Graphs and Calculations of SHM

What are Waves?

Amplitude, A, is how far the wave gets away from the equilibrium point of zero.
It is measured in meters.
It is equal to half of the difference between the maximum and minimum points of a wave.

Wavelength, λ, is the distance between two corresponding points on a wave and is measured in meters.

Period, T, is the time interval required by a wave to complete one cycle.
It is measured in seconds.
It is the difference between two corresponding points on a simple harmonic motion graph if the x-axis is time.

The phase difference is the difference between two waves that have the same frequency, but oscillate differently to one another.

Phase difference is generally measured in radians, and it is the difference between corresponding points on one wave and the other.
Keep in mind that waves of different amplitude can still be in phase, as long as their oscillations occur at the same times.

Example

The phase angle of an oscillation, given by ϕ, is the fraction of an oscillation (expressed in terms of π) that occurs between when it has zero displacement, and a sine wave which has zero displacement at time t=0.
It is effectively the phase difference between a specific wave and a standard sine wave.

While it can be easy to discern the phase angle if the wave is displaced by 2π or π/2, special equations are needed for more complex problems.
Phase angle is zero if the displacement is zero when time is equal to zero, as that is how a normal sine-wave looks.

A particle undergoing simple harmonic motion can be described using the phase angle.
Problems can be solved using:

Where x is vertical displacement, x₀ is amplitude, v is linear velocity, ω is angular frequency, t is time and ϕ is the phase angle.
These equations can also be rearranged to find the phase angle.

Previously we discussed how displacement, velocity and acceleration during simple harmonic motions appeared as similar sine wave graphs, yet shifted.
By using the phase angle, we can also determine the velocity and acceleration that occurs during simple harmonic motion.

As velocity is defined using cosine, we look at the phase angle between it and the standard cosine function.
As acceleration is angular velocity squared multiplied by displacement, the displacement can be substituted with its equation using phase angle.
The formula is negative as acceleration is in the opposite direction to displacement.

All mechanical oscillations involve a continuous exchange of energy between kinetic energy and some form of potential energy.
For example, in a mass-spring system, the maximum potential energy is Eₚ =1/2*kx² and the maximum kinetic energy is 1/2*mv².
Likewise, in a pendulum, the maximum potential energy is Eₚ = mgΔh and the maximum kinetic energy is 1/2*mv²
Energy will be continuously transferred between potential and kinetic energy.

The total energy of a simple harmonic motion system is proportional to the square of its amplitude.

E ∝ A²

Problems relating to energy levels in simple harmonic motion can be solved using:

Where v is velocity, ω is angular frequency, x₀ is amplitude and x is displacement.
Eₚ is potential energy and Eₜ is total energy.
In these calculations, energy is a function of displacement.

Notice how in the calculation for velocity, the answer can be either negative or positive.]
To visualize this, imagine an oscillating spring-mass system.
If the body is displaced by x, then the acceleration will be a = -kx.
If the body is displaced by -x, then the acceleration will be a = kx.
As acceleration is the same magnitude but with a different sign for displacements of the same magnitude and opposite signs, velocity should also be the same.
Thus with a displacement x, the velocity could be said to be v.
With displacement -x, the velocity would be -v.

The calculation for potential energy uses the formula for the potential energy for Hooke's law, where the constant k is substituted with angular frequency squared times mass.

As we can find velocity, we can also find kinetic energy at a any point by substituting velocity with its calculation.
As energy is scalar, the positive or negative sign in the velocity function isn't important.

Kinetic energy is at its highest when displacement, x is equal to zero, as then the potential energy is equal to zero, and all of the total energy consists of kinetic energy.

In more realistic situations, the total energy will decrease over time as some of it is lost as it is transformed into heat and sound.
The energy dissipation from an oscillating system is called damping.
In the graph showing energy changes for simple harmonic motion, it will result in a gradual decrease.

Sources