Te ‘eigenfunction solutions’ of the wave equation,


Z(z) = sin(γz), must be zero at depths z=0 and z=D, since the pressure here is zero. Te requirement Z(D)=0 gives the ‘modal equation’ for the idealised waveguide, γm or γm


D = mπ,


‘mode number’. Te γm = √k2


–γm2 with k2


= mπ/D, where m is an integer which designates the are known as the eigenvalues. Te


values of the horizontal component of the wavenumber are Km


≥ γm2 . In the idealised waveguide, the depth dependent eigenf uctions are simply Zm modes.


3.5.2 Normal Modes in the Ocean Tere are only a few relatively simple oceanic waveguides which allow us to obtain a closed analytical form of the solution describing sound propagation at long distances from the source. Pekeris (1948) is regarded as one of the pioneers in this field (see box below). Consider a water layer with thickness D and water speed c with source at depth zs receiver at depth z. Te pressure is effectively zero at the sea


and (z) = sin(mπz/D),


m=1,2,3,… Te sound pressure is the sum of the pressures in the


surface; the reflection coefficient is R=-1. It is well known that for plane wave incidence on the sea floor beyond the critical angle, there is a perfect reflection with an accompanying phase shift. Tis reflection can be represented with an equivalent reflection having R=-1 at a virtual pressure release interface displaced a distance below the sea floor. Terefore, when we study long-range sound propagation in a water layer over a real sediment, the simplest waveguide is the homogeneous water layer that has interfaces with vanishing pressure at the upper and lower boundaries. Te sound pressure is the sum of the pressures in the modes and given as (Jensen et al., 1994; Medwin, 2005):


where A depends on the source power, ρ is the ambient


density, and the summation is over all allowed modes m=1,…,M with real propagation wavenumbers; M increases with increasing frequency; am


is the modal excitation. Tis


normal mode expansion of the field in the waveguide is referred to as the Pekeris model and is useful to understand


Normal Modes – the Invention of Pekeris


Te normal modes concept was first introduced by Pekeris in 1948 for application of acoustic sound propagation caused by an explosive point source in shallow waters. It should be noted that Pekeris published a paper on normal modes in the theory of microwave propagation in 1946. A comprehensive description of normal modes generated within a layered half space is given in the well known textbook written by Ewing, Jardetsky and Press in 1957. Figure 3.33a is modified from their book and shows a liquid half space over an infinite liquid layer. It is possible to derive approximations for the recorded wave field at a large horizontal distance from the source, and the detailed derivation of this can be found in their book. In the analysis of the seismic data recorded at a long distance from a seismic vessel, the period equation is key to understanding the concept of normal modes:


c tankH c


1 2


2 1 =


1 2


1


1 2


2 c 1 2


2 2


Here c is the phase velocity, k is the wavenumber, H is the water depth, and α1 layer and α2


the first layer below the seabed and ρ1


denotes the P-wave velocity of the water and ρ2


denote densities for the corresponding layers, respectively. From this equation it is possible to determine the phase velocity (c) as a function of frequency for each mode. Since the left-hand side of equation (1) is periodic, we will get multiple solutions, leading to the harmonic (or normal) modes. Once the phase velocity for each mode is determined, it is possible to estimate the group velocity by taking the


derivative of the frequency with respect to the wave number k. An example of phase velocities (c) and corresponding group velocities are shown in Figure 3.33b. In the book of


112 (1)


Ewing there is no explicit formula given for the group velocity. It is shown by Landrø and Hatchell that the group velocity (U) can be derived directly from equation (1): 1


2 U c = c 1+ where 2 1Hk 1 = c


1 2


1 2 2 1 + 2 1 2 1


3 2


and 2 1


2 2


cos kH 1 2 2 = 1 c 2


2 2


From equation (2) we see that the group velocity approaches α₁ when c →α₁ and α₂ when c →α₂, as expected. From


Figure 3.33b we see that the phase velocity for each mode starts at the velocity for the second layer and asymptotically approaches the water velocity for higher frequencies. Te asymptotic behaviour is observed for the group velocity, apart from the fact that the group velocity reaches a local minimum for a given frequency for each mode. If the second layer is a solid layer, the shear velocity for


layer 2 (β₂) will enter into the period equation, as derived by Press and Ewing in 1950 (see Ewing et al., 1957 for a comprehensive derivation):


c tankH c


1 2


4 2 1 =


2 2 4


1c 1


1 2


2 c 1 2


2 2


Press and Ewing used the integral technique introduced


by Lamb in 1904 to derive approximate expressions for the wavefield at an arbitrary point in the water, including the effect of a solid layer below the seabed. Figure 3.33c shows an example of normal modes recorded by the permanent receiver array that was installed at the Valhall field in 2003, offshore Norway. Five modes can be identified from this figure, and the group velocity as given by equation (2) fits reasonably well to the observed data.


4 1 c 2


2 2


1 c


2 2


2 (2 c


2 2


2 ) 2 (3) (2)


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