Cool temperatures and showers during the first week of Roland Garros this year have had a mass of pundits and players talking about the “heavy” playing conditions. Some said those conditions favored players who hit the ball especially hard, or implied they needed to be “hit through”. Andy Murray described the conditions as both “heavy” and “slow”, implying that the balls moved more slowly than usual or that rallies tended to be longer.
Are these remarks consistent with the physics of how the ball moves through the air? Are they even consistent with each other? What specific effects do so-called “heavy conditions” have on the way a tennis match unfolds? Let’s examine how various atmospheric conditions really influence the game.
A Quick Introduction to Aerodynamics
First, we’ll need to understand a few basic things about how the air affects a ball’s flight path. Once it leaves a player’s racket, a ball naturally tends to fall due to gravity, but it also loses speed due to friction with the air. If the ball is moving anywhere near the speeds seen in a typical professional match, this drag force is greater than the force of gravity. It can be described by this formula:
CD is the ball’s drag coefficient
ρ is the air density
r is the ball’s radius, and
v is the ball’s speed, relative to the air
Notice that the drag force is proportional to the air density—a 10% increase in density, for example, causes a 10% increase in drag.
Also, notice that the drag force is proportional to the square of the speed, meaning that a 10% increase in speed causes a 21% increase in drag. One practical result of this is to level the playing field somewhat between players who hit with exceptional power and those who don’t. Balls might leave a player’s racket with 10% more speed than her opponent’s, but those shots experience disproportionately more drag, and arrive at the opposite baseline carrying only about 7% more speed. This equalizing effect is only magnified by slow conditions in which the drag force is stronger. Just like a slow court surface, slow dense air helps defensive players who play with less power by minimizing the advantage wielded by bigger-hitting opponents. Players trying to “hit through” dense air are likely to be frustrated—their shots will reach the other end of the court a bit faster, but generally not enough to justify the extra effort and loss of accuracy.
Why Spin Makes Balls Curve
When the ball is spinning, there is also a lift force that acts at right angles to the ball’s motion:
As you may have guessed, CL is the lift coefficient. Unlike the drag coefficient, which is fairly constant over the range of speeds tennis balls travel, the lift coefficient varies, roughly in proportion to the spin coefficient. If the ball’s rate of spin ω is expressed in terms of radians, the spin coefficient can be calculated simply as:
Most commonly in modern tennis, balls are hit with topspin, which causes them to curve downwards, allowing them to be hit with more net clearance and still land in play. Balls can also curve sideways due to sidespin—most often seen with slice serves and kick serves—but no matter what the force’s direction, aerodynamicists still call it the lift force. To see what causes this force and why it works in the direction it does, let’s look at how air flows around a spinning ball. In this diagram, based on photographs taken in a wind tunnel where streams of smoke were injected into the air, the blue areas represent layers of air moving around the ball:
As the air moves past the widest part of the ball, it rushes to fill the hole punched by the ball’s motion. The back of a sphere is blunt enough that if it’s moving with any speed, the air doesn’t have time to fill the space smoothly, with its layers intact. Instead, its flow becomes turbulent, tumbling chaotically like churning water in the wake of a speedboat, and leaves a low-pressure area behind the ball. The difference in pressure between the front and back sides of the ball is the main contributor to the drag force.
If the ball is spinning, the air moves faster relative to the ball’s surface on the side that’s spinning into the airflow. This causes the airflow to become turbulent earlier on that side, shifting the low pressure wake toward that side as well. On the other side of the ball, the airflow stays smooth longer, allowing it to follow some of the curve on the back surface of the ball. In the case of the topspin shown in the diagram, the net effect is to deflect the air upwards as it flows around the ball, and the reaction force pushes the ball downward.
This phenomenon is called the Magnus Effect, and it’s also responsible for curveballs in baseball and swing pitches in cricket (although in some sports, it’s triggered more by careful placement of seams or other textural elements on the ball surface than it is by spin). The fuzz on a tennis ball strengthens the Magnus Effect, by helping to break up the otherwise smooth airflow around the sides of the ball.
By creating more turbulence, the fuzz also causes tennis balls to experience more drag (higher drag coefficients) than smoother balls. On the other hand, by creating a relatively thick turbulent zone around the ball at a wide range of speeds, the fuzz also makes the drag coefficient of a tennis ball vary much less with changes in speed, compared to balls in other sports. This makes the flight path of a tennis ball easier for players to anticipate. (Interestingly, the dimples on the surface of a golf ball have the opposite effect: they create a thin, turbulent boundary layer, which helps pull the bulk of the airflow smoothly into the ball’s wake, and dramatically reduces the drag coefficient at higher speeds.)
Now, back to those atmospheric conditions.
When people in the sport talk about “heavy” playing conditions, humidity is the atmospheric variable they mention most often. It comes up during rainy weather like at this year’s Roland Garros, in the sauna-like conditions that often prevail on the US east coast during the hardcourt season, and has even been blamed for slowing down play when the roof closes at Wimbledon. Conversely, one popular justification for the widespread belief that balls tend to fly further than normal at Indian Wells is the dry desert air.
There are two problems with such theories. Most importantly, they’re backwards. Contrary to common intuition, humid air is less dense than dry air, and therefore creates less drag on a ball. This is a consequence of the ideal gas law, which implies that at a given pressure and temperature, a given volume of any gas contains the same number of molecules. Dry air consists almost entirely of nitrogen molecules (N2, molecular weight 28) and oxygen molecules (O2, molecular weight 32). Diluting it with molecules of water vapor (H2O, molecular weight 18) makes it lighter.
The other problem with blaming humidity for slowing down or speeding up play is that, in the conditions under which most tennis matches are played, the effect of humidity on air density is very small. At 70°F (21°C), a change in the humidity from 0% to 100% decreases the air density by a mere 1%, which increases the speed at which a fast groundstroke arrives at the opposite baseline by less than 0.3 mph (0.5 km/h).
At very high temperatures, the effect is larger because hot air at 100% humidity holds more water than cooler air at 100% humidity. But heat combined with high humidity is so hard on the human body that WTA rules, at least, require suspension of play under such conditions (see below).
The reason humid air feels heavy to our senses isn’t because of its density or viscosity—it’s because the more water is already in the air, the less sweat evaporates from our skin. That forces our bodies to sweat harder and do more work to keep cool, leaving less energy available for athletic performance. Taken to the extreme, it causes dehydration and excess body temperature, which can cause the body’s mechanisms to malfunction altogether.
Humidity does have some effect on the surface characteristics and mass of the ball, as well as the air density, but these effects turn out to be very small and largely cancel each other out. More on this later.
It seems Stephane Bohli described humidity’s effects on the game more accurately than most while playing a Challenger tournament in New York last year, saying “You feel like your racket is a little bit heavier… that your legs are a little bit heavier, everything is a little bit heavier.” It’s more about humidity’s effect on a player’s body than it is about the effect on the ball.
Temperature and Pressure
As you can see in the diagram above, temperature has a much stronger effect on the air density than humidity does. A change in temperature from 50°F (10°C) to 100°F (38°C) reduces the air density by 10%. That makes a fast groundstroke arrive at the opposite baseline traveling a noticeable 2–3 mph (3–5 km/h) faster, depending on the spin and depth it’s hit with. The effect of the cool temperatures this past week at Roland Garros dwarfs the small opposing effect of the moderately high humidity, and is undoubtedly the main contributor to the slow playing conditions many have observed.
Another significant contributor is an atmospheric variable I’ve never heard mentioned in tennis, whether by a player, coach, or commentator: the barometric pressure. The ideal gas law tells us that the density of a gas is directly proportional to its pressure. For most locations on Earth, the barometric pressure (corrected for altitude) stays between 29 and 31 in Hg (980 and 1050 mb) the vast majority of the time. (Much lower pressures have been measured, but in places like the eyes of hurricanes and typhoons, circumstances under which tennis is unlikely to be played.) Typical variations in pressure lead to a potential variation of nearly 7% in the density of the air.
Pressures have been moderate during Roland Garros so far this year, staying between 29.74 and 30.19 in Hg (1007 and 1022 mb), but particularly in the first week, they were higher than the pressures usually seen during wet weather, and that may have contributed to the perception that conditions have been unusually slow.
The biggest differences in air density from one tournament to the next are caused by altitude. Tennis commentators talk about altitude most often in reference to Madrid, the only Masters 1000/Premier Mandatory level tournament held at significant altitude. The Caja Mágica in Madrid is at 1860 ft (567 m), where the air density is more than 6% lower than at sea level, but it’s far from the highest venue on the tour. Here are the altitudes and relative air densities of the actual tournament sites for every currently- or recently-held tour-level event above 1000 ft, or 300 m:
|Tournament||Altitude, ft||Altitude, m||Air Density, % of sea level|
As you can see, events like Kitzbühel, São Paulo, and Bad Gastein are at significantly higher altitudes than Madrid, and the air density at these tournaments in very cold temperatures is as low as it would be in very hot conditions at sea level. Bogotá, Colombia and Quito, Ecuador host by far the highest events on the main tours, with air nearly three times thinner still.
Such altitudes make enough difference in the flight paths of tennis balls that we can easily see the effects visually, as well as in quantitative measurements. Using an evolved version of the computer model I developed for my analysis of court surface speeds, I examined how high altitude affects groundstrokes with varying kinds of spin. First, a flat shot with no spin (with the vertical scale magnified to emphasize differences in shape):
At high altitude, the ball experiences less drag, lands deeper in the court, and retains more speed. This shot, aimed to bounce 14 inches (36 cm) inside the baseline at sea level, lands nearly an inch (2 cm) beyond the baseline in Madrid, and over 4 feet (1.3 m) long in Bogotá.
Now, let’s look at a shot with heavy topspin:
Many people think topspin shots, with their higher net clearance, are inherently safer shots in a more general sense. But any shot played close to the lines or near the limits of a player’s ability is risky. Here we see that topspin actually makes a ball more sensitive to changes in air density. Topspin shots are struck at a higher initial angle, and rely on both aerodynamic drag and downforce from the spin to shorten their flight and pull them back down into the court. Thinner air weakens both of these forces. The topspin shot lands 19 inches (48 cm) long in Madrid and nearly 12 feet (3.6 m) long in Bogotá.
Finally, here’s a heavy slice:
A hard slice shot can actually be launched downward, relying on its aerodynamic lift to carry it over the net. As it continues into the opponent’s side of the court, drag slows it down, which sharply reduces the amount of lift it experiences, and it falls into the court. Since slices are aimed more directly in line which their intended landing point, they rely less on drag to keep them from going out of play. Instead, in thinner air, the loss of lift early in their flight makes balls land shorter in the court, even though they retain more speed. A hard slice aimed to land nearly 14 inches (34 cm) inside the baseline at sea level lands 2.7 feet (84 cm) inside the line in Madrid, even though it’s moving 2.2 mph (3.5 km/h) faster. In Bogotá, the shot just barely clears the net, and lands 8.9 feet (2.7 m) inside the baseline, carrying 9.7 mph (16 km/h) more speed.
What About Indian Wells?
Some of you may have noticed that Indian Wells does not appear in the table of tournaments at significant altitude. Indian Wells has an oft-repeated reputation as a place where balls fly further than players expect, and reputable news organizations, coaches, and star players like Murray and Maria Sharapova have all casually and erroneously asserted that this perceived behavior is due to high altitude. I’m not sure why. Because there are mountains visible on the horizon? Southern California has what the locals call a “high desert”, but it’s on the other side of those mountains to the north. The tournament site is at 145 ft (44 m), which accounts for a mere 0.5% difference in air density, compared to sea level. Nearby the Salton Sea, a salt lake with no outlet, lies 226 feet (69 m) below sea level.
In case you’re wondering, Roland Garros is the highest of the slam tournament venues, at 123 ft (37 m).
How Does Humidity Affect the Ball Itself?
Atmospheric factors alone simply cannot explain the perception that “balls fly” at Indian Wells, or the association of humidity with “heavy” conditions. It has been suggested that humidity might make the fuzz on the surface of the ball fluff up, increasing the ball’s effective diameter and therefore the drag it experiences.
I conducted my own experiments to test this idea. I opened cans of Wilson US Open Extra Duty balls at low and moderately high humidity levels and tumbled them in a clothes dryer, on the no-heat setting, to simulate the surface wear they experience during a match. I maintained the initial humidity inside the dryer throughout each test, and took high-resolution photographs of each ball before and after the tumbling process. After carefully measuring the effective diameter of the fuzz on each ball, I found that balls used at 61% humidity were effectively 0.6 mm smaller on average than balls used at 25% humidity. This difference is small, and just barely statistically significant—it’s only slightly larger than the typical variation between the individual balls I tested. But to the extent it affects balls’ behavior in the air, it only adds to the density-reducing effect of humidity, making balls travel further and faster.
Here are images and measurements of typical balls from the test. (The differences in color are due to different lighting conditions and the fact that I processed the images to optimize contrast of the fuzz fibers, not due to any difference in wear between the balls.)
High humidity has other effects, beyond this surprising reduction in the effective diameter of balls measured at rest. In humid conditions, the fuzz on a ball absorbs a measurable amount of water, increasing its mass slightly. At 27% humidity, I shaved as much fuzz off a ball as I could and gathered it into a small container. I weighed it on a high-precision scale, and then exposed it to 93% humidity for about 15 minutes, sealed the container, and weighed it again. It absorbed 0.04 grams of water, increasing its mass by 7%. This represents a mere 0.07% increase in the mass of the entire ball, which by itself isn’t enough to have a noticeable effect on the ball’s flight.
But it does affect the way the fuzz interacts with the air as the ball is in flight. Wind tunnel tests have shown that spinning balls have higher drag coefficients than balls with no spin. This is because the centrifugal effect makes the fuzz stand up off the surface of a spinning ball, creating extra disruption in the air flow around it. There are dramatic high-speed-camera photographs of this effect here (see Figure 12 in the link). By making the fuzz heavier, humidity enhances the centrifugal effect and makes the fuzz stand up higher.
I modified my computer model to take this into account and see how much it affects balls’ flight paths and speed in a match. The effect is measurable, but small. It’s just enough to nearly cancel out the effect of humid air’s lower density. The net result of all of these effects is that at 75°F (24°F), changing the humidity from 10% to 90% makes a deep topspin groundstroke land less than 11 inches (27 cm) deeper in the court, and increases its speed at the baseline by less than 0.6 mph (1 km/h). Other shots are affected significantly less. Once again, the results disprove the popular perception that humidity makes playing conditions slower, but the effect is so small it’s unlikely to make much difference in a match.
Do Heavy Balls Create Heavy Conditions?
Sometimes humid air is accompanied by rain, which potentially increases the mass of balls much more than humidity alone. On hard courts or grass, it doesn’t take much rain to stop play. But grass might still be damp enough when play resumes to get the balls wet. Saturated with water, but not actually dripping, a tennis ball weighs about 1 gram, or 1.8%, more than a dry ball. This further increases the degree to which the fuzz on a spinning ball stands up in flight, and the increased mass is also enough to reduce the speed with which balls come off a given player’s racket by nearly 1%.
This combination of increased drag and decreased initial speed is enough to noticeably slow the ball’s flight, but not as much as you might expect. To some extent, the extra mass of the ball helps it resist the tendency of the added drag to slow it down. In humid conditions, a wet ball hit deep with heavy topspin lands 8 inches (20 cm) shorter than a dry ball in dry air, takes 0.03 seconds longer to reach the baseline, and arrives traveling 1.7 mph (2.7 km/h) slower. Flat shots are affected less, but slices are affected even more, landing 4.6 feet (1.4 m) shorter, and reaching the baseline 0.05 seconds later with 2.5 mph (4.0 km/h) less speed.
On clay, matches can continue during light rain, and wet balls can pick up clay particles, adding somewhat more mass and slowing play even further. On the other hand, the significance of these effects in a match is limited by the fact that the balls don’t stay this wet for long. The impact of the racket strings and the spin flings much of the water off of them. If the humidity falls significantly after a shower, the air flowing around the ball in flight will dry the fuzz even further.
The added mass of a wet ball probably has a stronger effect on the way it feels on a player’s racket. If the ball is heavier, a player has to hit it harder to impart the same speed. More importantly, the water is all on the outer surface of the ball, where its added mass makes the greatest possible difference to the ball’s moment of inertia, or resistance to changes in spin. Adding 1 g of water increases a ball’s moment of inertia by 2.1%. The resulting 1% reduction in spin imparted to the ball by a given stroke doesn’t have a noticeable effect on the ball’s trajectory. But the extra resistance makes it noticeably more difficult to add spin to the ball, or even to neutralize the spin applied to it by an opponent’s racket and by its bounce off the court surface. Over the course of a match, this could help a strong player’s ability to wear down an opponent with a barrage of spin and power. The effect is felt as stronger recoil through the racket frame, contributing to greater fatigue in players’ arms. Here it actually makes sense to describe playing conditions as “heavy”, rather than slow or fast.
Like a lot of tennis lore, prevailing beliefs about how atmospheric conditions affect the game are a combination of truths, half-truths, and delusions made credible by repetition and recirculation. Scientific analysis reveals that many effects are more complicated than they appear, and often subtler than we expect. Often quantifiable reality differs from the tactile perceptions of players in the rush of battle. High altitude, high temperatures, and low pressures can all speed up play, in that order. Humidity has very little practical effect. Heavy balls, especially if they’re heavy because they’re wet, can make playing conditions heavier in a very real sense. But that’s a different kind of effect, sapping the strength of players more than it changes the flight of balls.