How do we combine two waves to create Lissajous figures?

Thread starter LCSphysicist
Start date Aug 13, 2020
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Superposition Superposition of waves Waves

In summary, superposition of waves is when two or more waves overlap and combine to form a new wave. This can result in either constructive or destructive interference, depending on the amplitude and phase of the waves. The resulting wave's amplitude is determined by the sum of the individual amplitudes of the overlapping waves, with in-phase waves resulting in a larger amplitude and out-of-phase waves resulting in a smaller amplitude. This phenomenon is applicable to all types of waves and is used in practical applications such as noise-cancelling headphones, music production, and medical imaging to manipulate and control waves for desired outcomes.

Aug 13, 2020

LCSphysicist

Homework Statement: All below

Relevant Equations: All below

To answer (a), i imagineed the oscillations parallels (say to z), so we simply add ξ1+ξ2
for b, i imagined two vectors ortogonais, representing the oscillations, so we should add √(ξ1² + ξ2²), is this right?

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Aug 14, 2020

anuttarasammyak

Gold Member

The particles move on xy plane as
[tex] \xi_1\ \mathbf{e_x}+\xi_2\ \mathbf{e_y}[/tex]
which draw Lissajous figures.

1. How do we combine two waves to create Lissajous figures?

To create a Lissajous figure, we need to combine two simple harmonic motions with different frequencies and amplitudes. These motions can be represented by sine and cosine functions, and the resulting Lissajous figure will be a closed curve.

2. What is the mathematical equation for creating Lissajous figures?

The mathematical equation for creating Lissajous figures is x = A*sin(a*t + δ) and y = B*sin(b*t), where A and B are the amplitudes, a and b are the frequencies, and δ is the phase difference between the two waves.

3. How does changing the phase difference affect the Lissajous figure?

Changing the phase difference between the two waves will result in a different Lissajous figure. A phase difference of 0 will create a straight line, while a phase difference of π/2 will create a circle. As the phase difference increases, the shape of the Lissajous figure will become more complex.

4. Can Lissajous figures be created with non-sinusoidal waves?

Yes, Lissajous figures can be created with any two periodic waves, as long as they have different frequencies and amplitudes. These waves do not have to be sinusoidal, but the resulting Lissajous figure may be more complex and difficult to predict.

5. What is the significance of Lissajous figures in science and engineering?

Lissajous figures have many practical applications in science and engineering. They are used to study the properties of waves, such as frequency, amplitude, and phase difference. They are also used in oscilloscopes to visualize and measure electrical signals. In addition, Lissajous figures have aesthetic value and are often used in art and music.