Why do open car windows make a throbbing noise?
Introduction
Have you ever opened a window in a car, only for your ears to be met with an annoying throbbing noise? It’s a rhythmic thumping sound, sometimes fast, sometimes slow. It can very quickly become uncomfortable (and the driver of the car or passengers may ask you to close the window!).
What causes the throbbing sound?
The throbbing sound is caused by a phenomenon named Helmholtz resonance. It describes how air passing through a gap into a cavity can resonate at a particular frequency.
When moving air passes through an open window, it passes into the car. The inside of the car can be thought of as a cavity.
When incoming air from the outside passes through the window, it causes a pressure difference at the window’s opening. The incoming air wants to push in, but the air already in the car wants to push the coming air out. This causes a repeated cycle of air moving in and out, in and out of the car, creating oscillations.
These oscillations cause pressure waves to form inside the car, which in turn cause an audible throbbing sound.
How does the throbbing sound get so loud?
If the speed of the incoming air is just right, the throbbing sound it produces matches the natural resonant frequency of the car’s interior. This in turn amplifies the throbbing sound, making it louder.
This is how a singer can break a glass just by singing. If the singer produces sound waves that match the resonant frequency of the glass, it causes the glass to vibrate back and forth, back and forth. If too much energy is put into the glass, it shatters.
In the case of a car, when the frequency of the throbbing sound waves and the car’s own internal resonant frequency match, the air pressure waves inside the car grow stronger. This creates a much louder sound.
The exact speed at which all this happens depends on the shape and size of the car’s interior, plus the size of the open window.
How do you calculate the Helmholtz resonance?
This is where maths comes in! You can calculate the Helmholtz resonance with this formula:
The formula
In this formula:
- \(f\) is the frequency of the Helmholtz resonance in hertz
- \(v\) is the speed of sound in air in metres per second (approx. 343 m/s)
- \(\pi\) is the mathematical constant Pi, approximately equal to 3.14159 (only it goes on forever, so this is just an estimation)
- \(A\) is the area of the opening (in the case of a car, this is the area of the open window) in m2
- \(V_o\) is the volume of the cavity (in the case of a car, this is the volume of the car's interior itself) in m3
- \(L\) is the length of the opening's neck (in the case of a car's window, this can be estimated as the thickness of the window) in metres
Example calculation
Let's do an example calculation. Say you are driving down a road and you open your car's window by 5 cm. The window itself measures 57 cm across.
The volume of the car can be hard to calculate, as it is a very irregular shape. However, we can estimate it.
Say you measure the car. The height is 1.3 m, the width is 1.4 m and the length (from the dash to the end of the boot, or trunk for my American readers) is 2.6 m.
This gives a volume of 1.3 × 1.4 × 2.6 = 4.732 cubic metres (m3).
Now for the area of the open window. If the length of the window is 57 cm (or 57 ÷ 100 = 0.57 m), and you roll it down by 5 cm (or 5 ÷ 100 = 0.05 m), the area of the open window (\(A\)) is approx. 0.57 × 0.05 = 0.0285 m2.
Now for the neck of the opening. This is the thickness of the car's window. For this example, let's estimate it as 0.7 cm (or 0.7 ÷ 100 = 0.007 m).
So, we now have the following values:
- \(A\) = 0.0285 m2
- \(V_o\) = 4.732 m3
- \(L\) = 0.007 m
We can now plug these into the formula to find the frequency at which the throbbing noise should occur (remmeber, the speed of sound in air, \(v\), is approximately 343 m/s):
So, this means that the frequency at which the pressure waves oscillate back and forth is approximately 50.6 hertz.
If this frequency matches the resonant frequency of the car's interior, this will get amplified.
How does the volume of the car's interior affect the frequency?
Due to the large volume of a car, the frequency at which the presure waves oscillate is low.
As the volume of the car's interior increases, the frequency of the pressure waves decreases. This can be explained when you look at the Helmholtz resonance formula:
Let's focus on the square root, which contains a fraction with the volume in the denominator:
If we increase the volume (\(V_o\) in the denominator), the denominator increases. As the fraction's denominator increases (as long as the numerator stays the same), the value of the fraction decreases.
This means the value of the square root also decreases, which in turn decreases the value of the entire formula (because the square root is being multiplied by the fraction \(\frac{v}{2\pi}\)).
Try the silder below. As you increase the volume, you should see the frequency is decreasing. If it doesn't work, you might have Javascript turned off.
Current Frequency:
What other things does Helmholtz resonance affect?
Helmholtz resonance not only applies to car windows.
Car exhausts
Helmholtz resonators can be used to dampen the sound coming from car exhausts.
The annoying, loud noises from the exhaust goes through a Helmholtz resonator. By carefully changing the neck length, opening area and volume of the cavity, the natural resonate frequency of the Helmholtz resonator can be tuned to destructively interfere with the incoming sound waves. This dampens the loud, annoying sounds of the exhaust.
Blowing bottles
Have you ever tried blowing over (or into) a bottle? Well, you may have noticed that it produces a tone when you do so. This is also due to Helmholtz resonance.
In this case, the opening area is the area of the bottle’s opening, the neck length is the narrow part leading up to the opening and the volume is the volume of the bottle itself.
Bottles usually have a higher frequency than open car windows, meaning they are a lot higher pitched.
Conclusion
In conclusion, the throbbing noise when you open a car’s window is caused by Helmholtz resonance.
This phenomenon occurs when air passing through an open window creates oscillating pressure waves inside the car, resonating at a particular frequency.
Helmholtz resonance also applies to many other things, including the opposite of the throbbing noise, dampening sound.
Interested in learning more about maths?
This blog post used various mathematical techniques, including solving equations. If you would like to learn more about these (and so much more!), then check out my eBook, A Beginner’s Guide to Algebra, Quadratics and Inequalities, available for just £4.99.
If you know someone else who might be interested in this eBook, feel free to share it with them. Thank you for reading, and have a great day!
This is rather interesting, our car’s windows can be quite noisy…
I don’t understand much about maths/equations, but if I was interested in maths I think that I would understand the equations etc.
Thanks for this post Interceptmaths.
Thank you James! Yeah, sometimes the noise can be quite annoying, but the maths is fascinating! – Tom.