The Wave Equation and Superposition Principle

A traveling sinusoidal wave is described by y(x, t) = A sin(kx − ωt), where k = 2π ÷ λ is the wave number and ω = 2πf is the angular frequency, so that the wave speed is v = ω ÷ k = fλ; the principle of superposition says that when waves overlap, the resultant displacement is the sum of the individual displacements. In brief: the wave moves in the +x direction if the argument is (kx − ωt), and in the −x direction if it is (kx + ωt). The wave speed on a stretched string is v = √(T ÷ μ). Each particle of the medium moves in SHM with amplitude A and maximum speed Aω, which is different from the wave speed. Superposition holds for linear waves.

The mathematical description of waves is the basis for interference, standing waves, sound and light. This guide introduces the wave function and its parameters, explains the wave equation and the speed on a string, states the superposition principle and its consequences, compares the quantities in a table, works through numerical examples and corrects common mistakes.

What is the wave function?

A sinusoidal wave traveling along the x-axis has a displacement that depends on both position and time:

y(x, t) = A sin(kx − ωt + φ).

• A is the amplitude (the maximum displacement).
• k = 2π ÷ λ is the wave number (rad/m): the phase changes by 2π over one wavelength.
• ω = 2πf = 2π ÷ T is the angular frequency (rad/s).
• φ is the phase constant, fixed by the initial conditions.
• The wave speed is v = fλ = ω ÷ k, since a point of constant phase (kx − ωt = const.) moves at dx/dt = ω ÷ k.

At a fixed position x, the particle oscillates in SHM in time; at a fixed time t, the shape of the wave is a sine curve in space. The velocity of the particle, ∂y/∂t = −Aω cos(kx − ωt), is perpendicular to the direction of travel (for a transverse wave) and its maximum value Aω is generally different from v.

What is the wave equation?

The function above satisfies the (linear) wave equation: ∂²y/∂t² = v² ∂²y/∂x². This equation arises from Newton’s second law for a small element of a stretched string: the net transverse force due to the tension T on an element of mass μ dx is T ∂²y/∂x² dx, giving the speed v = √(T ÷ μ), where μ is the mass per unit length. The speed increases with tension and decreases with the mass per unit length. Equivalent equations apply to sound waves (with v = √(B ÷ ρ)) and electromagnetic waves (with v = c). Any function of (x − vt), not only a sine wave, is a solution, which means that pulses of any shape travel without a change of shape in a non-dispersive medium.

What is the superposition principle?

Because the wave equation is linear, if y₁ and y₂ are solutions, so is their sum. When two or more waves occupy the same region of a medium, the total displacement at any point is the algebraic sum of the displacements that each wave would produce alone: y = y₁ + y₂ + …. After passing through each other, the waves continue unchanged. For two waves of the same frequency and amplitude with a phase difference Δφ, the resultant amplitude is 2A cos(Δφ ÷ 2): equal to 2A for waves in phase (constructive interference), and 0 for waves in antiphase (destructive interference). When the amplitudes are unequal, the amplitudes add (A₁ + A₂) in phase and subtract |A₁ − A₂| in antiphase. Superposition explains interference, standing waves, beats and diffraction, and it underlies the Fourier analysis of complex sounds and signals into sinusoidal components.

Wave function parameters compared

Worked example: reading a wave function

1. Parameters. A wave on a string is described by y = 0.030 sin(2.0x − 40t) (SI units). Amplitude A = 0.030 m. k = 2.0 rad/m, so λ = 2π ÷ k = 3.14 m. ω = 40 rad/s, so f = 40 ÷ 2π = 6.37 Hz and T = 0.157 s.
2. Speed and direction. v = ω ÷ k = 40 ÷ 2.0 = 20 m/s, in the +x direction (the signs of kx and ωt are opposite). Check: fλ = 6.37 × 3.14 = 20 m/s ✓.
3. Particle speed. The maximum transverse speed of a point of the string is Aω = 0.030 × 40 = 1.2 m/s, much less than the wave speed of 20 m/s.
4. Speed on a string. A string with tension T = 100 N and linear density μ = 0.010 kg/m: v = √(100 ÷ 0.010) = 100 m/s. If the tension is increased to 400 N, v = 200 m/s.
5. Superposition. Two waves of amplitudes 2.0 cm and 3.0 cm at the same point: in phase, the resultant amplitude is 5.0 cm; in antiphase, 1.0 cm. If two waves of equal amplitude 2.0 cm have a phase difference of 90°, the resultant amplitude is 2 × 2.0 × cos(45°) = 2.8 cm.
6. Displacement at a point. For the wave in item 1, at x = 0.50 m and t = 0.010 s: y = 0.030 sin(1.0 − 0.40) = 0.030 sin(0.60) = 0.030 × 0.565 = 0.017 m.

How was the wave equation developed?

Jean le Rond d’Alembert derived the wave equation for a vibrating string in 1746, and Leonhard Euler and Daniel Bernoulli extended the solutions. Bernoulli proposed that a string vibrates as a superposition of its harmonics, and Joseph Fourier’s work in 1822 showed that arbitrary periodic functions can be represented as sums of sines and cosines. James Clerk Maxwell showed in 1865 that electromagnetic fields satisfy a wave equation with speed c = 1/√(μ₀ε₀).

Which mistakes are common?

Mistake 1: Confusing wave speed and particle speed

The wave speed is ω ÷ k. The particle speed varies between 0 and Aω.

Mistake 2: Getting the direction wrong

(kx − ωt) is a wave moving toward +x; (kx + ωt) toward −x.

Mistake 3: Confusing wave number with frequency

k = 2π ÷ λ is a spatial quantity. ω = 2πf is a temporal quantity.

Mistake 4: Applying superposition to nonlinear waves

Very large waves (shock waves, breaking waves) are nonlinear and do not simply add.