# Will you get less wet running in the rain?

This is something I did one morning while waiting to take an exam that

afternoon on mathematical methods and fluid mechanics. I couldn't bring

myself to do any more revision but needed to keep my mind on maths. As

it was raining outside I started to wonder whether it was possible to

build a simple mathematical model for an object moving in the rain. What

follows is a write up of my thinking that morning.

Common sense would say that one will get less wet running in the rain

than walking, but this might not necessarily be true. The slower the

speed travelled the more water will land on one's your head, but the

faster the speed the greater the water hitting one's front. Given that,

for most people, the surface area pointing vertically upwards (top of

head, shoulders and chest) is much smaller than that of their surface

area pointing forwards (face, front of torso and limbs), it is possible

that travelling at greater speeds could in fact make one wetter. To

investigate this I build a simple mathematical model for an object

moving through rain and start by looking at the case when the rain is

falling directly downwards (there is no wind). Later I develop the model

to include the effects of wind.

## Assumptions

- Rain can be modelled by a constant density vector field.
- Rain travels at a constant velocity downwards.
- A person can be modelled by a regular cuboid.
- The person moves at constant velocity through the rain.
- Rain that comes in to contact with the person is assumed to be

perfectly absorbed. - Other than the person there are no other surface effects that need

to be considered.

The first assumption states that rather than modelling the rain as

individual drops it is possible to model it spread out evenly throughout

space. This appears reasonable as long as the rain droplets are

significantly smaller than the object moving through them and there is

no clumpiness; i.e. they are evenly spaced out. This the key assumption

of the model, without this the following maths is invalid.

The second assumption asserts that rain moves at a constant speed

towards the floor and doesn't change direction. While in the real world

the wind will affect the direction and speed of rain for the sake of

simplicity this model will ignore these effects. This assumption will

make the model far simpler than including wind direction, but is an

unrealistic assumption, any further models should include wind effect.

The third assumption appears to be fine for out basic initial model,

further investigation is needed to confirm this for more advanced

models.

The fourth assumption states that we won't be taking the acceleration

of the object into account. As the velocities involved are quite small I

think this is fair. Again, a more developed model may want to take this

into account.

The fifth and sixth assumptions are similar; none of the water hitting

the object bounces off and there is no spray onto the object from other

surfaces. Both of these are appear to be okay for our simplistic model,

but they don't realistically represent a person running in the rain

where there will be some spray effects.

## The Model

Let the rain be described by the vector field

where:

- $\mathbf{R}$ is the rain vector field
- and $\alpha $ is the downwards speed of the rain

The body, $O$, moving through the rain is a regular cuboid with corners located at $(0,0,0)$, $(x,0,0)$, $(0,y,0)$, $(x,y,0)$, $(0,0,z)$, $(x,0,z)$, $(0,y,z)$ and $(x,y,z)$.

The equation for the velocity of $O$ through the rain is

where:

- $\mathbf{s}$ is the displacement vector
- and $\beta $ is the speed

Given the above mathematical definitions it is possible to calculate the flux of the vector field $\mathbf{R}$ for each face of $O$ as it moves at velocity $\frac{d\mathbf{s}}{dt}$

by calculating the surface integrals, giving the volume flow rate

through each surface. In other words how wet each surface gets per unit

of time. The model has been created so only two surfaces need to be

considered; the top and the front faces. That is surface ${S}_{\mathrm{top}}$ with corners located at $(0,0,z)$, $(x,0,z)$, $(0,y,z)$ and $(x,y,z)$, and the surface ${S}_{\mathrm{front}}$ with corners located at $(x,0,0)$, $(x,y,0)$, $(x,0,z)$ and $(x,y,z)$.

The surface integral for a face $S$ is defined as

where:

- $\mathbf{F}$ is the vector field
- $\mathbf{n}$ is the unit normal vector for the surface
- and $A$ is the area of the surface

For our model it is necessary to calculate how wet each face gets through time. The equation for this is

where:

- ${w}_{S}$ is the wetness of surface $S$.

As the body is moving through a moving field the two vector equations describing velocity need to be combined, hence

For ${S}_{\mathrm{top}}$ the face normal points directly up, so

therefore

and

as ${w}_{\mathrm{top}}=0$ when $t=0$ hence $C=0$, the surface is dry at the start,

therefore

For ${S}_{\mathrm{front}}$ the face normal points directly outwards, so

therefore

and

as ${w}_{\mathrm{front}}=0$ when $t=0$ hence $D=0$, again, the surface is dry at the start,

therefore

The total wetness of the body if found by summing the wetness of all the faces.

So

The ratio of rain landing on the top surface to rain hitting the front surface is

## Initial Analysis

The model has produced two equations that will help to shed light on the

initial question of how the speed travelled in the rain effects how wet

a body gets. These are equation (1), giving the total wetness of the

body at a specified time, and equation (2), giving a ratio for the

proportion of rain landing on the top compared to the front of the body.

I'll start with an analysis of this surface wetness ratio.

### Ratio of Surface Wetness, $W_{ratio}$

Equation (2) gives a ratio for the proportion of rain landing on the top

compared to that hitting the front, and depends on 4 independent

variables: the speed of the rain, $\alpha$; the speed of the body,

$\beta$; the thickness of the body, $x$; and the height of the body,

$z$. This ratio shows that, according to the model, the two factors

affecting how wet the top of the body gets, compared to the front, are

the speed of the rain and the thickness of the body. Likewise, the two

factors affecting the front compared to the top are the height and the

speed of the body. By trying to find standard values for the rain speed,

and body width and height it is possible to see how this ratio changes

according to the speed the body travels at.

- According to

[1]

the speed of falling rain depends on the size of the raindrop.

Drizzle falls at 2 m/s, a 2 mm raindrop falls at 6.5 m/s, while a 5

mm raindrop falls at 9 m/s. To compare the effect of different rain

speed I'll take a low speed of 3 m/s, a mid speed of 6 m/s, and a

high speed of 9 m/s. - The mean height of an adult in England in 2008 was 1.68 m

[2].

To compare the effects of height I'll take a short person to be 0.3

m lower than this and a tall person 0.3 m higher. Given that people

aren't cuboid taking 80% of these values should give a fair value

for the height components. This gives the heights 1.10 m, 1.34 m,

and 1.58 m for a short, average and tall body, respectively. - To estimate people's thickness, finding statistics has been hard.

There is data for waist and chest circumferences, but turning these

into a reliable figure for thickness is nigh on impossible. The

figures I've chosen are arbitrary but hopefully represent a

reasonable estimate for the thickness of an individual. I will take

the thickness for a slim, average and large person as 0.2 m, 0.35 m

and 0.5 m, respectively. - The average walking speed is 3 miles per hour

[3],

or about 1.3 m/s. An Olympic sprinter can run 100m in under 10 s

[4],

an average speed of under 10 m/s.

## References

- [Weather Almanac for April 2003 - The Energy of a

Rainshower]{#ref1} - [NHS Health Survey for England

2008]{#ref2} - [Wikipedia article on

walking]{#ref3} - [IAAF list of fastest 100 m

runners]{#ref4}