Harmonic Wave Equation Calculator

A harmonic wave represents a continuous, smooth periodic oscillation that propagates through a medium over space and time. The fundamental mathematical model governing a traveling harmonic wave is expressed as:

y(x, t) = A * sin((2 * pi / λ) * (x - v * t) + φ)

Where:

• y(x, t) is the displacement of the wave at position x and time t.
• A is the maximum displacement or amplitude of the wave.
• λ (lambda) represents the spatial wavelength.
• v is the propagation velocity of the wave.
• φ (phi) denotes the initial phase shift.

Let us calculate the displacement of a harmonic wave given the following parameters: Amplitude (A) = 0.05 meters, Wavelength (λ) = 0.2 meters, Velocity (v) = 150 meters per second, Distance from source (x) = 2 meters, Time (t) = 0.01 seconds, and Phase shift (φ) = 0 radians.

Step 1: Compute the spatial and temporal components.
Multiply velocity by time: v * t = 150 * 0.01 = 1.5 meters.

Step 2: Determine distance relative to propagation.
Subtract that product from the distance: x - v * t = 2 - 1.5 = 0.5 meters.

Step 3: Apply the wave number scaling.
Divide 2*pi by the wavelength and multiply: (2 * pi / 0.2) * 0.5 = 10 * pi * 0.5 = 5 * pi radians.

Step 4: Calculate final displacement.
Take the sine of the angle and multiply by the amplitude: y = 0.05 * sin(5 * pi + 0) = 0.05 * 0 = 0 meters. At this precise spatial coordinate and moment in time, the wave displacement is zero.

• Check your angle units: Always ensure your calculator is set to radians when evaluating trigonometric functions within wave equations, as physics formulas rely on standard radian measure.
• Maintain consistent units: Convert all length measurements to meters and time to seconds before running computations to avoid scaling errors.
• Understand phase shifts: A positive phase shift moves the wave graph to the left, representing an advance in the initial oscillation cycle.