For a mathematical wave, the *phase constant* tells you how displaced a wave is from an equilibrium or zero position. You can calculate it as the change in phase per unit length for a standing wave in any direction. It's typically written using "phi," *ϕ*. You can use it to calculate how many oscillations a wave has undergone through its cycles.

To calculate the phase constant of a wave, use the equation 2π/λ for wavelength "lambda" λ. The wavelength is the length of a full cycle of the wave; for example, if you place a point at the top of a "peak" on a waveform and another point at an identical spot on an adjacent "peak" on the same waveform, the length between those two points is the wavelength. The phase constant does not change over time, and it describes the wave's displacement along the axis it travels.

The full equation for a harmonic wave with positions *x* and *y* with time *t* is:

y − y_{0} = A sin (2πt/T ± 2πx/λ + ϕ)

In which *y _{0}*

_{}is the

*y*position at

*x = 0*and

*t = 0*,

*A*is the amplitude, T is the period and "phi"

*ϕ*is the phase constant.

For this sinusoidal wave, the period *T* = 1/f for frequency (*f*), which is how many cycles of a wave pass over a given point per second. The left side *y − y _{0}* is the displacement of the wave in the

*y*direction from the initial position, and the value within the parentheses 2πt/T ± 2πx/λ + ϕ is the phase.

## Phase Constant and Phase Difference

Although you can calculate the velocity of the wave by multiplying its wavelength time frequency, v = fλ, you can also calculate velocity as the difference between two phases. For two different pairs of *x* and *t*, you can write the phases *ϕ _{1}* and

*ϕ*as 2πt

_{2}_{1}/T ± 2πx

_{1}/λ + ϕ and 2πt

_{2}/T ± 2πx

_{2}/λ + ϕ.

Subtracting one phase from the other and rewriting them gives you 2π(t_{2} − t_{1})/T ± 2π(x_{1} − x_{2})/λ = 0, which can be written with "delta" *Δx* and *Δt* for changes in position and time, respectively. This gives you 2πΔt/T ± 2πΔx/λ = 0.

Divide both sides of the equation by *2π* and rearrange it to get Δx/Δt = ∓λ/T. Because Δx/Δt is velocity (*v*), you end up with λ/T or λf for the velocity of a wave in either direction (given by the – or +).

Tbis derivation means scientists and engineers can use the phase difference between two waves for determining how far away two waves are from one another or how fast they are with respect to one another. In sonar and echolocation technologies, sound waves through different media, such as water or air, let scientists figure out the location of objects underwater.

## Excel Formula for Phase Constant

If you have a large amount of data about a wave, you can use Microsoft Excel's methods of calculation in determining phase constnat. Assign each variable to a specific column in an Excel spreadsheet, and use them to create a final column to calculate displacement. If you know the wavelength of the wave, you can calculate the phase constant as 2π/λ_._

As the phase constant can vary between different waves, it's helpful to use the formula in Excel to compare the differences. The percentage difference formula is one method of doing that.

If the phase constant varies over multiple waves, you can also use an Excel formula to calculate percentage of grand total displacement by summing the phase constants. You can then divide this by the number of waves you have to get the average wave phase constant. Then, you can use an Excel percentage difference formula by dividing the value of how much each wave differs from the average by the average.

References

About the Author

S. Hussain Ather is a Master's student in Science Communications the University of California, Santa Cruz. After studying physics and philosophy as an undergraduate at Indiana University-Bloomington, he worked as a scientist at the National Institutes of Health for two years. He primarily performs research in and write about neuroscience and philosophy, however, his interests span ethics, policy, and other areas relevant to science.