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# Conventions for Doppler Effect Equations

by Ron Kurtus

It is a good scientific practice to define conventions before stating equations for a physical phenomenon.

In the case of Doppler Effect equations, you need to define which direction is positive and which is negative. You also need to define restrictions on the motion of the waves, source, and observer.

Finally, there are certain relationships between terms.

Questions you may have include:

• What is the direction convention?
• What are assumptions about motion?
• What are the relationships between terms?

This lesson will answer those questions. Useful tool: Units Conversion

## Direction convention

It is important to have a clearly stated direction convention. Unfortunately, many sources of Doppler Effect equations don't clearly state which direction is positive and which is negative. This can create confusion in the stated equations. The following material is a logical convention for motion direction.

Consider the source and the observer placed on the x-axis, with everything moving in the x direction. Left-to-right is the standard direction

Left-to-right is considered the standard, positive (+) direction for the wavefront, source, and observer. If c, vS or vO moves the opposite direction—toward the left—its sign in the equation changes.

## Motion assumptions

Assume the velocities of the source, observer, and waveform are constant and do not vary during the time of consideration.

### With respect to medium of travel

The velocities are with respect to the medium in which they travel.

For example, in the case of sound waves, the velocities are with respect to the air. If the sound travels in moving air (wind), the velocities are still with respect to the air and not the ground.

In the case of electromagnetic waves—such as visible light—in ordinary cases, the velocities are with respect to the frame of reference. But at higher velocities, relativity effects take place.

Note: The equations for the Doppler Effect are the same for all waveforms, whether sound waves, light waves, water waves, or such.

### Source velocity less than wave speed

Also, assume that the speed of the source is less than the speed of the waves in their medium. In other words, if you are considering sound, the speed of the source is less than the speed of sound.

### Frequency independent of source velocity

The rate that the waves are emitted from the source—the frequency—is independent of the speed of the source. Likewise, the speed of the waveform is independent of the speed of the source.

### Line along the axis

Also, we are only looking at motion along a line between the source and observer. Motion at an angle brings in a cosine factor, and we will not consider that at this time.

## Relationships

The relationships between speed or velocity, wavelength, and frequency are:

c = λf

λ = c/f

f = c/λ

where

• c is the velocity of the waveform
• λ is the wavelength or distance between crests of the wave
• f is the frequency that the waves pass a given point (usually in Hz)

Note: Although c often denotes the speed of light, it is also used for the speed or velocity of other waveforms.

The period T of the wave motion is:

T = 1/f

This is the time between wave crests. For example, if f = 20 Hz, then T = 1/20 sec.

Also, from the above relationships:

T = λ/c

## Summary

It is important to define conventions before stating the Doppler Effect equations. You need to define which direction is positive and which is negative. You also need to define restrictions on the motion of the waves, source, and observer. Finally, your need to define the relationships between terms.

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