y) wave frequency spectra of the three data sets are shown is the downwind direction. To enhance the image of short- in Fig. 2b. The wave ®eld consists of fetch-limited wind er waves, a time derivative is applied to ␩(y, t). The time seas and swells generated, respectively, by the north- derivative image d[␩(y, t)]/dt (Fig. 3c) shows a very vivid westerly and southeasterly winds. The wave height of crosshatch pattern of crossing waves similar to that ob- wind seas increased from 0.37 to 0.63 m, with its peak served in airborne spatial measurements of a young wave wave periods decreasing from 0.42 to 0.32 Hz. The ®eld (Hwang and Wang 2001). inverse wave age U/Cp decreases from 2.38 to 1.67. The The wavenumber±frequency spectra in the along- swell energy peaks at about 0.14 Hz (7 s) and remains wind direction ␹(ky, ␻) are computed by applying the relatively steady during the deployment. Details of wind standard 2D fast Fourier transform (FFT) to shorter time and wave measurement results are listed in Table 2. segments. For the St. Andrew Bay deployment, each selected 2.7-min space±time dataset is divided into 16 3. Results shorter time segments of 20 ϫ 256 data points (20 gauges at 25-Hz sampling rate). For the deployment in the a. Wavenumber±frequency spectra of the along-wind Gulf of Mexico, each 5.4-min dataset is divided into 32 direction shorter time segments of 20 ϫ 512 data points (20 gaug- Figure 3 shows a sample of 20-s space±time wave es at 50-Hz sampling rate). To reduce spectral leakage elevation measurement ␩(y, t) in the along-wind direction from background long waves, a prewhitenning proce- by the wave gauge array during the St. Andrew Bay dure that applies a temporal and spatial derivative on

FIG. 2. Wave frequency spectra of the selected datasets for the deployments in (a) St. Andrew Bay and (b) the Gulf of Mexico. The wave frequency spectra are averaged over the wave frequency spectra of the central ®ve wire gauges of the along-wind array. The spectral slope of the dashed lines is Ϫ4.

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TABLE 2. Wind and wave conditions of the three selected datasets from the WGA Gulf of Mexico deployment (16 Feb 2003).

Time U ␪u Hs (wind sea) fp Air temp Water temp Ϫ1 (UTC) (m s ) (Њ) (m) (Hz) U/Cp ka (ЊC) (ЊC) 1700 8.8 288 0.37 0.42 2.38 0.125 15.9 17.9 2000 8.1 270 0.51 0.37 1.93 0.119 15.6 17.7 2300 8.5 300 0.63 0.32 1.77 0.122 1.36 17.4 the shorter segments prior to the 2D FFT computation dependent phase velocity is computed as C(␻) ϭ ␻/k(␻). is performed. The ®nal wavenumber±frequency spectra Both k(␻) and ␻(k) are extracted from the 2D wavenum- are the ensemble average of the calculated spectra. ber±frequency spectra in the positive wavenumber side The 2D wavenumber±frequency spectra for the St. (downwind direction), based on the approach by Gotwols

Andrew Bay and the Gulf of Mexico deployments are and Irani (1980). The k(␻i) is de®ned as the wavenumber shown in Figs. 4 and 5, respectively. The spectra are associated with the peak in ␹(k, ␻i) along the frequency normalized by the 2D spectral peak. The interval of the band ␻i and the ␻(ki) is de®ned as the frequency asso- contour line is 2 dB. Elgar and Guza (1985) show that ciated with the peak in ␹(ki, ␻) along the wavenumber the directional spreading could also cause the increase band ki. of phase velocity. Without analyzing directional spread- The 2D spectrum of short waves ␹(k, ␻) in a random ing of short waves, the dispersion relations for an iso- wave ®eld consists of components from fundamental tropic distribution (k ϭ 1.414ky) and unidirectional dis- waves and higher harmonics of nonlinear waves of low- tribution in the along-wind direction (k ϭ ky) are used er frequencies (Komen 1980). The fundamental wave to approximate, respectively, the upper and lower limits components ␹ f (k, ␻) follow the linear dispersion rela- of the directional spreading effect. Most of the wave tion in the 2D plane, while the higher harmonics com- energy appears in the positive wavenumber side (i.e., ponents ␹ (k, ␻) (often called bound waves in the lit- the downwind direction) of the 2D spectra. At lower h Ϫ1 erature) do not. This splits the energy distribution in the wavenumbers and frequencies (k Ͻ 20 rad m and ␻ 2D plane (JaÈhne 1989). To illustrate this, a schematic Ͻ 15 rad sϪ1), the energy distribution closely follows 2D wavenumber±frequency spectrum ␹(k, ␻) consisting the curves representing the dispersion relations for the of ␹ (k, ␻) and ␹ (k, ␻) is shown in Fig. 6a. The 2D two directional distributions. At higher wavenumbers f h and frequencies, the energy distribution becomes sig- spectrum is displayed as a function of normalized wavenumber k/k and frequency / , where k and are, ni®cantly broadened and split into regions away from d ␻ ␻d d ␻d the linear dispersion relations. respectively, the dominant wavenumber and frequency It is also noted that there is a small amount of energy resolvable by the array. In the 2D plane, the distribution appearing in the negative wavenumber side of the 2D of fundamental wave components ␹ f (k, ␻) follows the spectra from the St. Andrew Bay deployment. The en- linear dispersion relation represented by the dashed ergy peak in the negative wavenumber side is about curve. The higher harmonic component of the dominant 20% of that of the positive wavenumber side. This ap- waves ␹h(k, ␻) moves at the same phase speed as that pearance of wave energy with negative wavenumbers of the dominant waves and is represented by the dotted implies the existence of upwind-traveling wave energy, straight line through the origin and the dominant wave which is also indicated by the crossing wave patterns (k/kd ϭ 1 and ␻/␻d ϭ 1). in the 2D image (Fig. 3c). There is an increasing amount The coexistence of ␹ f (k, ␻) and ␹h(k, ␻) affects the of energy appearing in the negative wavenumber side determination of ␻(k) and k(␻). As an example, referring of the 2D spectra from the Gulf of Mexico deployment to the 2D spectrum in Fig. 6a, k(␻) at frequency band

(Fig. 5). The peak energy in the negative wavenumber ␻ ϭ 2␻d depends on the relative magnitude between side is about 50% of that in the positive wavenumber ␹h(2kd,2␻d) and ␹ f (4kd,2␻d), marked by a cross and side, much higher than the 20% for the St. Andrew Bay circle, respectively. In the case of a larger ␹ f (4kd,2␻d), deployment. The observed stronger upwind-traveling k(␻) is determined by the location of ␹ f (4kd,2␻d) and wave activity in open oceans than in the sheltered bay k(␻) ϭ 4kd. In the case of a larger ␹h(2kd,2␻d), k(␻) is interesting. Further study is needed to identify the is determined by the location of ␹h(2kd,2␻d), and k(␻) source of these wave actions. The upwind-traveling ϭ 2kd. Similarly, the harmonic component ␹h(2kd,2␻d) waves are also reported by Plant and Wright (1980) and also affects the determination of ␻(k) at wavenumber Hara et al. (1998). In the following study, we focus the band k ϭ 2kd. Depending on the relative magnitude analysis on the positive wavenumber side (downwind between ␹ (2k ,2␻ ) and ␹ (2k , 1.414␻ ), marked by direction) of the 2D spectra. h d d f d d a cross and square, ␻(k) is either 1.414␻d or 2␻d. Be- cause the fundamental wave energy decrease with an in- b. Determining the phase velocity from wavenumber± creasing frequency (wavenumber) following ␻Ϫ4 (kϪ2.5) frequency spectra (Phillips 1985), the fundamental component ␹ f (4kd,2␻d)

In this study, the wavenumber-dependent phase veloc- is signi®cantly smaller than ␹ f (2kd, 1.414␻d). This im- ity is computed as C(k) ϭ ␻(k)/k and the frequency- plies that the determination of k(␻) is more likely dom-

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FIG. 3. A selected 20-s sample of the space±time wave measurements ␩(y, t) by the along-wind array from the St. Andrew Bay deployment. (a) Time series of wave elevation ␩(t) from the gauges at the upperwind end (solid line), the middle (dashed-dotted line), and the downwind end (dashed line). For a better illustration, the time series data are offset by 0.2 m for the upperwind and the downwind gauges. Also shown are the 2D images of (b) ␩(y, t) and (c) d␩(y, t)/dt.

inated by harmonic component ␹h(2kd,2␻d), because ence of higher harmonics ␹h(k, ␻) and deviates from of a smaller fundamental component ␹ f (4kd,2␻d). Sim- the linear dispersion relation. Figure 6b shows the 2D ilarly, the determination of ␻(k) is likely dominated by spectra from Fig. 5a and the extracted k(␻) (squares) the larger fundamental component ␹ f (2kd, 1.414␻d). As and ␻(k) (circles). As expected, ␻(k) closely follows the a result, k(␻) is more signi®cantly affected by the pres- dashed curve representing the linear dispersion relation

FIG. 4. Contour plots of normalized 2D wavenumber±frequency spectra in the along-wind

direction for the selected datasets from the St. Andrew Bay deployment; (a) 1500 UTC, U/Cp ϭ

4.65; (b) 1530 UTC, U/Cp ϭ 3.57; (c) 1600 UTC, U/Cp ϭ 2.74; (d) 1700 UTC, U/Cp ϭ 2.32. The contour lines cover a range of 12 dB and the interval is 2 dB. The dashed and dotted curves represent, respectively, the linear dispersion relations for the unidirectional and isotropic distributions.

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FIG. 5. Same as Fig. 4 but for the Gulf of

Mexico deployment: (a) 1700 UTC, U/Cp ϭ

2.38; (b) 2000 UTC, U/Cp ϭ 1.93; (c) 2300

UTC, U/Cp ϭ 1.77.

while k(␻) is closer to that of higher harmonics repre- deployment. Figures 7c and 7d show the scatterplots of sented by the straight line through the peak of dominant C(␻) and C(k). The dashed and dotted curves, respec- wave and origin. Consequently, the phase velocity C(k) tively, represent the results according to the dispersion based on ␻(k) is consistent with that of the linear dis- relation for the unidirectional and isotropic distributions. persion relation while the phase velocity C(␻) based on The variation of k(␻) and C(␻)at␻ Ͻ 15 rad sϪ1 closely k(␻) is larger and closer to that of higher harmonics. In follows the linear dispersion relations (Figs. 7a,c). At ␻ the following analysis, both C(␻) and C(k) are extracted Ͼ 15 rad sϪ1, k(␻) becomes smaller than that of the linear and presented. dispersion relations and C(␻) becomes more independent Figures 7a and 7b show the scatterplots of k(␻) and of the increasing frequency. The deviation of k(␻) and ␻(k) extracted from the 2D spectra of the Gulf of Mexico C(␻) from the linear dispersion relation becomes more

FIG. 6. (a) Schematic 2D wavenumber±frequency spectrum as a function of normalized wavenumber and frequency. (b) The contour plot of the 2D spectrum in Fig. 5a and the k(␻) and ␻(k) extracted from the spectrum. The dashed curves represent the linear dispersion relation for the unidirectional distribution. The dotted straight line represents the relation of wavenumber and

frequency having the same phase speed as that of the dominant wave ␻/k ϭ Cp.

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FIG. 7. Scatterplots of (a) k(␻), (b) ␻(k), (c) C(␻), and (d) C(k) for the datasets from the Gulf of Mexico deployment. The dashed and dotted curves represent, respectively, the dispersion relations for the unidirectional and isotropic distributions. signi®cant with increasing frequency. For ␻(k) and C(k), the linear dispersion relation, the phase velocities C(␻) they agree well with that of the linear dispersion relation and C(k) are normalized by the phase velocity Cp at the at wavenumbers up to 60 rad mϪ1 (Figs. 7b,d). It is noted wind wave spectral peak, according to the linear dispersion that Gotwols and Irani (1980) show that the phase ve- relation shown in (2) and (3). Figure 8 shows C(␻)/Cp locity C(k) in the along-wind direction agrees well with and C(k)/Cp versus the normalized wave frequency ␻/␻p that of the linear dispersion relation at wavenumbers up and wavenumber k/kp. Also shown are dashed curves rep- Ϫ1 Ϫ0.5 Ϫ1 to 12 rad m . resenting CL(k)/Cp ϭ (k/kp) and CL(␻)/Cp ϭ (␻/␻p) To further quantitatively illustrate the deviation from based on (2) and (3). For C(␻), the decrease of C(␻)/Cp with increasing frequency ␻/␻p becomes signi®cantly different from the linear dispersion relation (dashed line) at

higher frequencies, ␻/␻p Ͼ 6. The difference seems slightly larger for younger waves (larger U/Cp). For C(k), the decreasing of C(k)/Cp with increasing k/kp is very consis- Ϫ0.5 tent with the linear dispersion relation (k/kp) up to k/kp ϭ 100. Very similar results are also shown in the datasets from the St. Andrew Bay deployment.

4. Concluding remarks The effect of wave nonlinearity on phase velocity can be observed in two ways (Mitsuyasu et al. 1979; Komen 1980; Laing 1986). One is the increase of the phase velocity of fundamental waves, which is caused by the weakly nonlinear wave±wave interaction (Lon- guet-Higgins and Phillips 1962; Laing 1986). The increase caused by the weak dynamic interaction is very limited. The other is due to the effect of harmonics of Fourier decomposition of nonlinear wave pro®les. The presence of the higher harmonics can signi®cantly in- FIG. 8. (a) Normalized phase velocity C(␻)/Cp vs normalized fre- crease the frequency-based phase velocity C(␻), es- quency ␻/␻p and (b) normalized phase velocity C(k)/Cp vs normalized wavenumber k/k , for the Gulf of Mexico deployment. The dashed lines pecially at higher frequencies (Komen 1980). With the p use of space±time measurements to provide a direct represent that, according to the linear dispersion relation, CL(k)/Cp ϭ Ϫ0.5 Ϫ1 (k/kp) and CL(␻)/Cp ϭ (␻/␻p) . examination on the relation between wave frequency

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FIG. A1. The 2D normalized wavenumber±frequency spectra from (a) ®xed-buoy deployment and (b) free-drifting-buoy deployment. (c) The phase velocity C(k) from the ®xed-buoy (square) and free-drifting-buoy (circle) deployments. The dotted and dashed curves, respectively, represents the phase velocities according to the linear dispersion relation without and with the Doppler shift Ϫ1 correction using an assumed current speed Uc ϭ 0.1ms .

and wavenumber from 2D spectra, it has been shown oratory), Richard Smith (Neptune Science, Inc.), and that the higher harmonics can have a more signi®cant Jeff Jones (Planning System, Inc.) helped to prepare, effect on frequency-based phase velocity C(␻) than on improve, and deploy the WGA system during the two wavenumber-based phase velocity C(k), as explained ®eld studies. in section 3b and Fig. 6a. The effects of higher harmonics are more likely observed in frequency-based APPENDIX wave variables C(␻) and k(␻) from temporal measurements. Space±Time Measurements from Free-Drifting In this study, we present results of in situ space±time and Fixed Platforms measurements of short gravity waves by the linear wave gauge arrays mounted on a wave-following buoy during We carried out space±time wave measurements with two ®eld deployments. The 2D wavenumber±frequency the WGA buoy in a long and narrow canal during a spectra directly derived from the high-resolution space± cold front passage. The canal is about 400 m long and time data provide a reliable means to examine the dis- 100 m wide and is oriented in the northwest±southeast persion relation between the wavenumber and frequency direction with the northern end closed and the southern of short wind waves. Both phase velocities C(␻) and end open to the main navigation canal system in the C(k) are extracted from the spectra. The effect of higher Stennis Space Center, Mississippi. The space±time wave harmonics revealed in the 2D spectra is shown to be data are collected from a fetch-limed wave ®eld under very prominent in C(␻). The wavenumber-based C(k) a steady northwesterly wind blowing in the longitudinal is less affected and is consistent with that of the linear direction of the canal. The measurements are ®rst carried dispersion relation. Our results show that the short wind out with the buoy aligned into the wind and stabilized waves (wavenumbers ranging from 6.18 to 61.8 rad by two long pipes and two mooring lines attached to mϪ1) are dispersive and follow the linear dispersion the buoy's four columns and anchored at the bottom of relation. The larger frequency-based phase velocity the canal (at about 200 m to the northern end of the C(␻) is biased by the higher harmonics inherent in the canal). Later, the wave measurements are carried out computation and does not re¯ect objectively the dis- with the buoy deployed near the northern end of the persive nature of short wind waves. canal and in free drift southeastward carried by the wind and surface current. Acknowledgments. This work is sponsored by the Of- The 2D wavenumber±frequency spectra for the ®ce of Naval Research (Naval Research Laboratory PE space±time wave data from the ®xed and free-drifting 61153N and 62435N). Ray Burge (Naval Research Lab- deployments are shown, respectively, in Figs. A1a and

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A1b. Most wave energy in the 2D spectra appears in surface waves'' by A. Ramamonjiarisoa and E. Mollo-Christen- the positive wavenumber side (downwind direction). sen. J. Geophys. Res., 86, 2073±2075. ÐÐ, and C. C. Tung, 1976: The dispersion relation for a nonlinear For the 2D spectrum from the ®xed-buoy deployment random gravity wave ®eld. J. Fluid Mech., 75, 337±345. (Fig. A1a), the energy-containing region is generally Hwang, P. A., and D. W. Wang, 2001: Directional distribution and above the dashed curve representing the linear disper- mean square slopes in the equilibrium and saturation ranges of sion relation for the unidirectional distribution. The en- the wave spectrum. J. Phys. Oceanogr., 31, 1346±1360. ÐÐ, and ÐÐ, 2004: Field measurements of duration-limited growth ergy-containing regions in the 2D spectrum from the of wind-generated ocean surface waves at a young stage of de- free-drifting deployment closely follow the dispersion velopment. J. Phys. Oceanogr., 34, 2316±2326. relation (Fig. A1b). Figure A1c shows the extracted ÐÐ, S. Atakturk, M. A. Sletten, and D. B. Trizna, 1996: A study wavenumber-based phase velocity C(k) from the 2D of the wavenumber spectra of short water waves in the ocean. spectra in Figs. A1a and A1b. The free-drifting C(k) J. Phys. Oceanogr., 26, 1266±1285. Irani, G. B., B. L. Gotwols, and A. W. Bjerkaas, 1986: The 1978 agrees generally with the linear dispersion relation ocean wave dynamics experiments: Optical and in situ mea- (dashed curve). The ®xed-buoy C(k) is larger than that surement of the phase velocity of wind waves. Wave Dynamics of the free-drifting C(k) and closely follows the dotted and Radio Probing of the Ocean Surface, O. M. Phillips and K. curve representing the phase velocity of the liner dis- Hasselmann, Eds., Plenum Press, 165±179. 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