WHEN A FISHERMAN CASTS A fly, how should he move the rod and throw the line? The fly is strongly affected by air drag; how then can it continue to move forward after the rod stops? Does the bending of the rod propel the fly as a bow propels an arrow? Why does one change rods and lines according to the kind of fish one is seeking? How does the angler resist the pull of a hooked fish and reel it in?

I have looked into these questions and others on the physics of fishing, drawing on manuscripts from several researchers bank and William W. Buchman of Los Angeles published a paper on the dynamics of fly casting. Mosser has also written several unpublished papers on the subject. Graig A. Spolek of Portland State University and Steve C. Fry of Hawthorne, Calif., have independently sent me manuscripts in which a fly cast is studied mathematically.

Spolek analyzed a forward, overhead cast of a fly. Initially the length of line extending from the tip of the rod is short. To lengthen the line and to prepare for the final cast the angler makes a series of false casts forward and back in which the fly does not touch the water. In each false cast the angler moves the tip of the rod overhead and somewhat to his rear and then rapidly forward. The line follows. Although the rod may not pass appreciably beyond the vertical in the back cast, the line is stretched out horizontally (because of its rearward motion) just as the forward cast begins.

In each false forward cast the angler prevents the fly from touching the water by pulling backward on the rod. During the false back cast a righthanded fisherman uses his left hand to remove additional line from the reel and to hold it in a loop. During the false forward cast he releases the extra line as the forward momentum of the moving line pulls it through the line guides on the rod. In this way the total length of moving line increases.

Once enough line has been released the angler again brings the rod tip and the line rearward and then makes a final forward cast, this time stopping the rod in a forward direction and letting the fly land on the water. Spolek analyzed the flight of the line and fly during this final forward cast. The photograph below shows a view in stroboscopic light of a forward cast made by Mosser.

The ability of the angler to propel a fly for any significant distance is puzzling because of the air drag on the fly. Imagine trying to hurl a fly without a line. To equal the distance commonly achieved with a cast you would have to make a heroic throw. Spolek calculates that to throw a typical fly 65 feet from an initial height of five feet the launching speed must be more than 300 miles per hour. Since the fly is attached to a line that also encounters air drag, the theoretical value for the initial speed is seemingly higher. Even if one argues that the long rod amplifies the throw, the air drag on the line and fly should still eliminate the possibility of a long cast.

This puzzle has been solved by many people who have studied the physics of casting. A forward cast involves a subtle mechanism that accelerates the line and fly after the rod has launched them. All skilled anglers know about the mechanism, but Spolek has shown by mathematical analysis why it is so successful.

To explain the mechanism and Spolek's analysis I shall follow the steps of the forward cast. The analysis begins at the end of the back cast, a point at which the line extends straight toward the rear. As the angler rotates the handle and moves it forward the flexible rod bends toward the rear, delaying the forward motion of the rod tip, line and fly. Once they begin to move the line and fly have the same speed as the rod tip until near the end of the rod's forward motion. As the angler slows the rod to stop it in its most forward position the line travels at the speed it had when the speed of the rod tip was highest. This is the launching speed.

As the rod slows, the line gains on it and sails over the tip, forming a loop that is convex in the forward direction. Between the loop and the tip the line is stationary because the tip is stationary. Between the loop and the fly the line continues to travel forward. Initially most of it is above the loop and moving, but because the loop moves at half the speed of the top part of the line, that part gradually passes through the loop and becomes stationary. When the end of the line reaches the loop, the fly flips over and the line straightens. From that point the fly and line fall to the water.

The loop is the propelling mechanism in a long cast. It concentrates kinetic energy in the continuously decreasing portion of line above the loop. The transfer of energy can be analyzed by starting with the line just before the loop forms. It has a kinetic energy equal to the product of half of the line's mass and the square of the line's speed (the launching speed). This energy has been supplied by the angler.

To simplify the analysis ignore for a moment the air drag on the line and fly. Once the line is launched its kinetic energy cannot change because no work is done on it. The kinetic energy remains constant. Since the amount of moving mass decreases, the speed must increase in order to conserve the kinetic energy. Therefore the part of the line above the loop and the fly attached to it accelerate forward. The acceleration ceases only when the end of the line passes through the loop. Fly casting is actually an exercise in the conservation of kinetic energy that yields an acceleration as the amount of moving mass decreases.

Spolek considered how air drag alters this simple model. Drag diminishes the acceleration because the moving line and the fly lose energy as they do work on the passing air, but the loss is not sufficient to prevent a long cast.

Much of the air drag on the line derives from the broad cross section the loop presents to the air. Skilled anglers know that a long, accurate cast requires forming a loop with a small diameter (about three feet). Larger loops mean greater losses of energy and shorter, less accurate casts. Can the angler reduce the diameter of the loop to less than three feet? Probably not, because the loop is enlarged by the overshoot of the rod when it oscillates around its final resting position.

To investigate the mechanics of a cast Spolek calculated the launching speed required when the line is 65 feet long and the final speed of the fly is 98 feet per second before it flips over. He postulated a uniformly dense line that had a mass typical of fishing line. He also assumed the line develops a three-foot loop. He found that if air drag is neglected, the line and fly must have a launching speed of approximately 20 feet per second.

Spolek also made a calculation for a rig called a double-taper line. It is narrow and uniform in diameter for the first two feet, increases steadily in diameter for the next 10 feet and then becomes uniform again. It has the same mass as the uniform line for the first 30 feet. Spolek found the launching speed for the double-taper line and fly to be less than 20 feet per second. Because the diameter of the double-taper line decreases toward the end, the amount of moving mass declines faster than it would decline if the line were uniform. If the final speed is 98 feet per second for both lines, the greater acceleration afforded by the double-taper line requires less launching speed.

When Spolek included air drag in these calculations, he discovered some interesting changes in the acceleration. If the

fly is attached to a uniform line, it must be launched at more than 98 feet per second to attain a speed of 98 feet per second. More interesting, he found that the speed gradually decreases at first and then, near the end of the path, rapidly increases to its final value. Spolek explains these changes in terms of the competition between the concentration of kinetic energy in the decreasing amount of moving mass and the removal of energy by air drag. Just after the loop forms, the rate of energy being lost to air drag is greater than the rate at which energy is being concentrated. The line and fly slow down. Near the end of the cast the rate of concentration begins to dominate, accelerating the line and fly.

A double-taper line encountering air drag requires a higher launching speed than a uniform line. As in the case of a line of uniform diameter, the speed of the line and fly initially decreases, but near the end of the cast it suddenly increases and then decreases. Some anglers describe the acceleration as a "kick", because the fly pulls on the line

For the sake of calculation Spolek invented two experimental lines tapered along their full length. Each has the same mass as the uniform line for the first 30 feet. One of them, called a long-taper line, has the same rate of change in diameter as the tapered part of the double-taper line. The other one, called an experimental-taper line, has a more gradual taper. According to Spolek's calculations, the speed of a fly on the end of a long-taper line increases at first and then decreases near the end of the trip. If an experimental line is used, the speed of the fly increases throughout the trip.

The fly requires the lowest launching speed with the long-taper line and only slightly more with the experimental line. Each is lower than the final speed of 98 feet per second. With the uniform and double-taper lines the launching speeds exceed the final speed. The differences may not matter because all the launching speeds are within the angler's capability.

A more important input might be the initial energy the fisherman must give the line. If he makes many casts during an outing, he wants to minimize the energy required for each one. Spolek calculates that the uniform line requires the least initial kinetic energy followed (in order) by the experimental-taper, the double-taper and the long-taper lines. The long-taper line requires almost twice as much launching energy as the uniform line.

Fry's analysis dealt with the dynamics of the line and rod during the forward drive. In his model an external force (from the angler) compresses a spring (the rod) attached to a mass (the mass of the rod). The spring drives forward a second mass (the line).

The dynamics of the model can be ascertained only approximately. Much of the complication derives from air drag, which affects the rod as it moves forward and also as it oscillates at the beginning and end of that movement.

The force on the line rises rapidly and then falls during the forward drive of the rod. Initially the drag on the system is small because all components are moving slowly. The drag increases as their speed rises. Fry found that midway through the forward drive much of the energy supplied by the angler goes into accelerating the line whereas somewhat less goes into accelerating the rod. Much less energy is lost to the air drag on the rod and an even smaller amount of energy is lost to the air drag on the line.

Mosser and Buchman studied how an angler should execute a forward cast to give the line and fly a large amount of

energy. If the back cast has not directed the line straight back, part of the rod's forward motion is wasted in straightening it, leaving less time for the angler to do work to give it kinetic energy. Another common error is to rotate the rod without also driving it forward. Then the line and fly move in an arc instead of a straight line. Again less work is done on them and they derive less kinetic energy.

Some anglers believe the rearward bending of the rod during the final back cast stores energy in it, enabling it to function like a bow. According to studies done by Mosser, Buchman and Fry, not much of this stored energy ends up in the kinetic energy of the line and fly.

The bending of the rod does serve a subtle purpose in a forward cast. It allows the line and fly to follow a path that is almost horizontal and therefore increases the efficiency of the angler's effort. If the fly moves in an arc, part of the angler's effort goes into pulling the fly upward and then downward along the arc, a trajectory that deprives it of forward speed.

The proper size of rod is partially determined by the size of the fish being sought and also by what is comfortable. The advantage of a long rod is that a given forward motion of the angler's hand moves the rod tip through a greater distance than the same motion would move a shorter rod. The angler may then have more time in which to work on the line and fly, giving them more kinetic energy. The disadvantage of a long rod is that it is harder to move and rotate. If the rod is too long, the angler is forced to move and rotate it too slowly to make a cast. An angler usually winds up choosing a rod that achieves a suitable compromise between the extremes.

When a fish is on the line, what determines the force on the hook? Does it depend on the length or bending of the rod? Mosser argues that the bending does not determine the force on the hook and that a long rod may be a disadvantage in a fight with a strong fish. If the pull by the fish exceeds the total resistance of the line, additional line unwraps from the reel. At such a time the fish is said to be "running." When a fish is running, the force on the hook exceeds the resistance of the line, but otherwise the force is never greater than the resistance no matter how much the rod is bent.

With a fish on the line the angler feels a direct pull and a torque that attempts to rotate the rod about an axis near his hand. The lever arm for the torque is a line that runs from the axis of rotation on the grip to the rod tip, where the force from the fish is applied. The force from the fish can be split into two components, one component parallel to the lever arm and the other perpendicular to it. The perpendicular component produces the torque, which is the product of that component and the lever arm.

Two factors determine the size of the components: the force from the fish and the orientation of its pull in relation to the lever arm. If the angler points the rod toward the fish, the torque vanishes because no perpendicular component of force is exerted on the rod. In this case the entire force exerted by the fish is delivered parallel to the lever arm, and the angler pulls in the opposite direction to keep the rod stationary. If the angler rotates the rod vertically in such a way as to make the force from the fish entirely perpendicular to the lever arm, the torque is at its maximum.

To hold the rod stationary the angler must resist both the pull and the torque created by the fish. To counter the torque he must generate a torque that works to rotate the rod in the opposite direction. The torque is produced by a force from the angler's hand. The lever arm associated with the force is short because the hand is near the point about which the torques attempt to rotate the rod. Because of the short lever arm, the angler must provide a large force if his torque is to match that of the fish. Here the bending of the rod serves a purpose because it reduces the length of the lever arm affected by the fish's torque and therefore also reduces the torque the angler is required to supply.

If the angler must fight a large fish pulling strongly, a long rod means the torque from the fish is large. To reduce e the torque the angler should use a shorter rod so that the lever arm of the torque exerted by the fish is reduced.

To reel in a fish an angler usually rotates the rod vertically so that the fish is drawn toward him. Then he winds line onto the reel as he lowers the rod tip. The procedure is repeated, with the fish being pulled closer when the tip is raised and remaining in place when the tip is lowered. In this way the angler provides the work needed to move the fish through the water.

Many questions remain to be answered about the physics of fishing. You might want to investigate other types of fishing lines that are available e commercially or to invent new ones, as Spolek did. You might also follow the analysis by Mosser and Buchman on how the flexibility of the rod, which is called the rod action, changes the cast. For example, if the rod is highly flexible and vibrates at the end of its forward motion, is the transfer of energy to the line altered?

You could also study several other types of cast. Mosser and Buchman have analyzed a lure cast in which the forward momentum of a somewhat heavy lure pulls line from the reel. Does the bending of the rod during the back cast supply kinetic energy for the launching of the lure? What is the e proper method of moving the hand to make this cast?

I am also intrigued by the roll cast, which no one seems to have analyzed mathematically. This cast is employed when obstacles to the rear prevent the angler from making the back cast have described. After he plays out about 15 feet of line on the water he brings the rod tip from a forward, eye-level position to a high, slightly rearward position. As the line is partially pulled from the water, it begins to pass by the side of the angler, bowed rearward. The angler then quickly moves the rod tip forward. This motion creates a complete loop in the line that travels toward the fly. After the loop reaches the fly the line is straight on the water in front of the angler.

How is the loop maintained as it travels along the line? What becomes of the energy the angler gives the line? How does the speed of the line at the top of the loop change as the loop moves? When the loop reaches the fly, is more line pulled from the rod? What role does the surface tension of the water play in the cast? Mosser, Buchman, Fry, Spolek and I should like to hear what you find out.

Bibliography

ON THE DYNAMICS OF A BULL WHIP. B. Bernstein, D. A. Hall and H. M. Trent in The Journal of the Acoustical Society of America, Vol. 30, No. 12, I pages 1112-1115; December, 1958.

THE DYNAMICS OF A FLYCAST. Ed Mosser and William W. Buchman in The Flyfisher, Vol. 13, No. 4, pages 5-9; 1980.

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