If you really want to get into the materials and construction of bows I highly recommend the Bowyer's Bible series of books. They're reasonably short, reasonably inexpensive and they go into great detail on the different materials, methods, and designs used over the years; as well as covering the theory and physics behind the designs. They make for a great starting point.

For materials, in medieval Europe, Yew was the preferred wood for war bows. However, other woods were used as well, such as oak, elm, and ash. high draw weight war bows could be made from any of these other woods, but there were some drawbacks in using them. In the ideal bow you want to be able to store as much energy as possible when drawing the bow back, then have as much of that energy directed into the arrow when releasing. Wood like oak is very strong, but rather heavy for its strength. So a bow made of oak uses more energy to move the limbs (it's inertia); energy that doesn't get used to make the arrow go. Woods like ash and elm are not so strong or flexible, so the only way to make a goo strong bow from them is to make them wider, so each bit of wood doesn't get stressed as much. But if you're making a bow out of a round log, a wide bow is a bit of a problem, you can't get many staves wide enough out of a single log; a real problem if you're trying to outfit a whole bunch of people. There are bow designs that can overcome these drawbacks and make exceptional bows from these woods, but they were not used during the middle ages.
So they went with yew, it had just the right mix of properties. It was strong, flexible, but also light, so it could store lots of energy and less of it was needed to overcome inertia and get the limbs moving. Also, since the sap wood is strong in tension and the heartwood is very strong in compression, a bow made of yew could be made narrow, but with a thick cross-section. This was really handy because you can get many more narrow staves out of a log, making them easier to make in large numbers.

As far as the length, the longer the flexing portion of the bow, the less stress is put on any particular piece. So generally speaking, longer bows are easier to make, and will usually more efficiently store the energy of the draw. Of course you can't go nuts with the length. Longer bows start to get unwieldy, also, longer limbs mean more weight to get moving, and longer working limbs can cause problems with efficiently transferring the energy to the arrow. The cross-section of yew longbows effectively compromises on this. A narrow width lessens the weight, and the deep circular or D shaped section shape pushes the wood to get the most out of it.

For draw weight and it's effects on the power of the bow; simplifying it, the higher the draw weight, the more energy the bow can store during the draw. This is why war bows usually had over 100lb draw weights, with some from the period above 130lbs.
For a little more detail, it comes down to the area under the force / draw curve. The larger the area under the curve, the more energy the bow can store. Obviously, the higher the peak draw weight, and the longer the draw length, the more energy it will store. But the design of the bow can tweak the curve of the curve. A concave f/d will store less energy then a convex curve for the same draw length and weight. Different things like the deflex or reflex, the angle of the string at full draw, materiel properties, bunch of stuff can all impact the area under the curve.

/u/Skookum_J has given a good summary of bow wood characteristics (although a more complete one can be found here, while /u/Grad_Phi has given a good run down of the archaeological, textual and visual evidence. My goal is to weld it all together into some broad guides to performance for you.

Types of Bow

There were two types of bows commonly used in England and on the Continent through to the early fourteenth century: the longbow and the shortbow. The shortbow itself survived on the Continent until at least 1346, as it appears in the Leugemeete Fresco. Clifford Rogers, who examined more than 1000 pieces of medieval art, found that they existed as the primary bow in Continental sources into the second half of the fifteenth century. In England, however, longbows show up abruptly in the early 14th century, replacing the "short" (3-4ft when strung) and "medium" (4'9"-4'11") bows, though they still were in evidence to a lesser degree into the middle of the fourteenth century.

There is some controversy over whether or not the shortbow existed. Robert Hardy, Jim Bradbury and Matthew Strickland have all argued strenuously that it did not exist, and that the old argument of English superiority over other nations in the 14th century was in part due to the introduction of the longbow is invalid. There are a number of well documented longbow finds from late Antiquity (Vimose, Nydam) and the Early Middle Ages (Hedeby, Ballinderry), all of which have the same basic ratio of width and depth as the Mary Rose bows.

However, there is also some clear evidence of shortbows. On intact shortbow - found with a military style arrow - and two half length fragments have been found in Ireland, in addition to several fragments which feature the same cross section and decurved tip. As I've just learned today, there's also the Pineuil Bow which, although it has a handle and lacks any deflex, has the same length and cross section as the Waterford bow. This suggests a degree of continuity in design between the 11th and the 13th centuries, and provisionally validates the existence of the shortbow.

/u/Grad_Phi might be willing to fill us in on more information about the proportion of longbow to shortbow in the other complete/mostly complete finds I wasn't aware about until today.

Construction of the bows varied. Longbows, so far as archaeological evidence has shown, were almost always of a "D" or elliptical shape (center row + bottom right are most typical), but there was some variation in cross section. They generally didn't have handles, although there are a couple of examples from the Mary Rose which do appear to have handles, in addition to this very unusual flatbow from 15th century Germany, linked by /u/Grad_Phi.

Longbows, for the most part, were longer than 5'8" (172cm), although most authors accept Bradbury's definition of them being 5ft or longer. In general, they did not exceed 200cm in length Short bows were somewhere between 120cm and 140cm. The short end is represented by the Waterford and Pineuilh bows (126cm and 124cm respectively), while the bow used in the Skeffington murder and the shortbow used in the murder of Robert de Esnyngton were both approximately 140cm. The two longest fragments from Ireland would most likely belong to bows of 120-130cm and 130-150cm in length.

I've mentioned decurved bows before, and I should probably explain them now. A decurve is where the limbs of the bow are bent back towards the user. The most common, natural, form of this is called "set" and happens natural as the bow gradually deforms during use in the direction it is being drawn. However, the tips of a bow can be deliberately decurved back to the user. There are two possible reasons for this. With the Hedeby bow, the decurve was most likely introduced in order to protect the soft yew wood from the string when strung. One credible reproduction has shown that it might have drawn up to 100lbs and, without a horn tip, the string would quickly bite into the yew unless decurve was introduced.

The other use for decurve is to allow a bow to be strung for longer periods of time. A normal straight bow will quickly begin to "follow the string" (take on a temporary set) and, if strung for long enough, the string follow will become permanent. This reduces the power of a bow and, if strung for too long, will ruin a good bow (Ascham talks about this in Toxophilus). However, a bow with decurved tips will be under less strain when strung, allowing it to stay strung for longer. I suspect that this was the overriding intention with the Anglo-Norman bows in Ireland, as they were military bows. The decurved does reduce the power of the bow, but the trade off is a bow you can leave strung almost indefinitely. While not common, a couple of the Mary Rose bows also have decurved tips (eg 81A1599).

In contrast to decurved, there is recurve. Recurved tips curve away from the user, and the result is that the bow is under extra strain when fully strung. This increases the cast - the speed at which the arrow leaves the bow - at the cost of a slightly increased risk of the bow breaking and slightly increasing the eventual set of the bow. The risk would be most pronounced with the stiffest, densest Italian Yew, but it's not likely to be a serious risk to the user if they care for the bow well. It just shortens the overall lifespan of the bow.

Finally, there is the reflex. In a reflex bow, both limbs of the bow are bent away from the user, which has a similar effect to recurving the tips, only more so. A surprising number of longbows have this feature. Two from the Nydam ship and the slight majority of Mary Rose bows exhibited this. The Ballinderry bow, despite the decurved tips, also has reflexed limbs. The reflexed bow provides an even faster cast than the recurved bow, with another marginal decrease in the life of the bow.

Performance

The exact performance of a Waterford style shortbow is unknown as, so far as I know, no one has made a replica of it yet for testing. It's likely to be somewhere between 40 and 50lbs at 22-24" if made with yew, ash or elm. A laburnum shortbow of this style should be able to manage around 50lbs at 26". This means that they will be performing considerably less well than a longbow of a similar draw weight. I would estimate the effective range with a 25-30g arrow to be 80-100 meters. A really well made bow, made with yew and reflexed might manage 140 meters, and a 15-20g flight arrow could probably reach to 160-170 meters. The energy of the bow using a 25-30g arrow would probably be around 15-20 joules, based on American Indian shortbows. This is sufficient to kill a man at close range, and penetrate thick clothing, but thick leather or a basic aketon will prevent fatal wounds.

The longer, 140cm bows, based on a speculative reconstruction by Lars Persson of the English Warbow Society, will reach around 140m with a ~30g arrow. However, that bow was using laburnum and drew to 28". Laburnum allows for a longer draw length for a given bow length (example ), so a yew bow would draw slightly shorter, perhaps 26". The range of a yew bow might only reach 120m. This would go a long way to explaining why laburnum was quite common in 12th and 13th century literature and documentation. Not only did it look good, it provided extra performance. The energy at discharge would probably be 25-30j, enough to inflict serious - but not necessarily lethal - wounds through thick leather or an aketon, but not both.

Now we come to the longbow. The most commonly quoted draw weight for the Mary Rose bows is 150-160lbs, but I believe this is too high. Weapons of Warre draws directly on the original study (which used computer modelling verified by actual bows), and Clifford Rogers also drew on a presentation of the paper in 1991 in his 1993 article on the military revolution of the Hundred Years War. Both state that the most common bows were between 100 and 120lbs. 63.5% were in this range, and 77.6% were between 90 and 120lbs. This puts the maximum range at 230-250 yards for a 45g arrow, and 200-220 yards for a 70g arrow. The 45g arrow would likely be the standard war arrow, made from poplar, with a light bladed or bodkin head. The 70g arrow would be the armour piercing arrow, made from ash with a heavier bodkin head.

The energy is harder to extrapolate, as the only good data we have is for 70lb and 140-150lb bows, but I would estimate it to be somewhere in the region of 70-90j. This is sufficient to penetrate mail and an aketon, an aketon and a gambeson, or quite probably a jack, but it will have difficulty with plate armour. Only the thinner pieces of plate (arm harness, visor, leg harness) will be penetrated and, if it's of the finest quality, this might not happen either. The issue of angle comes into play as well, as even at point blank range 30% off the arrows will be deflected. At 180m, almost all will ricochet. Adding boiled rawhide to the mail or textile armour will make penetration iffy. (1/2)

The classic English longbow was made of yew. The heartwood is very strong under compression and the sapwood is good under tension, making it superb bow material. Longbows were about as tall as the person wielding them, and draw weights of 100 lbs were common. When examining the remains of people aboard the Mary Rose, a Tudor ship carrying archers, researchers found many of them had abnormalities in line with drawing very heavy bows (e.g., pronounced hyperdevelopment of bone on one side, indicating outsized muscle attachment).

The length of the bow doesn't directly mean more range. The speed and power of a bow are determined by the stiffness of the limbs, essentially, but how much the wood can bend without breaking depends on the type of wood and even how it grew. There is quite a lot of complicated math here about possible draw weight, wood flexibility, etc. However a bow that "snaps" is not fun to shoot; they hurt. There is great skill in making bows flex "just right" and its hard to describe. Bowyery is pretty popular; you might find someone in your area who can show you some bows and let you get hands-on.

The modern theory of bow efficiency has already been covered, so I’ll be focusing mainly on what period sources have to say on the matter to explain what was known at the time, and then on some of the archaeological examples to cover the much greater range of bow designs that existed at the time than just the stereotypical ‘English Longbow’.

Also, although you say ‘medieval’ which is defined as 5-15th century, a lot of this is going to include early 16th cent sources, as that was when treasties on archery really started to be written. I’ll also be focusing predominantly on Western Europe and particularly Britain, with a few notes on Scandinavia as this is what I specialise in.

How do we know about medieval bows?

There are three ways we can study medieval bows:

• Archaeological Finds, which for Western Europe and Scandinavia means:

• 4-5th cent Wremen-Fallward, Denmark, one find

• 5-6th cent Kragehul, Denmark, one find

• 5-8th cent Asby, Sweden, one find

• 5-9th cent Jokkmok, Sweden, one find

• 6th cent, Chessel Down, Isle of Wight, evidence of rotted wooden object assumed to be a bow

• 7th cent Oberflacht, Germany, fifteen finds

• 7th cent Altdorf, Switzerland, one find

• 8th cent Aalsum, Netherlands, one find

• 8-10th cent Waasener, Netherlands, one find

• 8-11th cent Karasmaki, Finland, one find

• 9-11th cent Hedeby, Denmark, seven finds

• 10th cent Ballinderry, Ireland, one find

• 11th cent Vibbymoren, Sweden, one find

• 11-13th cent Waterford, Ireland, seven finds

• 11-13th cent Dublin, Ireland, eight finds

• 11-13th cent L’anse aux Medowes, from the Scandinavian settlement on Newfoundland, one find

• 11th cent Pineuilh, France, five finds

• 12th cent Burg Elmendorf, Germany, one find

• 13-14th cent Viitsaari, Finland, one find

• 15th cent (potentially) Hedgley Moor, England, one find

• 16th cent Mary Rose, England, over 100 finds

• 16-17th cent Mendelsham, England, one find

• Undated Desmond Castle, Ireland, one find

• Undated Abo, Finland, one find

• Oslo, Bergen and Trondheim – dated but I don’t have enough sources on them to discuss.

• Of note but not discussed here are several finds from Russia, Poland and Czechoslovakia and Eastern Europe, of composite construction

• Textual descriptions, the most useful of which are treatises, but also other accounts:

• Livre de Chasse, a 14th cent. French hunting manual that includes instructions on suitable bows for use.

• Le livre du Roy Modus et de la Royne Racio, another 14th cent. French hunting manual with a section on archery

• L’arte d’archerie, likely written in the 15th cent, but printed in 1515, this is a French treatise on archery, describing the equipment used in detail

• Toxophilus, Ascham’s 1545 treatise covering archery in detail. It has some notes on bow construction, but much is left secret at behest of the Guild of Bowyers. That which regards treatment and considerations of what to buy and use for what purpose is likely good however.

• The de Banco Rolls, these are a collection of court pleas that on occasion contain the period version of coroner’s reports, including detailing the weapons used. Two of interest are- the murder of Simon Skeffington in 1298 and the murder of Robert in Staffordshire in 1313.

• The Journey through Wales, a 12th century book by Gerald of Wales that includes a passage on the archers of southern wales.

• Period illustrations, which are too numerous to list. Notable examples include the Harley, Utrecht and Stuttgart Psalters and the Bayeux Tapestry.

From these, it is then a case of piecing together the facts, by cross-referencing the texts to the finds and by studying how the finds relate to each other.

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