Displacement of Harmonic Motion Calculator

This calculator determines the displacement of a harmonic wave at a given point and time. Harmonic waves are a foundational concept in physics that describes the propagation of waves with a sinusoidal shape. 
CalculatorInputs﻿﻿A
:1.00m​
﻿
﻿﻿Φ
:1.00rad​
﻿
﻿﻿v
:2.00m/s​
﻿
﻿﻿λ
:5.00m​
﻿
﻿﻿x
:3.00m​
﻿
﻿﻿t
:1.00s​
﻿
Output﻿﻿y
:0.77m​
﻿
﻿﻿y=Asin[λ2π​ (x−vt)+ϕ]
 
Output graphA plot showing how the displacement ﻿y
 at a fixed position ﻿x
 changes as time progresses.
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﻿
ExplanationHarmonic waves are fundamental to our understanding of many physical phenomena, from the simple oscillations of a pendulum to the complex propagation of electromagnetic waves. These waves are distinguished by their sinusoidal pattern, exhibiting smooth and continuous oscillations that repeat at regular intervals. 
💡Harmonic Motion vs Simple Harmonic Motion
﻿
Displacement vs time graph of harmonic motion
The displacement, ﻿y
 of a point on a harmonic wave at a given time ﻿t
 and position  ﻿x
 is mathematically represented by the equation:
﻿
y=Asin⁡[2πλ (x−vt)+ϕ]y=A \sin \left[ \dfrac{2π}{λ}\space (x−vt)+ϕ \right]y=Asin[λ2π​ (x−vt)+ϕ]
Where:
﻿﻿y
 is the displacement of the wave at a given point and time, measured in meters (﻿m
).
﻿﻿A
 is the amplitude of the wave, signifying the maximum displacement from the rest position, measured in meters (﻿m
). It reflects the wave's energy.
﻿﻿λ
 denotes the wavelength, the distance over which the wave's shape repeats, measured in meters (﻿m
). It is inversely related to the frequency, affecting how tight or spread out the oscillations are.
﻿﻿v
 is the wave velocity, indicating how fast the wave propagates through the medium, measured in meters per second (﻿m/s
). This determines the speed at which the wave's energy travels. 
﻿﻿x
 is the position along the wave measured in meters (﻿m
) and signifies a specific location along the medium through which the wave propagates.
﻿﻿t
 indicates a specific moment in time, measured in seconds (﻿s
).
﻿﻿ϕ
 is the initial phase of the wave, measured in radians (﻿rad
). It specifies the wave's starting position within its cycle at ﻿t=0
, affecting where on the sinusoidal curve the wave begins.
Displacement in a harmonic wave follows a sinusoidal pattern, peaking at the amplitude during oscillation crests or troughs and hitting zero at equilibrium. This cycle, dictated by the harmonic wave equation, is key for understanding wave dynamics, showing energy transmission and interaction with the medium. 
Harmonic waves are crucial in various fields, including acoustics, optics, and electromagnetic theory, offering insights into the behavior of waves and their interaction with different media. This calculator has applications such as:
Physics and Engineering: For designing and analysing systems that utilise or are affected by wave phenomena.
Acoustics: In the study of sound waves, their propagation, and their interaction with different materials.
Optics: For understanding light waves and designing optical devices.
Seismology: To model the propagation of seismic waves and predict their effects on structures.
Related ResourcesDamped Harmonic Motion Energy Loss Calculator
Simple Harmonic Motion Calculator
Frequency of a Simple Harmonic Motion Calculator
Time Period of a Simple Harmonic Motion Calculator
ReferencesHalliday, D., Resnick, R., & Walker, J. (2014). Fundamentals of Physics (10th ed.). Wiley.
Tipler, P. A., & Mosca, G. (2008). Physics for Scientists and Engineers (6th ed.). W.H. Freeman and Company.
Young, H. D., & Freedman, R. A. (2012). University Physics with Modern Physics (13th ed.). Pearson.