# Talk:Ursell number

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A question about the limit for which linear wave theory applies: it's down as 3 / (32 π2). I went back to the references Dingemans (1997) and Stokes (1847) and it seems to come from ka << (kh) 3. Assuming H = 2a and ${\displaystyle k=2\pi /\lambda }$, this gives a linearity limit of 8 π2. Could you tell me if I have got this right or am I missing something? This is my first time inside Wikipedia so please excuse me if I'm not adhering to any of the rules!

--cheers! --Ally --13:52, 7 July 2009 (UTC)

Hi Alexandra. See Eq. (2.421a) on page 179 of Dingemans (1997). The ratio of the second-order amplitude A2 to the first-order amplitude a is:
${\displaystyle {\frac {A_{2}}{a}}=ka{\frac {3-\sigma ^{2}}{4\sigma ^{3}}}}$   with   ${\displaystyle \sigma =\tanh \,(kh),}$
i.e. the factor in front of cos 2χ.
In shallow water, kh≪1, we have asymptotically σ ≈ kh≪1, and further that the wave height H=2a according to 2nd-order Stokes theory, so
${\displaystyle {\frac {A_{2}}{a}}\approx ka{\frac {3}{4(kh)^{3}}}={\frac {3a}{4k^{2}\;h^{3}}}={\frac {3H}{8\left({\frac {2\pi }{\lambda }}\right)^{2}\;h^{3}}}={\frac {3}{32\pi ^{2}}}{\frac {H\lambda ^{2}}{h^{3}}}={\frac {3}{32\pi ^{2}}}U.}$
As a result, linear theory is applicable if A2a, which is the case if U≪32π2/3. Best regards, Crowsnest (talk) 16:43, 7 July 2009 (UTC)

hi Crowsnest Thanks for your prompt reply: I haven't have time to look at it yet, but will head to the library this week with the page printed off to get my head around it. -cheers -Ally-- 14:03, 20 July 2009 (UTC) —Preceding unsigned comment added by AlexandraPrice (talkcontribs)

hello again Crowsnest. Thanks for the pointers: I've been through the maths and I understand where I was getting stuck. From ${\displaystyle {\frac {A_{2}}{a}}=ka{\frac {3-\sigma ^{2}}{4\sigma ^{3}}}}$ it follows that ${\displaystyle {\frac {3}{4}}ka<<(kh)^{3}}$, and from this it follows that linearity is defined by ${\displaystyle U<<{\frac {32\pi ^{2}}{3}}}$.

Stokes referred to ka << (kh) 3, presumably because he was talking the rough order of magnitude, and from this it follows that linearity is defined by ${\displaystyle U<<8\pi ^{2}}$. - cheers - Ally -- 09:13, 27 July 2009 (UTC)