Superposition of Waves

Soundwaves

Waves are all around us. Assume you’re on a boat and you hear a ship’s siren. In this instance, you’ll be able to hear both the direct sound wave from the ship’s siren and the sound wave reflected by the seawater.

Introduction to superposition of waves

Let us use the example of a string wave to define the superposition theorem-based principle of superposition of waves. According to this, the algebraic totality of the displacements induced by each wave is equal to the net displacement of any component on the string for a given time. As a result, the principle of superposition refers to the procedure of adding distinct waveforms to determine the net waveform.

When two or more waves arrive at the same location at the same time, they superimpose on each other. When waves collide, the disturbances of the waves are superimposed, a phenomenon is known as the superposition of waves. Each disruption is associated with a force, which adds up. If the disturbances all follow the same path, the final wave is simply the sum of the individual waves’ disturbances—that is, their amplitudes add up.

Two opposite-direction wave pulses superimposed

Two Gaussian wave pulses travel in the same medium but different directions, as shown in the animation at left. The waves pass through each other unaffected, and the net displacement is equal to the total of their separate displacements.

It’s also worth noting that, because Gaussian wave pulses don’t change shape as they travel, this medium is nondispersive (all frequencies move at the same pace). The shape of the waves would vary if the medium was dispersive.

Solitons are nonlinear waves that interact with each other but do not obey the principle of superposition.

Constructive and destructive Interference

Two waves are travelling in the same direction (with the same amplitude, frequency, and wavelength). The resultant wave displacement may be expressed as: When focusing on the principle of superposition of waves, the resulting wave displacement can be written as:

y(x,t) = ymsin⁡(kx−ωt) + ymsin⁡(kx−ωt+ϕ)

y(x,t) = 2ymcos⁡(ϕ/2)sin⁡(kx−ωt+ϕ/2)

This is a travelling wave whose amplitude is proportional to its phase. When the two waves are in phase (=0), they constructively interfere, resulting in a wave with twice the amplitude of the separate waves. When the phase of two waves is 180 degrees apart, they interfere destructively and cancel each other out.

The animation on the left shows how, depending on their relative phase, two sinusoidal waves of the same amplitude and frequency can add destructively or constructively.

(NOTE: This animation does not portray actual wave propagation in a medium; it simply serves to demonstrate the effect of adjusting the phase shift between two waves and the ensuing constructive or destructive interference.)

With time, the phase difference between the two waves widens, revealing the impacts of both constructive and destructive interference. The consequence is a big amplitude when the two individual waves are exactly in phase. The sum wave is 0 when the two grey waves are perfectly out of phase.

A standing wave is formed by two sine waves Moving in directions opposite to each other

A travelling wave moves from one location to another, whereas a standing wave looks to be still and vibrates in the same spot. Two waves that are travelling in opposite directions in this animation. The resultant wave amplitude may be written as: Using the principle of superposition of waves, the resulting wave amplitude can be written as:

y(x,t) = y_msin⁡(kx−ωt) + ymsin⁡(kx+ωt)

y(x,t) = 2y_mcos⁡(ωt)sin⁡(kx)

Since the position and time dependency have been separated, this wave is no longer a travelling wave. 2y_msin⁡(kx) is the wave amplitude as a function of position (kx). This amplitude does not move; instead, it remains stationary and oscillates up and down following cos⁡(ωt). Standing waves have locations with maximum displacement (antinodes) and locations with zero displacements (zero displacements) (nodes).

When two sinusoidal waves of the same frequency (and wavelength) and amplitude travel in opposing directions in the same medium, the net displacement of the medium is equal to the sum of the two waves using the superposition of waves. The two waves cancel each other when they are 180° out of phase with each other, and they add together when they are perfectly in phase with each other, as shown in the video. The net effect of the two waves passing through each other fluctuates between zero and some maximum amplitude. However, because this pattern simply oscillates rather than moving to the right or left, it is referred to as a “standing wave.”

Two sine waves with different frequencies: Beats

In the same direction, two waves of identical amplitude are travelling. Although the frequencies and wavelengths of the two waves are different, they both travel at the same pace. The resultant particle displacement can be represented as: Using the principle of superposition of waves, the resulting particle displacement can be written as:

y(x,t) = y_msin⁡(k₁x−ω₂t) + ymsin⁡(k₂x−ω₂t)

y(x,t) = 2y_mcos⁡[x(k₁−k₂)/2−t(ω1−ω2)/2]sin⁡[x(k₁+k₂)/2−t(ω1+ω2)/2]

The resultant particle motion is the result of two waves travelling in opposite directions. One component is a sine wave with an average frequency of f = ½(f1 + f2). This is the frequency that a listener perceives. The other component is a cosine wave with a differential frequency of f = ½(f1 – f2). This word describes how the wave’s amplitude “envelope” is controlled and how “beats” are perceived, fbeat = (f1 – f2).

The two waves travel at the same pace since they are in the same medium. The superposition sum wave moves in the same direction and at the same speed as the two component waves, but its local amplitude is governed by whether the two individual waves have the same or opposite phase. The “beat” wave oscillates at the average frequency, with the differential frequency affecting the amplitude envelope.

Conclusion

When two (or more) waves pass through a dedicated medium, the principle of superposition of waves can be applied to them. The waves are undisturbed as they travel through each other. The sum of the individual wave displacements is the net displacement of the medium at any location in space or time. This holds for both finite-length waves (wave pulses) and continuous sine waves.