# Tutorial Four: Trains and the Doppler effect

The Doppler effect (also called Doppler shift) is one of the most widely–observed phenomena associated with moving objects. In fact, we often observe it when we hear the sound coming from moving vehicles like cars, trucks, and trains.

Imagine that a tape recorder is placed on a railroad track. The recorder’s microphone records the sound of the horn of an approaching train. It also records that same train’s horn once the locomotive has passed and is moving away (or receding) from the recorder. Anyone who has stood next to a train track and has listened to the sound of a train’s horn as it approach and moves away knows that you hear two different sounds. The sound that you hear as the train approaches has a higher frequency than after it has passed and is moving away from your location. The sound that you hear changes, even though the speed of the train or the sound that is actually emitted by the locomotive’s horn remains unchanged.

In 1842, 37 years before Einstein’s birth, Austrian physicist Christian Doppler published a paper describing this behavior, which today we call the Doppler effect. While you don’t need to worry about how these equations are developed, we do need to show you what they are so that you can see what they look like, because we are going to use them in upcoming tutorials.

The sound of the train’s horn is described in terms of wavelength. If the horn emits a blast at wavelength x’, then you will hear a wavelength less than x’ when the train approaches. This is called the approaching Doppler wavelength (or simply, the approaching wavelength) and is mathematically defined as:

approaching_wavelength = x’ / (1 + v / w)

Similarly,  you will hear a wavelength greater than x’ when the train recedes. This is called the receding Doppler wavelength (or simply, the receding wavelength) and is mathematically defined as:

receding_wavelength = x’ / (1 – v / w)

In the Doppler equations, x’ represents the wavelength coming from the train’s horn, v is the speed of the train, and w is the speed at which the wave propagates (eg, the speed that sounds travels through the air). The approaching and receding wavelengths are the sounds you hear (or record with the tape recorder).

In the 1800s, scientists were just beginning to understand the characteristics of the electromagnetic force. They used what they knew about Doppler shifts and applied it to light, electricity, and radio waves. They expected light to behave to same way that sound waves or water waves might behave, but when they put their ideas to the test, they were unable to show that this was the case. This created a crisis, because scientists did not have a theory the would explain what they were seeing. This gap in what we could explain using classical mechanics was later believed to be filled by Einstein’s theory of relativity.

At the end of the 19th century, we were just beginning to scratch surface in our understanding of this new and mysterious force. Today we take the electromagnetic force, the building blocks of modern communication and computing, for granted.

Fact: The Doppler effect was developed long before Einstein’s birth.

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