🌊 Harmonic Wave Equation Calculator
Calculate wave displacement
How to Use This Calculator
Enter Amplitude
Input the amplitude (A) in meters (m). This is the maximum displacement of the wave from its equilibrium position. For example, a wave with amplitude 1 m oscillates between -1 m and +1 m.
Enter Frequency
Input the frequency (f) in hertz (Hz). This is how many complete wave cycles occur per second. For example, a sound wave with frequency 440 Hz completes 440 oscillations per second.
Enter Wavelength
Input the wavelength (λ) in meters (m). This is the distance between two consecutive identical points on the wave (e.g., peak to peak). Wavelength is related to frequency by v = fλ, where v is wave speed.
Enter Position and Time
Input the position (x) in meters where you want to calculate the wave displacement, and the time (t) in seconds. The wave displacement varies with both position and time, creating the wave pattern.
Calculate and Review
Click the "Calculate" button to compute the wave displacement at the specified position and time. The result shows how the wave oscillates in space and time, following a sinusoidal pattern.
Formula
y(x,t) = A sin(kx - ωt)
k = 2π/λ, ω = 2πf
Where:
• y(x,t) = Wave displacement at position x and time t (m)
• A = Amplitude (m)
• k = Wave number (rad/m) = 2π/λ
• x = Position along wave (m)
• ω = Angular frequency (rad/s) = 2πf
• t = Time (s)
• λ = Wavelength (m)
• f = Frequency (Hz)
Example 1: Wave at Specific Point
A harmonic wave has amplitude 2 m, frequency 1 Hz, and wavelength 4 m. Calculate the displacement at position x = 1 m and time t = 0.5 s.
Given:
• Amplitude (A) = 2 m
• Frequency (f) = 1 Hz
• Wavelength (λ) = 4 m
• Position (x) = 1 m
• Time (t) = 0.5 s
Solution:
k = 2π/λ = 2π/4 = π/2 rad/m
ω = 2πf = 2π × 1 = 2π rad/s
y(x,t) = 2 × sin((π/2) × 1 - 2π × 0.5)
y(x,t) = 2 × sin(π/2 - π) = 2 × sin(-π/2)
y(x,t) = -2 m
Example 2: Wave Propagation
A wave with amplitude 1 m, frequency 2 Hz, and wavelength 3 m. What is the displacement at x = 0 and t = 0.25 s?
Given:
• Amplitude (A) = 1 m
• Frequency (f) = 2 Hz
• Wavelength (λ) = 3 m
• Position (x) = 0 m
• Time (t) = 0.25 s
Solution:
k = 2Ï€/3 rad/m
ω = 4π rad/s
y(x,t) = 1 × sin(0 - 4π × 0.25)
y(x,t) = 1 × sin(-π) = 0
y(x,t) = 0 m
Frequently Asked Questions
What is a harmonic wave?
A harmonic wave is a wave that follows a sinusoidal (sine or cosine) pattern. It's characterized by its amplitude, frequency, and wavelength. Harmonic waves are fundamental in physics because they represent pure oscillations and can be used to describe more complex waves through Fourier analysis. Examples include sound waves, light waves, and water waves.
What does the wave equation y(x,t) = A sin(kx - ωt) represent?
This equation describes a harmonic wave traveling in the positive x-direction. The term (kx - ωt) is called the phase. When kx - ωt = 0, the wave is at its starting point. As time increases, the wave propagates to the right. The amplitude A determines the maximum displacement, k controls the spatial variation, and ω controls the temporal variation.
What's the relationship between wave number, frequency, and wavelength?
Wave number (k) is related to wavelength by k = 2π/λ. Angular frequency (ω) is related to frequency by ω = 2πf. The wave speed is v = fλ = ω/k. These relationships connect the spatial properties (wavelength, wave number) with the temporal properties (frequency, angular frequency) of the wave.
Why does the wave displacement vary with both position and time?
The displacement depends on position because the wave has a spatial pattern - different points along the wave have different displacements at any given moment. It depends on time because the wave propagates - the pattern moves through space over time. At a fixed position, you see the wave oscillating up and down. At a fixed time, you see the wave's spatial pattern.
What does the negative sign in (kx - ωt) mean?
The negative sign indicates wave propagation in the positive x-direction. If the equation were y(x,t) = A sin(kx + ωt), the wave would travel in the negative x-direction. The sign convention determines the direction of wave propagation relative to the coordinate system.
Where are harmonic waves used in real-world applications?
Harmonic waves are fundamental in: acoustics (sound waves), optics (light and electromagnetic waves), seismology (earthquake waves), signal processing and communications, music and audio engineering, quantum mechanics (wave functions), and oceanography (water waves). Understanding harmonic waves is essential for analyzing any periodic wave phenomenon.
About Harmonic Wave Equation Calculator
The harmonic wave equation calculator computes the displacement of a harmonic wave at a given position and time using y(x,t) = A sin(kx - ωt). Harmonic waves are fundamental in physics, describing sinusoidal oscillations that propagate through space and time.
This calculator is essential for students studying wave physics, engineers working with signals and communications, and anyone analyzing oscillatory phenomena. Understanding harmonic waves helps explain everything from sound and light to quantum mechanics and signal processing.