Wave

A wave is a disturbance that travels in a medium, transferring energy and momentum from one place to another. The direction of the propagation of the wave is the direction of energy transfer. There is no large-scale motion of the medium itself.

Fundamentals

We define several key parameters:

• The length of a complete oscillation is the wavelength $\lambda$.
• The time of one complete oscillation is the period $T$.
• The frequency $f$ is the reciprocal of the period, measured in hertz (Hz); i.e., how many periods we have within one second.
• The angular frequency is given by $2\pi /T=2\pi f$
• The speed of a wave is given by $v=f\lambda$.
  • This only depends on the properties of the medium it passes through and not how it’s produced.

Waveforms

Waves are important fundamentals in electrical circuit design. Common input signals to circuits include sine waves, square waves, triangle waves, or sawtooth waves; these are produced with function generators.

The duty cycle is a parameter of the wave that describes how long it is active (in percent). A 75% duty cycle square wave, for instance, is at its peak voltage for 75% of its period and at its trough 25% of the time. We have a few more mechanisms for producing unusual waveforms.

• One is by using op amps in saturation mode as a comparator. When a sinusoidal signal is sent through, when it is positive $v_{out}=+VCC$ and when it is negative $v_{out}=-VCC$.
• Triangle waves are similar, but we use capacitors to charge and discharge over time.
• Astable multivibrator circuits can be used to produce a square wave only with DC sources, by tracking op amp supply rails.

Square waves can be approximated as the sum of sinusoids with a Fourier series:

$$x(t)=\frac{4}{\pi }\sum _{n=1,3,5,\cdots }^{\infty }\frac{1}{n}\sin \frac{2\pi nt}{T}$$

Simulation

In LTspice, triangle waves can be simulated using the PULSE function. For a given source, right click and select from the function options. Vinitial and Von can define the minimum and maximum values of the source output. The Trise and Tfall should be half of the period.¹

To adjust the scale of the $x$-axis, we can right click on axis labels.

Extensions

Some types of wave motion (specifically for travelling waves) are governed by the one-dimensional wave equation:

$$\frac{{\partial }^{2}u}{\partial t^2}=c^2\frac{{\partial }^{2}u}{\partial x^2}$$

See also

• Oscillatory systems

Footnotes

1. From the Analog Devices documentation. ↩