Read all of this first and prepare your tools and materials.
Woodwind repair materials can be obtained from Wind Craft Ltd.
Clean away all old cork, grease and glue residue. Use a knife blade and scrape the wood clean. It does not matter if you remove some varnish.
Measure the thickness of the original cork. (Use callipers if you have them.) 1.5mm is usually suitable if you are uncertain. (See the note at the bottom of this page for help with the measurement.)
Select a suitable piece of cork and cut it to fit the groove using a very sharp blade (scalpel) and a steel rule. Trim it to length carefully, slightly too long is better than too short. Cut the end clean and square. A good butt joint is entirely satisfactory.
Mix a small quantity of epoxy (preferably quick setting, the common sort) on a scrap of waste card. (A pea sized total mix is usually enough.)
Spread a thin layer of the epoxy glue to the groove in the wood and the butt end of the cork. A large saxophone spring is an ideal tool for this. A large darning needle is a domestic substitute.
I have recently started using 'Gorilla' glue for small repair jobs. It is expensive but economical in use as no mixing is needed. Read the instructions carefully! This polyurathane glue cures in the presence of water and foams, so that it penetrates the work. It has very good adhesion to synthetic cork. I find that a small trail of glue about 1mm. across, straight out of the bottle, will spread over the usual joint area. You can always add a bit more if necessary, but don't over do it. Damp the inner face of the cork, and one end. Don't forget to apply glue to the other end of the cork strip.
Place the cork round the joint groove and bind it into place. Make very sure that the ends of the cork are butted together tightly. There is a problem with using thread (button thread) to bind the cork as it tends to leave indentations. I have found that gift wrap tape made of shiny plastic (or ribbon) is ideal as it is about 5mm wide and does not leave any mark. Fasten the end with a short length of sticky tape while the glue sets. You will need about 2 metres of thread for a treble. Work quickly.
Leave the joint to set (at least an hour, but see the instructions on the glue).
Remove the binding and try the fit. Do not apply grease if you anticipate sanding afterwards. If it is too tight, carefully sand the cork down with glass paper wrapped round a support. It is usual for cork lapping to be either flat of slightly convex.
However, it is not easy to make joints fit 'easily' without also making them prone to wobble. One answer to this problem is to sand the cork using glass paper wrapped round a circular section support, a piece of dowelling or a bottle cork. Keep to the middle of the cork and sand a shallow groove all the way round, adjust the width as well as the depth. This will reduce the area of contact and so, the friction when the joint is turned, without reducing the effective length of the joint. The result has much in common with joints that have two separated narrow bands of cork and it is a good way of dealing with a stiff tenor of bass recorder joint. Do not mark the wood.
Natural cork becomes much more flexible and manageable if it is hammered before use. You can do this to alter the thickness. Use many light blows on the pre-cut strip. It will grow in width and shrink in thickness. Use the distortion of its shape as a guide to your hammering and try to keep it a true rectangle. Re-trim when the thickness is right.
This same technique works very well for the synthetic cork substitute.
Glue the cork onto the instrument fairly quickly after hammering otherwise it will creep back to its original thickness.
If you have callipers you can judge the required cork thickness by halving the difference in diameter of the two parts of the joint and choosing cork the next half millimeter size larger.
If the tenon (the plug in part) is cracked, you can fill the crack with glue and then bind the wood with fine wire, cotton or polyester thread and glue before hiding it with cork. (Cut away the ends after the glue sets, there should be no knots.) You can also hide a packing layer of paper under the cork if you need a bit of extra thickness.
I far prefer a vegetable grease to vaseline (petroleum jelly) as a cork lubricant. Clear or white lipstick works well too. Joint grease for orchestral woodwind is entirely suitable and much better than the concoction provided by most recorder makers.
If you do not have callipers to measure the diameter there is a better way than by holding a ruler across the end. Cut a strip of paper and wind it round the part you want to measure. Don't line the edge up pefectly at the overlap. At some place where the strip overlaps itself make a neat mark with a sharp pencil across the paper from the upper part to the lower part. Unwind the paper and measure the distance between the two marks with a ruler, and note it down. Do you remember the maths you did at school, but have never used? Forgive me... all you have to do now is to divide your measurement by π ('pi'). 3.1412 will do if you have a basic calculator without it available as a constant. If you are working with pencil and paper or a slide rule, 22 divided by 7 is a good enough value of π for all practical domestic purposes. (To divide by this, multiply by the 7 and divide by the 22.) If you want to know more (there is a lot more!) key 'pi' into Google.