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Maybe you should try those.
Melatonin. With low doses it improves sleep quality, while in high doeses it can make you feel quite drowsy. Also if taking really high doses (10mg - 15mg) it can make you feel tired the next day. Study
Lavender too allows me to relax more and appears slightly to enhance sleep quality. Lavender appears to have some sort of synergism with lemon balm in regards to better sleep. Study
Valerian also has relaxing properties and high doses of it seem to cause mild sedation. Study
This is my personal experience and of a few of my friends.
Some other supplements that seem to improve overall sleep quality not sure about knocking you right out. Those include magnesium, ginko bilboa, l-theanine. If you make yourself a c**ktail of all above items I'm sure you will notice at least some improvement.
I have always had problems falling asleep on uncomfortable journeys, wooden floors or loud bus rides. Little blue valium can be a life saver in those infrequent occasions.
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I don't have a definitive answer, and it might simply be anecdotal.
I've done this for many flights from Montreal to Europe (Paris, Rome, London) where you leave in the evening (and usually leave after "normal" diner time).
I don't think I ever took sleeping aid (natural or chemical), so I cannot help you for that.
Lately, I found that I sleep a lot better when I am cold. For my next trip in April, I will skip on the blanket.
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I would not spend a lot of effort trying to get to sleep when I know that I can't sleep well on that single occasion. Just close your eyes for a while and if you don't feel like sleeping, do something else. If you just say to yourself after you arrive in London that whatever sleep you got was good enough because had you really needed a lot more sleep you would have slept more, then you will do a lot better than if you focus on how drowsy you feel and link that to not getting much sleep on the plane. See also this study.
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The following does not constitute medical advice.
I find diphenhydramine hydrochloride is useful in my travels for when I absolutely must get some sleep before arrival. It is an antihistamine usually sold under a trade name such as "Nytol" (UK) or "Benadril" (South Africa, USA). In my experience it plays well with food and the moderate quantity of alcohol that attends any dinner (although excessive alcohol consumption should be avoided). It also has a relatively short biological half life so I do wake feeling quite refreshed. However on the eastbound transatlantic you will struggle to get in more than six hours so you may be drowsy for a few hours after landing.
The only side effect is that, as an antihistamine it does leave your mouth very dry, and this is redoubled by the effect of the low humidity air onboard airliners. Have plenty of water ready for when you wake up. There is also a mild dependence aspect from repeated use but this should not be a problem unless you are going to be shuttling between the USA and Europe several times a week.
Obviously, read the packaging carefully and if necessary consult with your doctor.
Aside from this, my advice is eat in the lounge or the terminal well before boarding, so once on board you can lay out and fall asleep shortly after take off. You will see the weekly commuters on the NYC-LON route are asleep before take off and only wake on landing. Use the provided ear plugs and blindfolds too, even if they feel uncomfortable at first.